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in @xcite the problem of how to give an explicit combinatorial definition of weak higher categories was solved , and the development of a conceptual framework for their further analysis was begun . in the aftermath of this , the expository work of other authors , most notably street @xcite and leinster @xcite , contributed greatly to our understanding of these ideas . the central idea of @xcite is that the description of any @xmath0-dimensional categorical structure @xmath2 , may begin by starting with just the underlying @xmath0-globular set , that is , the sets and functions @xmath3^{t }   \ar@<-1ex>[l]_{s } & { x_2 } \ar@<1ex>[l]^{t }   \ar@<-1ex>[l]_{s } & { x_3 } \ar@<1ex>[l]^{t }   \ar@<-1ex>[l]_{s } & { ... } \ar@<1ex>[l]^{t }   \ar@<-1ex>[l]_{s } & { x_n } \ar@<1ex>[l]^{t }   \ar@<-1ex>[l]_{s}}\ ] ] satisfying the equations @xmath4 and @xmath5 , which embody the the objects ( elements of @xmath6 ) , arrows ( elements of @xmath7 ) and higher cells of the structure in question . at this stage no compositions have been defined , and when they are , one has a globular set with extra structure . in this way the problem of defining an n - categorical structure of a given type is that of defining the monad on the category @xmath8 of @xmath0-globular sets whose algebras are these structures . as explained in the introduction to @xcite , this approach works because the monads concerned have excellent formal properties , which facilitate their explicit description and further analysis . the @xmath0-operads of @xcite can be defined from the point of view of monads : one has the monad @xmath9 on @xmath8 whose algebras are strict @xmath0-categories , and an @xmath0-operad consists of another monad @xmath1 on @xmath8 equipped with a cartesian monad morphism @xmath10 . the algebras of this @xmath0-operad are just the algebras of @xmath1 . strict @xmath0-categories are easily defined by iterated enrichment : a strict @xmath11-category is a category enriched in the category of strict @xmath0-categories via its cartesian product , but are too strict for the intended applications in homotopy theory and geometry . for @xmath12 the strictest structure one can replace an arbitrary weak @xmath13-category with  and not lose vital information  is a gray category , which is a category enriched in @xmath14 using the gray tensor product of 2-categories instead of its cartesian product @xcite . this experience leads naturally to the idea of trying to define what the higher dimensional analogues of the gray tensor product are , so as to set up a similar inductive definition as for strict @xmath0-categories , but to capture the appropriate semi - strict @xmath0-categories , which in the appropriate sense , would form the strictest structure one can replace an arbitrary weak @xmath0-category with and not lose vital information . crans in @xcite attempted to realise this idea in dimension 4 , and one of our main motivations is to obtain a theory that will deliver the sort of tensor product that crans was trying to define explicitly , but in a conceptual way that one could hope to generalise to still higher dimensions . our examples([ex : gray ] ) and ( [ ex : crans ] ) embody the progress that we have achieved in this direction in this paper . in @xcite the theory of the present paper is used to show that the _ funny tensor product _ of categories  which is what one obtains by considering the gray tensor product of @xmath15-categories but ignoring what happens to 2-cells  generalises to give an analogous symmetric monoidal closed structure on the category of algebras of any higher operad . from these developments it seems that a conceptual understanding of the higher dimensional analogues of the gray tensor product is within reach . fundamentally , we have two kinds of combinatorial objects important for the description and study of higher categorical structures @xmath0-operads and tensor products . in @xcite a description of the relationship between tensor products and @xmath0-operads was begun , and @xmath11-operads whose algebras involve no structure at the level objects were canonically related with certain lax tensor products on @xmath8 . under this correspondence the algebras of the @xmath11-operad coincide with categories enriched in the associated lax tensor product . sections([sec : eg - lmc])-([sec:2-functoriality ] ) of the present paper continue this development by studying , for a given category @xmath16 , the passage @xmath17 where @xmath18 is the category of graphs enriched in @xmath16 , in a systematic way . this analysis culminates in section([sec:2-functoriality ] ) where the above assignment is seen as the object part of a 2-functor @xmath19 where @xmath20 is a sub 2-category of the 2-category of lax monoidal categories , and @xmath21 is as defined by the formal theory of monads @xcite . from this perspective , one is able to describe in a more efficient and general way , many of the previous developments of higher category theory in the globular style . for instance , in section([ssec : induction ] ) we give a short and direct explicit construction of the monads @xmath9 for strict @xmath0-categories from which all their key properties are easily witnessed . in sections([ssec : general - op - mult ] ) and ( [ ssec : induction ] ) we give shorter and more general proofs of some of the main results of @xcite . in section([ssec : monmonad - distlaw ] ) using a dual version of our 2-functor @xmath22 and the formal theory of monads @xcite , we obtain a satisfying general explanation for how it is that monad distributive laws arise in higher category theory see @xcite @xcite . in sections([ssec : tci ] ) and ( [ ssec : tcii ] ) we apply our theory to simplifying many aspects of @xcite .    the correspondence between @xmath11-operads and certain lax monoidal structures on @xmath8 given in @xcite , associates to the 3-operad @xmath23 for gray categories , a lax tensor product on the category of 2-globular sets . however the gray tensor product itself is a tensor product of 2-categories . any lax monoidal structure on a category @xmath16 comes with a `` unary '' tensor product , which rather than being trivial as is the usual experience with non - lax tensor products , is in fact a monad on @xmath16 . for the lax tensor product induced by @xmath23 , this is the monad for 2-categories . in section([sec : lift - mult ] ) we solve the general problem of lifting a lax monoidal structure , to a tensor product on the category of algebras of the monad defined by its unary part . this result , theorem([thm : lift - mult ] ) , is the main result of the paper , and provides also the sense in which these lifted tensor products are unique . in practical terms this means that in order to exhibit a given tensor product on some category of higher dimensional structures as arising from our machinery , it suffices to exhibit an operad whose algebras are categories enriched in that tensor product . in this way , one is able see that the usual gray tensor product and that of crans , do so arise .    moreover applying this lifting to the lax tensor products on @xmath8 associated to general @xmath11-operads ( over @xmath24 ) , one exhibits the structures definable by @xmath11-operads as enriched categories whose homs are some @xmath0-dimensional structure . in this way the globular approach is more closely related to some of the inductive approaches to higher category theory , such as that of tamsamani @xcite .    in section([ssec : a - infinity ] ) we describe two applications of the lifting theorem . in theorem([thm : a - infinity - app ] ) we construct a tensor product of @xmath25-algebras . as explained in @xcite the problem of providing such a tensor product is of relevance to string theory , and it proved resistant because of the negative result @xcite which shows that no `` genuine '' tensor product can exist . however this result does not rule out the existence of a _ lax _ tensor product , which is what we were able to provide in theorem([thm : a - infinity - app ] ) . it is possible to see the identification by joyal and street @xcite , of braided monoidal categories as monoidal categories with a multiplication as an instance of theorem([thm : a - infinity - app ] ) . another instance is our second application given in corollary([cor : coh - bm2c ] ) , namely an analogous result to that of joyal and street but for braided monoidal 2-categories . a weak @xmath0-category is an algebra of a _ contractible _ @xmath0-operad . in section([sec : contractibility ] ) we recall this notion , give an analogous notion of contractible lax monoidal structure and explain the canonical relationship between them .    in this paper we operate at a more abstract level than in much of the previous work on this subject . in particular , instead of studying monads on the category of @xmath0-globular sets , or even on presheaf categories , we work with monads defined on some category @xmath18 of enriched graphs . as our work shows , the main results and notions of higher category theory in the globular style can be given in this setting . so one could from the very beginning start not with @xmath24 as the category of @xmath26-categories , but with a nice enough @xmath16 . for all the constructions to go through , such as that of @xmath9 , the correspondence between monads and lax tensor products , their lifting theorem , as well as the very definition of weak @xmath0-category , it suffices to take @xmath16 to be a locally c - presentable category in the sense defined in section([ssec : lcpres ] ) .    proceeding this way one obtains then the theory of @xmath0-dimensional structures enriched in @xmath16 . that is to say , the object of @xmath0-cells between any two @xmath27-cells of such a structure would be an object of @xmath16 rather than a mere set . some alternative choices of @xmath16 which could perhaps be of interest are : ( 1 ) the ordinal @xmath28=\{0<1\}$ ] ( for the theory of locally ordered higher dimensional structures ) , ( 2 ) simplicial sets ( to obtain a theory of higher dimensional structures which come together with a simplicial enrichment at the highest level ) , ( 3 ) the category of sheaves on a locally connected space , or more generally a locally connected grothendieck topos , ( 4 ) the algebras of any @xmath0-operad or ( 5 ) the category of multicategories ( symmetric or not ) . the point is , the theory as we have developed it is actually _ simpler _ than before , and the generalisations mentioned here come at _ no _ extra cost . given a topological space @xmath2 and points @xmath29 and @xmath30 therein , one may define the topological space @xmath31 of paths in @xmath2 from @xmath29 to @xmath30 at a high degree of generality . in recalling the details let us denote by @xmath32 a category of `` spaces '' which is complete , cocomplete and cartesian closed . we shall write @xmath33 for the terminal object . we shall furthermore assume that @xmath32 comes equipped with a bipointed object @xmath34 playing the role of the interval . a conventional choice for @xmath32 is the category of compactly generated hausdorff spaces with its usual interval , although there are many other alternatives which would do just as well from the point of view of homotopy theory . let us denote by @xmath35 the reduced suspension of @xmath2 , which can be defined as the pushout @xmath36 { i{\times}x}="tr " [ d ] { \sigma{x}.}="br " [ l ] { 1{+}1}="bl " " tl"(:"tr":"br",:"bl":"br " ) " br " [ u(.3)l(.3 ) ] ( : @{-}[r(.15)],:@{-}[d(.15)])}\ ] ] writing @xmath37 for the category of bipointed spaces , that is to say the coslice @xmath38 , the above definition exhibits the reduced suspension construction as a functor @xmath39 in a sense this functor is the mother of homotopy theory  applying it successively to the inclusion of the empty space into the point , one obtains the inclusions of the @xmath27-sphere into the @xmath0-disk for all @xmath40 , and its right adjoint @xmath41 is the functor which sends the bipointed space @xmath42 , to the space @xmath31 of paths in @xmath2 from @xmath29 to @xmath30 . this adjunction @xmath43 is easy to verify directly using the above elementary definition of @xmath44 as a pushout , and the pullback square @xmath45 { x^i}="tr " [ d ] { x^{1{+}1}}="br " [ l ] { 1}="bl " " tl"(:"tr":"br"^{x^i},:"bl":"br"_-{(a , b ) } ) " tl " [ d(.3)r(.3 ) ] ( : @{-}[l(.15)],:@{-}[u(.15)])}\ ] ] where @xmath46 is the inclusion of the boundary of @xmath34 . the collection of spaces @xmath31 is our first example of an enriched graph in the sense of    [ def : enriched - graph ] let @xmath16 be a category . graph @xmath2 enriched in @xmath16 _ consists of an underlying set @xmath6 whose elements are called _ objects _ , together with an object @xmath31 of @xmath16 for each ordered pair @xmath47 of objects of @xmath2 . the object @xmath31 will sometimes be called the _ hom _ from @xmath29 to @xmath30 . a morphism @xmath48 of @xmath16-enriched graphs consists of a function @xmath49 together with a morphism @xmath50 for each @xmath47 . the category of @xmath16-graphs and their morphisms is denoted as @xmath18 , and we denote by @xmath51 the obvious 2-functor @xmath52 with object map as indicated .    the 2-functor @xmath51 is the mother of higher category theory in the globular style  applying it successively to the inclusion of the empty category into the point ( ie the terminal category ) , one obtains the inclusion of the category of @xmath27-globular sets into the category of @xmath0-globular sets . in the case @xmath53 this is the inclusion with object map @xmath54 { ... } = " m " [ r ] { x_{n{-}1}}="r " " r":@<1ex>"m":@<1ex>"l " " r":@<-1ex>"m":@<-1ex>"l " } } & \mapsto & { \xygraph{{x_0}="l " [ r ] { ... } = " m " [ r ] { x_{n{-}1}}="r " [ r ] { \emptyset}="rr " " rr":@<1ex>"r":@<1ex>"m":@<1ex>"l " " rr":@<-1ex>"r":@<-1ex>"m":@<-1ex>"l " } } \end{array}\ ] ] and when @xmath55 this is the functor @xmath56 which picks out the empty set . thus there is exactly one @xmath57-globular set which may be identified with the empty set . it is often better to think of @xmath51 as taking values in @xmath58 . by applying the endofunctor @xmath51 to the unique functor @xmath59 for each @xmath16 , produces the forgetful functor @xmath60 which sends an enriched graph to its underlying set of objects . this manifestation @xmath61 has a left adjoint which we shall denote as @xmath62 for reasons that are about to become clear . the functor @xmath62 is a variation of the grothendieck construction . to a given functor @xmath63 it associates the category @xmath64 with objects triples @xmath65 where @xmath29 is an object of @xmath1 , and @xmath66 is an ordered pair of objects of @xmath67 . maps are just maps in @xmath1 which preserve these base points in the obvious sense . it is interesting to look at the unit and counit of this 2-adjunction . given a category @xmath16 , @xmath68 is the category of bipointed enriched graphs in @xmath16 . the counit @xmath69 sends @xmath42 to the hom @xmath31 . when @xmath16 has an initial object @xmath70 has a left adjoint given by @xmath71 . given a functor @xmath63 the unit @xmath72 sends @xmath73 to the enriched graph whose objects are elements of @xmath67 , and the hom @xmath74 is given by the bipointed object @xmath65 . consider the case where @xmath75 and @xmath76 is the representable @xmath77 . then @xmath64 may be regarded as the category of endo - cospans of the object @xmath26 , that is to say the category of diagrams @xmath78 and a point of @xmath73 is now just a map @xmath79 . when @xmath1 is also cocomplete one can compute a left adjoint to @xmath80 . to do this note that a graph @xmath2 enriched in @xmath64 gives rise to a functor @xmath81 where @xmath6 is the set of objects of @xmath2 . for any set @xmath82 , @xmath83 is defined as the following category . it has two kinds of objects : an object being either an element of @xmath82 , or an ordered pair of elements of @xmath82 . there are two kinds of non - identity maps @xmath84 where @xmath66 is an ordered pair from @xmath82 , and @xmath83 is free on the graph just described . a more conceptual way to see this category is as the category of elements of the graph @xmath85 { z}="r " " l":@<1ex>"r " " l":@<-1ex>"r"}\ ] ] where the source and target maps are the product projections , as a presheaf on the category @xmath86 { = } [ r(1.25 ) ] { \xybox{\xygraph{0 [ r ] 1 " 0":@<1ex>"1":@<1ex>@{<-}"0 " } } } * \frm{-}}\ ] ] and so there is a discrete fibration @xmath87 . the functor @xmath88 sends singletons to @xmath75 , and a pair @xmath66 to the head of the hom @xmath89 . the arrow map of @xmath88 encodes the bipointings of the homs . one may then easily verify    let @xmath75 , @xmath77 and @xmath1 be cocomplete . then @xmath90 has left adjoint given on objects by @xmath91 .    in the exposition thus far we have focussed on building an analogy between the reduced suspension of a space and the graphs enriched in a category . now we shall bring these constructions together . as we have seen already to each space @xmath2 one can associate a canonical topologically enriched graph whose homs are the path spaces of @xmath2 . denoting this enriched graph as @xmath92 , the assignment @xmath93 is the object map of the composite right adjoint in @xmath94 { \ca g(\top_{\bullet})}="m " [ r ] { \ca g\top}="r " " l":@<-1.5ex>"m"_-{\eta}|{}="b1":@<-1.5ex>"l"|{}="t1 " " t1":@{}"b1"|{\perp } " m":@<-1.5ex>"r"_-{\ca gh}|{}="b2":@<-1.5ex>"m"_-{\ca g\sigma}|{}="t2 " " t2":@{}"b2"|{\perp}}.\ ] ] as explained by cheng @xcite , this functor @xmath95 is a key ingredient of the trimble definition of weak @xmath0-category . the important properties of @xmath51 are apparent because of the close connection between @xmath51 and the @xmath96 construction . a very mild reformulation of the notion of @xmath16-graph is the following : a @xmath16-graph @xmath2 consists of a set @xmath6 together with an @xmath97-indexed family of objects of @xmath16 . together with the analogous reformulation of the maps of @xmath18 , this means that we have a pullback square @xmath98_{(-)_0=\ca gt_v } \save \pos?(.3)="lpb " \restore \ar[r]^- { } \save \pos?(.3)="tpb " \restore & { \fam{v } } \ar[d]^{\fam(t_v ) } \save \pos?(.3)="rpb " \restore \\ { \set } \ar[r]_-{(-)^2 } \save \pos?(.3)="bpb " \restore & { \set } \pos " rpb " ; " lpb " * * @ { } ; ? ! { " bpb";"tpb"}="cpb " * * @ { } ; ? * * @{- } ; " tpb " ; " cpb " * * @ { } ; ? * * @{-}}}\ ] ] in @xmath99 , and thus a cartesian 2-natural transformation @xmath100 . from @xcite theorem(7.4 ) we conclude    [ prop : gfam2fun ] @xmath51 is a familial 2-functor . in particular it follows from the theory of @xcite that @xmath51 preserves conical connected limits as well as all the notions of `` grothendieck fibration '' which one can define internal to a finitely complete 2-category . moreover the obstruction maps for comma objects are right adjoints . see @xcite for more details on this part of 2-category theory . we shall not use these observations very much in what follows . more important for us is    [ lem : g - em - object ] @xmath51 preserves eilenberg - moore objects .    given a monad @xmath101 on a category @xmath16 , we shall write @xmath102 for the category of @xmath101-algebras and morphisms thereof , and @xmath103 for the forgetful functor . we shall denote a typical object of @xmath102 as a pair @xmath104 , where @xmath2 is the underlying object in @xmath16 and @xmath105 is the @xmath101-algebra structure . from @xcite the 2-cell @xmath106 , whose component at @xmath104 is @xmath107 itself has a universal property exhibiting @xmath102 as a kind of 2-categorical limit called an _ eilenberg - moore object_. see @xcite or @xcite for more details on this general notion . the direct proof that for any monad @xmath101 on a category @xmath16 , the obstruction map @xmath108 is an isomorphism comes down to the obvious fact that for any @xmath16-graph @xmath109 , a @xmath110-algebra structure on @xmath109 is the same thing as a @xmath101-algebra structure on the homs of @xmath109 , and similarly for algebra morphisms . let us recall the notions of lax monoidal category and category enriched therein from @xcite . for a category @xmath16 , the free strict monoidal category @xmath111 on @xmath16 has a very simple description . an object of @xmath111 is a finite sequence @xmath112 of objects of @xmath16 . a map is a sequence of maps of @xmath16  there are no maps between sequences of objects of different lengths . the unit @xmath113 of the 2-monad @xmath114 is the inclusion of sequences of length @xmath33 . the multiplication @xmath115 is given by concatenation . a _ lax monoidal category _ is a lax algebra for the 2-monad @xmath114 . explicitly it consists of an underlying category @xmath16 , a functor @xmath116 , and maps @xmath117 for all @xmath82 , @xmath118 from @xmath16 which are natural in their arguments , and such that @xmath119 ^ -{u\ope\limits_i } \ar[d]_{1 } \save \pos?(.4)="domeq " \restore & { e_1\ope\limits_iz_i } \ar[dl]^{\sigma } \save \pos?(.4)="codeq " \restore \\ { \ope\limits_iz_i } \pos " domeq " ; " codeq " * * @ { } ; ? * { = } } } } ; ( 4,0)*{\xybox{\xymatrix @c=1.5em { { \ope\limits_i\ope\limits_j\ope\limits_kz_{ijk } } \ar[r]^-{{\sigma}\ope\limits_k } \ar[d]_{\ope\limits_i\sigma } \save \pos?="domeq " \restore & { \ope\limits_{ij}\ope\limits_kz_{ijk } } \ar[d]^{\sigma } \save \pos?="codeq " \restore \\ { \ope\limits_i\ope\limits_{jk}z_{ijk } } \ar[r]_-{\sigma } & { \ope\limits_{ijk}z_{ijk } } \pos " domeq " ; " codeq " * * @ { } ; ? * { = } } } } ; ( 8,0)*{\xybox{{\xymatrix @c=1em { { \ope\limits_ie_1z_i } \ar[dr]_{\sigma } \save \pos?(.4)="domeq " \restore & { \ope\limits_iz_i } \ar[d]^{1 } \save \pos?(.4)="codeq " \restore   \ar[l]_-{\ope\limits_iu } \\ & { \ope\limits_iz_i } } \pos " domeq " ; " codeq " * * @ { } ; ? * { = } } } } \endxy\ ] ] in @xmath16 . as in @xcite we use either of the expressions @xmath120 as a convenient yet precise short - hand for @xmath121 , and we refer to the endofunctor of @xmath16 obtained by observing the effect of @xmath122 on singleton sequences as @xmath123 . the data @xmath124 is called a _ multitensor _ on @xmath16 , and @xmath125 and @xmath126 are referred to as the unit and substitution of the multitensor respectively . given a multitensor @xmath124 on @xmath16 , a _ category enriched in @xmath122 _ consists of @xmath127 together with maps @xmath128 for all @xmath40 and sequences @xmath129 of objects of @xmath2 , such that @xmath130{e_1x(x_0,x_1)}^-{u } : [ d]{x(x_0,x_1)}="bot"^{\kappa } , : " bot"_{\id } ) } } } [ r(5)][d(.15 ) ] { \xybox{\xygraph{!{0;(2.75,0):(0,.5 ) : : } { \ope\limits_i\ope\limits_jx(x_{(ij)-1},x_{ij } ) } ( : [ r]{\ope\limits_{ij}x(x_{(ij)-1},x_{ij})}^-{\sigma } : [ d]{x(x_0,x_{mn_m})}="bot"^{\kappa},:[d]{\ope\limits_ix(x_{(i1)-1},x_{in_i})}_{\ope\limits_i\kappa } : " bot"_-{\kappa})}}}}\ ] ] commute , where @xmath131 , @xmath132 and @xmath133 . since a choice of @xmath46 and @xmath134 references an element of the ordinal @xmath135 , the predecessor @xmath136 of the pair @xmath137 is well - defined when @xmath46 and @xmath134 are not both @xmath33 . with the obvious notion of @xmath122-functor ( see @xcite ) , one has a category @xmath138 of @xmath122-categories and @xmath122-functors together with a forgetful functor @xmath139 the notation we use makes transparent the analogy between multitensors and monads , and categories enriched in multitensors and algebras for a monad . in particular the unit and subtitution for @xmath122 provide @xmath123 with the unit and multiplication of a monad structure . moreover , any object of the form @xmath140 is canonically an @xmath123-algebra , as is the hom of any @xmath122-category , and the substitution maps of @xmath122 are @xmath123-algebra morphisms ( see @xcite lemma(2.7 ) ) . thus in a sense , any multitensor @xmath124 on a category @xmath16 is aspiring to be a multitensor on the category @xmath141 of @xmath123-algebras , but of course there is no meaningful way to regard @xmath125 as living in @xmath141 , except in the boring situation when @xmath123 is the identity monad , that is , when @xmath125 is an identity natural transformation . the multitensors with @xmath125 the identity are called _ normal_.    [ def : lift ] let @xmath124 be a multitensor on a category @xmath16 . a _ lift _ of @xmath124 is a normal multitensor @xmath142 on @xmath141 together with an isomorphism @xmath143 which commutes with the forgetful functors into @xmath144 .    in @xcite we explained how to associate normalised @xmath145-operads and @xmath0-multitensors , which are multitensors on the category of @xmath0-globular sets . in the present paper we shall explain why any @xmath0-multitensor has a canonical lift . at an abstract level much of this paper is about the interplay between the theory of monads on categories of enriched graphs , and the theory of multitensors . it is time to be more precise about which monads on @xmath18 we are interested in . [ def : nmnd ] let @xmath16 be a category . a monad _ over @xmath24 _ on @xmath18 is a monad on @xmath146 in the 2-category @xmath58 . that is , a monad @xmath147 on @xmath18 is over @xmath24 when the functor @xmath101 does nt affect the object sets , in other words @xmath148 for all @xmath127 and similarly for maps , and moreover the components of @xmath149 and @xmath150 are identities on objects . in this section we will describe how such a monad , in the case where @xmath16 has an initial object denoted as @xmath151 , induces a multitensor on @xmath16 denoted @xmath152 . let us describe this multitensor explicitly . first we note that @xmath153 enables us to regard any sequence of objects @xmath112 of @xmath16 as a @xmath16-graph . the object set is @xmath154 , @xmath155 for @xmath156 , and all the other homs are equal to @xmath151 . then we define @xmath157 and the unit as @xmath158 before defining @xmath159 we require some preliminaries . given objects @xmath160 of @xmath16 where @xmath156 , and @xmath161 denote by @xmath162 the obvious subsequence inclusion in @xmath18 : the object map preserves successor and @xmath163 , and the hom maps are identities . now given objects @xmath118 of @xmath16 where @xmath164 and @xmath132 , one has a map @xmath165 given on objects by @xmath166 and @xmath167 for @xmath164 , and the hom map between @xmath168 and @xmath46 is @xmath169 . with these definitions in hand we can now define the components of @xmath159 as @xmath170 ^ -{\{t\tilde{\tau}\}_{0,k } } & & { t(z_{11}, ... ,z_{kn_k})(0,n_{\bullet } ) } \ar[rr]^-{\mu_{0,n_{\bullet } } } & & { \tbar\limits_{ij}z_{ij}}}.\ ] ] from now until the end of ( [ ssec : tbar ] ) we shall be occupied with the proof of    [ thm : tbar ] let @xmath16 be a category with an initial object @xmath151 and @xmath147 be a monad over @xmath24 on @xmath18 . then @xmath152 as defined in ( [ eq : tbar])-([eq : mubar ] ) defines a multitensor on @xmath16 . in principle one could supply a proof of this result immediately by just slogging through a direct verification of the axioms . instead we shall take a more conceptual approach , and along the way encounter various ideas that are of independent interest . for most of the time we will assume a little more of @xmath16 : that it has finite coproducts , to enable our more conceptual approach . in the end though , we will see that only the initial object is necessary . we break up the construction of @xmath152 into three steps . first in ( [ ssec : nmonad->monmonad ] ) , we describe how @xmath171 acquires a monoidal structure and @xmath147 induces a monoidal monad @xmath172 on @xmath171 . then we see that this monoidal monad induces a lax monoidal structure on @xmath171 , which in turn can be transferred across an adjunction to obtain @xmath152 . these last two steps are very general : they work at the level of the theory of lax algebras for an arbitrary 2-monad ( which in our case is @xmath114 ) . so in ( [ ssec : laxalg - const1 ] ) and ( [ ssec : laxalg - const2 ] ) we describe these general constructions , and in ( [ ssec : tbar ] ) we finish the proof of theorem([thm : tbar ] ) . finally in ( [ ssec : path - like ] ) we present a condition on @xmath101 which ensures that @xmath101-algebras and @xmath173-categories may be identified . from now until just before the end of ( [ ssec : tbar ] ) we shall assume that @xmath16 has finite coproducts . we now describe some consequences of this . first , the functor @xmath174 which sends an enriched graph to its set of objects becomes representable . we shall denote by @xmath26 the @xmath16-graph which represents @xmath175 . it has one object and its unique hom is @xmath151 . the second consequence is that @xmath171 inherits a natural monoidal structure and any normalised monad @xmath147 on @xmath18 can then be regarded as a monoidal monad @xmath172 . the explanation for this begins with the observation that the representability of the underlying set functor @xmath175 enables a useful reformulation of the category @xmath171 as the category of endocospans of @xmath26 as in section([ssec : enriched - graphs ] ) . the usefulness of this is that such cospans can be composed , thus endowing @xmath171 with a canonical monoidal structure . the presence of @xmath151 in @xmath16 enables one to compute coproducts in @xmath18 . the coproduct @xmath2 of a family @xmath176 of @xmath16-graphs has object set given as the disjoint union of the object sets of the @xmath177 , @xmath178 when @xmath107 and @xmath179 are objects of @xmath177 , and all the other homs are @xmath151 . with finite coproducts available one can also compute pushouts under @xmath26 , that is the pushout @xmath180 of maps @xmath181 ^ -{y } \ar[l]_-{x } & y}\ ] ] in @xmath18 is described as follows . the object set of @xmath180 is the disjoint union of the object sets of @xmath2 and @xmath182 modulo the identification of @xmath107 and @xmath179 , and let us write @xmath183 for this special element of @xmath180 . the homs of @xmath180 are inherited from @xmath2 and @xmath182 in almost the same way as for coproducts . that is if @xmath29 and @xmath30 are either both objects of @xmath2 or both objects of @xmath182 and they are not both @xmath183 , then their hom @xmath184 is taken as in @xmath2 or @xmath182 . the hom @xmath185 is the coproduct @xmath186 . otherwise this hom is @xmath151 . note that in the special case where the homs @xmath187 and @xmath188 are both @xmath151 , one only requires the initial object in @xmath16 to compute this pushout . given a sequence @xmath189 of doubly - pointed @xmath16-graphs , one defines their _ join _ @xmath190 where @xmath191 is the colimit of @xmath192_-{b_1 } \ar[r]^-{a_2 } & { ... } & 0 \ar[l]_-{b_{n-1 } } \ar[r]^-{a_n } & { x_n}}\ ] ] in @xmath18 which can be formed via iterated pushouts under @xmath26 . this defines a monoidal structure on @xmath171 whose tensor product we denote as @xmath193 . given a functor @xmath194 over @xmath24 , one has a functor @xmath195 whose object map is indicated on the right in the previous display . when both @xmath16 and @xmath196 have an initial object , one defines for each sequence of doubly - pointed @xmath16-graphs a map @xmath197 as follows . write @xmath198 for the components of the colimit cocones ( [ eq : join ] ) . using the unique map @xmath199 , there is a unique map @xmath200 such that @xmath201 . by the unique characterisation of these maps , they assemble together to provide the coherence 2-cell @xmath202 ^ - { * } \ar[d]_{mt_{\bullet } } \save \pos?="dom " \restore & { \ca gv_{\bullet } } \ar[d]^{t_{\bullet } } \save \pos?="cod " \restore \\ { m\ca gw_{\bullet } } \ar[r]_- { * } & { \ca gw_{\bullet } } \pos " dom " ; " cod " * * @ { } ? ( .35 ) \ar@{=>}^{\tau } ? ( .65)}}\ ] ] for a lax monoidal functor , and for @xmath203 ^ -{s } & { \ca gv } \ar[r]^-{t } & { \ca gw}}\ ] ] over @xmath24 one has @xmath204 as monoidal functors . moreover any natural transformation @xmath205 over @xmath24 defines a monoidal natural tranformation @xmath206 . in fact , denoting by @xmath207 the 2-category whose objects are categories with initial objects , a 1-cell @xmath208 in @xmath207 is a functor @xmath194 over @xmath24 , and a 2-cell between these is a natural tranformation also over @xmath24 , we have defined a 2-functor @xmath209 applying @xmath210 to monads gives    [ prop : nmnd->monmnd ] if @xmath16 has finite coproducts and @xmath147 is a monad over @xmath24 on @xmath18 , then @xmath172 is a monoidal monad on @xmath171 , whose monoidal structure is given by pushout - composition of cospans . finally we note that the functor @xmath211 has a left adjoint @xmath212 which we shall now describe . given @xmath213 the underlying @xmath16-graph of @xmath214 , which we shall denote as @xmath215 , has object set @xmath216 and the distinguished pair is @xmath217 . as for the homs , @xmath218 is just @xmath82 itself , and all the other homs are @xmath151 . formally it is the composite @xmath219 ^ -{ml } & { m\ca gv_{\bullet } } \ar[r]^- { * } & { \ca gv_{\bullet } } \ar[r]^-{u } & { \ca gv}}\ ] ] where @xmath220 is the obvious forgetful functor , which enables us to view a sequence of objects of @xmath16 as a @xmath16-graph , as in ( [ ssec : defnmonad ] ) above . now and in ( [ ssec : laxalg - const2 ] ) let @xmath221 be a 2-monad on a 2-category @xmath222 . suppose that we are given a monad _ in _ @xmath223 . let us write @xmath224 for the underlying lax @xmath225-algebra , @xmath226 for the lax @xmath225-algebra endomorphism of @xmath16 , and @xmath46 and @xmath227 for the unit and multiplication respectively . then one obtains another lax @xmath225-algebra structure on @xmath16 with one cell part given as the composite @xmath228 ^ -{e } & v \ar[r]^-{f } & v}\ ] ] and the 2-cell data as follows @xmath229 { sv}="tr " [ d ] { v}="br " [ l ] { v}="bl " " tl":"tr"^-{\eta}:"br"^{e}:"bl"^{f } " tl":"br"|{1 } " tl":"bl"_{1 } " tl " [ d(.35)r(.6 ) ] : @{=>}[r(.2)]^{\iota } " tl " [ d(.7)r(.2 ) ] : @{=>}[r(.2)]^{i } } } } [ r(3 ) ] { \xybox{\xygraph{{s^2v}="l1 " [ d ] { sv}="l2 " [ d ] { sv}="l3 " [ r ] { v}="m " [ r ] { v}="r3 " [ u ] { v}="r2 " [ u ] { sv}="r1 " " l1":"l2"_{se}:"l3"_{sf}:"m"_-{e}:"r3"_-{f } " l1":"r1"^-{\mu}:"r2"^{e}:"r3"^{f } " l2":"r2"^-{e}:"m"_{f } " l1 " [ d(.5)r(.85 ) ] : @{=>}[r(.3)]^{\sigma } " l2 " [ d(.5)r(.5 ) ] : @{=>}[r(.3)]^{\phi } " m " [ u(.2)r(.5 ) ] : @{=>}[r(.3)]^{m}}}}}\ ] ] the verification of the lax algebra axioms is an easy exercise that is left to the reader . now suppose we are given a lax s - algebra @xmath224 together with an adjunction @xmath230 w " v":@<-1.2ex>"w"_-{r}:@<-1.2ex>"v"_-{l } " v":@{}"w"|-{\perp}}\ ] ] with unit @xmath125 and counit @xmath231 . one can then induce a lax @xmath225-algebra structure on @xmath196 . the one - cell part is given as the composite @xmath232 ^ -{sl } & { sv } \ar[r]^-{e } & v \ar[r]^-{r } & w}\ ] ] and the 2-cell data as follows @xmath233 { v}="m " [ r ] { sv}="r " [ l(.5)d ] { v}="br " [ l ] { w}="bl " " m " [ u ] { sw}="t " " l":"t"^-{\eta}:"r"^-{sl}:"br"^-{e}|{}="e":"bl"^-{r } " l":"m"^-{l}:"br"_{1}|{}="onev " " m":"r"^-{\eta } " l":"bl"_{1}|{}="onew " " m":@{}"t"|(.4)*{= } " onew":@{}"onev"|(.35){}="d1"|(.65){}="c1 " " d1":@{=>}"c1"^-{u } " onev":@{}"e"|(.2){}="d2"|(.8){}="c2 " " d2":@{=>}"c2"^-{\iota } } } } [ r(4 ) ] { \xybox { \xygraph{!{0;(1,0):(0,.8 ) : : } { s^2w}="l1 " [ dl ] { s^2v}="l2 " [ d ] { sv}="l3 " [ d ] { sw}="l4 " [ r ] { sv}="ml " [ r ] { v}="mr " [ r ] { w}="r4 " [ u ] { v}="r3 " [ u ] { sv}="r2 " [ ul ] { sw}="r1 " " l1":"l2"_{s^2l}:"l3"_{se}:"l4"_{sr}:"ml"_{sl}:"mr"_{e}:"r4"_{r } " l1":"r1"^-{\mu}:"r2"^{sl}:"r3"^{e}:"r4"^{r } " l3":"ml"^{1}|{}="onemv " " mr":"r3"^{1}|{}="onev " " l2":"r2"^-{\mu } " l1 " [ d(2)r(.35 ) ] : @{=>}[r(.3)]^{\sigma } " l3 " [ d(.7)r(.15 ) ] : @{=>}[r(.25)]^{sc } " r4 " [ u(.25)l(.35 ) ] { = } " l1 " [ d(.5)r(.5 ) ] { = } } } } } \ ] ] the reader will easily verify that the lax @xmath225-algebra axioms for @xmath196 follow from those of @xmath16 and the triangle identities of the adjunction . let us now put together ( [ ssec : nmonad->monmonad])-([ssec : laxalg - const2 ] ) . given a monad @xmath147 on @xmath18 over @xmath24 such that @xmath16 has finite coproducts , we obtained the monoidal monad @xmath172 on @xmath171 in proposition([prop : nmnd->monmnd ] ) . in other words @xmath172 is a monad in @xmath234 . applying ( [ ssec : laxalg - const1 ] ) for @xmath235 gives us a multitensor on @xmath171 , and then applying ( [ ssec : laxalg - const1 ] ) to this last multitensor and the adjunction @xmath236 { v}="w " " v":@<-1.2ex>"w"_-{\varepsilon}:@<-1.2ex>"v"_-{l } " v":@{}"w"|-{\perp}}\ ] ] gives us a multitensor on @xmath16 which we denote as @xmath152 . we shall now unpack this multitensor to see that it does indeed agree with that of theorem([thm : tbar ] ) . the one cell part @xmath173 is the composite @xmath219 ^ -{ml } & { m\ca gv_{\bullet } } \ar[r]^- { * } & { \ca gv_{\bullet } } \ar[r]^-{t_{\bullet } } & { \ca gv_{\bullet } } \ar[r]^-{\varepsilon } & v}\ ] ] which agrees with equation([eq : tbar ] ) and one also easily reconciles equation([eq : etabar ] ) for the unit . as for the substitution unpacking the @xmath159 of our conceptual approach gives the following composite @xmath237 { mv}="p12 " [ dr ] { m\ca gv_{\bullet}}="p22 " [ l(3 ) ] { m^2\ca gv_{\bullet}}="p21 " [ dl ] { m\ca gv_{\bullet}}="p31 " [ d ] { m\ca gv_{\bullet}}="p41 " [ d ] { mv}="p51 " [ r(1.25)d ] { m\ca gv_{\bullet}}="p52 " [ r(1.25)d ] { \ca gv_{\bullet}}="p53 " [ r(1.25)u ] { \ca gv_{\bullet}}="p54 " [ r(1.25)u ] { v}="p55 " [ u ] { \ca gv_{\bullet}}="p42 " [ u ] { \ca gv_{\bullet}}="p32 " " p11":"p12"^-{\mu}|{}="mv1 " " p21":"p22"^-{\mu}|{}="mv2 " " p31":"p32"^-{*}|{}="mv3 " " p51":"p52"_-{ml}:"p53"_-{*}:"p54"_-{t_{\bullet}}:"p55"_-{\varepsilon } " p11":"p21"_-{m^2l}:"p31"_-{m*}:"p41"_-{mt_{\bullet}}:"p51"_-{m\varepsilon } " p12":"p22"^{ml}:"p32"^{*}:"p42"^ { t_{\bullet}}:"p55"^{\varepsilon } " p41":@/^{2pc}/"p52"^{1}|{}="mh1 " " p32":@/_{2.5pc}/"p53"_{t_{\bullet}}|{}="mh2 " " p42":@/_{.5pc}/"p54"_{1}|{}="mh3 " " mv1":@{}"mv2"|*{=}:@{}"mv3"|*{\iso } " p51":@{}"mh1"|{}="mid1":@{}"mh2"|{}="mid2":@{}"mh3"|{}="mid3":@{}"p55"|*{= } " mid1 " [ l(.15 ) ] : @{=>}[r(.3)]^{mc } " mid2 " [ l(.15 ) ] : @{=>}[r(.3)]^{\tau } " mid3 " [ l(.15 ) ] : @{=>}[r(.3)]^{\mu_{\bullet}}}\ ] ] which we shall now unpack further . the counit @xmath231 of the adjunction @xmath238 is described as follows . for a given doubly - pointed @xmath239-graph @xmath42 , the corresponding counit component @xmath240 is specified by insisting that the hom map between @xmath26 and @xmath33 is @xmath241 . define @xmath242 by @xmath243 { t\{(0,t(x_{11}, ... ,x_{1n_1}),n_1)* ... *(0,t(x_{k1}, ... ,x_{kn_k}),n_k)\}}="tl " [ d(.4)r ] { t^2(x_{11}, ...... ,x_{kn_k})}="tr " [ d(.6 ) ] { t(x_{11}, ...... ,x_{kn_k})}="br " " bl"(:"tl"^{t\{c* ... *c\}}:"tr"_-{t\tau}:"br"^{\mu},:"br"_-{\tilde{\mu } } ) " bl":@{}"tr"|*{=}}\ ] ] and then @xmath159 is the effect of @xmath242 on the hom between @xmath26 and @xmath244 . but one may easily verify that the composite @xmath245 is just @xmath246 described in ( [ ssec : defnmonad ] ) . this completes the proof of theorem([thm : tbar ] ) for the case where @xmath16 has finite coproducts . the general case is obtained by observing that only the joins of doubly - pointed @xmath16-graphs @xmath42 such that @xmath247 and @xmath248 are actually used in the construction of the multitensor and in its axioms , and these only require an initial object in @xmath16 . we shall now give a condition on a normalised monad @xmath101 which ensures that categories enriched in @xmath173 are the same thing as @xmath101-algebras . let @xmath249 and consider a sequence @xmath250 of objects of @xmath2 such that @xmath251 and @xmath252 . define the @xmath16-graph @xmath253 and we have a map @xmath254 given on objects by @xmath255 for @xmath256 , and the effect on the hom between @xmath168 and @xmath46 is the identity for @xmath156 .    [ def : path - like ] let @xmath16 be a category with an initial object and @xmath196 be a category with all small coproducts . a functor @xmath194 over @xmath24 is _ path - like _ when for all @xmath249 , the maps @xmath257 for all @xmath40 and sequences @xmath250 such that @xmath251 and @xmath252 , form a coproduct cocone in @xmath196 . a normalised monad @xmath147 on @xmath258 is _ path - like _ when @xmath101 is path - like in the sense just defined . [ ex : cat - monad - path - like ] let @xmath259 and @xmath101 be the free category endofunctor of @xmath260 . for any graph @xmath2 and @xmath261 , the hom @xmath262 is by definition the set of paths in @xmath2 from @xmath29 to @xmath30 . each path determines a sequence @xmath250 of objects of @xmath2 such that @xmath251 and @xmath252 , by reading off the objects of @xmath2 as they are visited by the given path . conversely for a sequence @xmath250 of objects of @xmath2 such that @xmath251 and @xmath252 , @xmath263 identifies the elements @xmath264 with those paths in @xmath2 from @xmath29 to @xmath30 whose associated sequence is @xmath107 . thus @xmath101 is path - like . [ prop : pl - alg<->cat ] let @xmath16 have small coproducts and @xmath147 be a path - like monad on @xmath18 over @xmath24 . then @xmath265 . let @xmath2 be a @xmath16-graph . to give an identity on objects map @xmath266 is to give maps @xmath267 . by path - likeness these amount to giving for each @xmath40 and @xmath250 such that @xmath268 and @xmath269 , a map @xmath270 since @xmath271 , that is @xmath272 . when @xmath273 , for a given @xmath274 , @xmath107 can only be the sequence @xmath275 . the naturality square for @xmath149 at @xmath276 implies that @xmath277 , and the definition of @xmath278 says that @xmath279 . thus to say that a map @xmath266 satisfies the unit law of a @xmath101-algebra is to say that @xmath29 is the identity on objects and that the @xmath280 described above satisfy the unit axioms of a @xmath173-category .    to say that @xmath29 satisfies the associative law is to say that for all @xmath274 , @xmath281 ^ -{\{\mu_x\}_{y , z } } \ar[d]_{tx_{y , z } } & { tx(y , z ) } \ar[d]^{a_{y , z } } \\ { tx(y , z ) } \ar[r]_-{a_{y , z } } & { x(y , z)}}\ ] ] commutes . given @xmath250 from @xmath2 with @xmath268 and @xmath269 , and @xmath282 from @xmath283 with @xmath284 and @xmath285 , consider the composite map @xmath286 ^ -{t\overline{w}_{0,k } } & { t^2x^*x(0,n ) } \ar[r]^-{t\overline{x}_{0,n } } & { t^2x(y , z)}}\ ] ] and note that by path - likeness , and since the coproduct of coproducts is a coproduct , all such maps for @xmath107 and @xmath287 such that @xmath268 and @xmath269 form a coproduct cocone . precomposing ( [ eq : assoc1 ] ) with ( [ eq : copr ] ) gives the commutativity of @xmath288 ^ -{\overline{\mu } } \ar[d]_{\tbar\limits_ia } & { \tbar\limits_{ij}x(x_{ij-1},x_{ij } ) } \ar[d]^{a_x } \\ { \tbar\limits_ix(x_{w_{i-1}},x_{w_i } ) } \ar[r]_-{a_w } & { x(y , z)}}\ ] ] and conversely by the previous sentence if these squares commute for all @xmath107 and @xmath287 , then one recovers the commutativity of ( [ eq : assoc1 ] ) . this completes the description of the object part of @xmath265 . let @xmath289 and @xmath290 be @xmath101-algebras and @xmath291 be a @xmath16-graph morphism . to say that @xmath292 is a @xmath101-algebra map is a condition on the maps @xmath293 for all @xmath274 , and one uses path - likeness in the obvious way to see that this is equivalent to saying that the @xmath294 are the hom maps for a @xmath173-functor . the proof of proposition([prop : pl - alg<->cat ] ) is not new : exactly the same argument was used in the second half of the proof of theorem(7.6 ) of @xcite , although in that case the setting was far less general . the real novelty is the generality of definition([def : path - like ] ) which is crucial for section([sec : lift - mult ] ) . the general way of obtaining a monad from a multitensor , which is the topic of this section , applies to multitensors which conform to    [ def : dist - mult ] let @xmath16 be a category with small coproducts . then a multitensor @xmath124 is _ distributive _ when the functor @xmath122 preserves coproducts in each variable . that is to say , for each @xmath40 , the functor @xmath295 obtained by observing @xmath122 s effect on sequences of length @xmath0 , preserves coproducts in each of its @xmath0 variables . the first step in associating a monad to a distributive multitensor is to identify the bicategory @xmath296 which has the property that monads in @xmath296 are exactly distributive multitensors in the sense of definition([def : dist - mult ] ) . there is also a useful reformulation of the notion of monad on @xmath18 over @xmath24 : as a monad in another 2-category @xmath297 where @xmath298 denotes the 2-comonad on @xmath99 induced by the adjunction @xmath299 of section([ssec : enriched - graphs ] ) . our monad - from - multitensor construction is then achieved by means of a pseudo functor ( homomorphism of bicategories ) @xmath300 which as a pseudo functor sends monads to monads . in fact for a distributive multitensor @xmath122 , @xmath301 is path - like and so @xmath302 . the objects of @xmath296 are categories with coproducts . a morphism @xmath303 in @xmath296 is a functor @xmath304 which preserves coproducts in each variable . a 2-cell between @xmath122 and @xmath305 is simply a natural transformation between these functors @xmath306 . vertical composition of 2-cells is as for natural transformations . the horizontal composite of @xmath307 and @xmath308 , denoted @xmath309 , is defined as a left kan extension @xmath310 ^ -{\mu } \ar[d]_{mf } \save \pos?="dom " \restore & { mv } \ar[d]^{e \comp f } \save \pos?="cod " \restore \\ { mw } \ar[r]_-{e } & { x } \pos " dom " ; " cod " * * @ { } ? ( .35 ) \ar@{=>}^{l_{e , f } } ? ( .65)}}\ ] ] of @xmath311 along @xmath150 . computing this explicitly gives the formula @xmath312 where @xmath164 and @xmath132 on the right hand side of this formula , and we denote by @xmath313 ^ -{\opc\limits_{ij } } & { \opeof\limits_{ij}x_{ij}}}\ ] ] and also by @xmath314 , the corresponding coproduct inclusion . the definition of horizontal composition is clearly functorial with respect to vertical composition of 2-cells . [ prop : dist ] @xmath296 is a bicategory and a monad in @xmath296 is exactly a distributive multitensor . it remains to identify the coherences and check the coherence axioms . the notation we have used here for the coproduct inclusions matches that used in section(3 ) of @xcite . the proof of the first part of proposition(3.3 ) of @xcite interpretted in our present more general setting , is the proof that @xmath296 is a bicategory . the characterisation of distributive multitensors as monads in @xmath296 is immediate from the definitions and our abstract definition of horizontal composition in terms of kan extensions . what in @xcite was called @xmath315 is here the hom @xmath316 . because of the adjunction @xmath299 and the definition of the comonad @xmath298 , a normalised monad on @xmath18 is the same thing as a monad on @xmath16 in the kleisli 2-category @xmath297 of the ( 2-)comonad @xmath298 . the reason this is sometimes useful is that it expresses how our monads over @xmath24 only involve information at the level of homs . the validity of this reformulation is most plainly seen by realising that the factorisation of @xmath51 as an identity on objects 2-functor followed by a 2-fully - faithful 2-functor , can be realised as @xmath317 ^ -{r } & { \kl { \ca g_{\bullet } } } \ar[r]^-{j } & { \cat/\set}}\ ] ] where @xmath318 is the right adjoint part of the kleisli adjunction for @xmath298 . thus applying @xmath210 sends a monad in @xmath297 on @xmath16 to a monad on @xmath18 over @xmath24 , and the definition of @xmath51 and the 2-fully - faithfulness of @xmath210 ensures that any such monad arises uniquely in this way . it is the identity on objects . given a distributive @xmath304 , @xmath301 is defined as a left kan extension @xmath319 ^ - { * } \ar[d]_{m\varepsilon } \save \pos?="dom " \restore & { \ca g_{\bullet}v } \ar[d]^{\gamma{e } } \save \pos?="cod " \restore \\ { mv } \ar[r]_-{e } & { w } \pos " dom " ; " cod " * * @ { } ? ( .35 ) \ar@{=>}^{\gamma_e } ? ( .65)}}\ ] ] of @xmath320 along @xmath321 . we shall now explain why this left kan extension exists in general , give a more explicit formula for @xmath301 in corollary([cor : explicit - gamma ] ) , and then with this in hand it will become clear why @xmath22 is a pseudo - functor . of course one could just define @xmath301 via the formula in corollary([cor : explicit - gamma ] ) . we chose instead to give the above more abstract definition , because it will enable us to attain a more natural understanding of why @xmath22 produces _ path - like _ monads from distributive multitensors in ( [ ssec : distmult->monad ] ) . conceptually , the reason why the left kan extension @xmath322 involves only coproducts in @xmath196 is that @xmath321 is a local left adjoint . a functor @xmath323 is a _ local left adjoint _ when @xmath324 is a local right adjoint . this is equivalent to asking that for all @xmath73 the induced functor @xmath325 between coslices is a left adjoint . we shall explain in lemma([lem : spancomp - lra ] ) why and how pullback - composition of spans can be seen as a local right adjoint , and the statement that @xmath321 is a local left adjoint is just the dual of this because @xmath321 is defined as pushout - composition of cospans in @xmath18 . in lemma([lem : lla - ple ] ) we give a formula for computing the left kan extension along a local left adjoint , and this will then be applied to give our promised more explicit description of @xmath22 s one - cell map . [ lem : spancomp - lra ] let @xmath1 be a category with finite products and let @xmath29 be an object of @xmath1 such that @xmath326 also has finite products . then the one - cell part @xmath327 of the monoidal structure on @xmath328 given by span composition is a local right adjoint . note that for all @xmath40 the slices @xmath329 , where @xmath330 is the @xmath0-fold cartesian product of @xmath29 , have finite products , so the statement of the lemma makes sense and all the limits we mention in this proof exist . in general a functor out of a coproduct of categories is a local right adjoint iff its composite with each coproduct inclusion is a local right adjoint . thus it suffices to show that @xmath0-fold composition of spans @xmath331 is a local right adjoint for all @xmath40 . the case @xmath332 may be exhibited as a composite @xmath333 ^ -{1_a } & { a / a } \ar[r]^-{\delta _ ! } & { a / a{\times}a}}\ ] ] of local right adjoints and thus is a local right adjoint . the case @xmath273 may be regarded as the identity . it suffices to verify the case @xmath334 because with this in hand an easy induction will give the general case . the pullback composite of spans as shown on the left @xmath335 b ( [ dl ] { a}="a1 " , [ dr ] { a}="a2 " ) , [ dr ] c [ dr ] { a}="a3 " ) " d":"b"_{p}:"a1"_{w } " d":"c"^{q}:"a3"^{z } " b":"a2"^{x } " c":"a2"_{y } " d " [ d(.3)l(.3 ) ] [ d(.1)r(.1)]:@{-}[d(.2)r(.2)]:@{-}[u(.2)r(.2 ) ] } } [ r(4 ) ] * \xybox{\xygraph{!{0;(1.2,0):(0,.7 ) : : } d [ r ] { b{\times}c}="bc " [ d ] { a^4}="a4 " [ l ] { a^3}="a3 " [ d ] { a^2}="a2 " " d"(:@{.>}"a3"_{(wp , xp = yq , zq)}(:@{.>}"a2"_{(\pi_1,\pi_3)},:"a4"_-{a{\times}\delta{\times}a } ) , : " bc"^-{(p , q)}:"a4"^{(w , x){\times}(y , z ) } " d " [ d(.3 ) ] [ r(.1)]:@{-}[r(.2)]:@{-}[u(.2)]}}}\ ] ] may be constructed as the dotted composite shown on the right in the previous display , and so binary composition of endospans of @xmath29 is encoded by the composite functor @xmath336 ^ -{\textnormal{prod}_{(a^2,a^2 ) } } & { a/(a^4 ) } \ar[r]^-{(a{\times}\delta{\times}a)^ * } & { a/(a^3 ) } \ar[r]^-{(\pi_1,\pi_3 ) _ ! } & { a / a{\times}a}}\ ] ] where @xmath337 denotes the ( right adjoint ) functor @xmath338 which sends a pair to its cartesian product ( and @xmath339 is the slice of @xmath337 over the pair @xmath340 ) . the constituent functors of this last composite are clearly all local right adjoints . recall @xcite @xcite that local right adjoints can be characterised in terms of generic factorisations . the dual characterisation is as follows . a functor @xmath323 is a local left adjoint iff for all @xmath341 the components of the comma category @xmath342 have terminal objects . a map @xmath343 which is terminal in its component of @xmath342 is said to be _ cogeneric _ , and by definition any @xmath76 can be factored as @xmath344 ^ -{fh } & { fc } \ar[r]^-{g } & b}\ ] ] where @xmath345 is cogeneric . this factorisation is unique up to unique isomorphism and is called the _ cogeneric factorisation _ of @xmath76 . [ lem : lla - ple ] let @xmath323 be a local left adjoint . suppose that @xmath1 has a set @xmath346 of connected components , each of which has an initial object , and that @xmath109 is locally small . for a functor @xmath347 where @xmath348 has coproducts , the left kan extension @xmath349 of @xmath23 along @xmath292 exists and is given by the formula @xmath350 where @xmath351 denotes the initial object of the component @xmath352 , and @xmath353 ^ -{fh_f } & { fa_f } \ar[r]^-{g_f } & b}\ ] ] is a chosen cogeneric factorisation of @xmath354 .    by the general formula for computing left kan extensions as colimits , it suffices to identify the given formula with the colimit of @xmath355 ^ -{p } & a \ar[r]^-{g } & d}\ ] ] where @xmath356 is the obvious projection . this follows since the components of @xmath342 are indexed by pairs @xmath357 , where @xmath352 and @xmath354 , and the component of @xmath342 corresponding to @xmath357 has terminal object given by @xmath358 , which is mapped by @xmath356 to @xmath359 .    in the case of @xmath193 note the initial objects of the components of @xmath360 are of the form @xmath361 and a map @xmath362 is a map @xmath363 which amounts to a sequence of elements of @xmath2 of length @xmath145 starting at @xmath29 and finishing at @xmath30 . the reader will easily verify that @xmath364 & { x^*x } \ar[r]^-{\overline{x } } & { x}}\ ] ] is a cogeneric factorisation of the map associated to @xmath250 . so we have given a conceptual explanation of @xmath365 and @xmath276 which were used in ( [ ssec : path - like ] ) , as well as completed the proof of    [ cor : explicit - gamma ] for @xmath16 and @xmath196 with coproducts and @xmath304 , the defining left kan extension of @xmath301 exists and we have the formula @xmath366    we shall identify @xmath22 with the composite @xmath367 , which amounts to identifying @xmath368 with its mate @xmath369 by the adjunction @xmath299 . then for @xmath127 and @xmath261 , we have for each @xmath40 and for each sequence @xmath250 such that @xmath251 and @xmath252 a coproduct inclusion @xmath370 by corollary([cor : explicit - gamma ] ) . the components of the coherence natural transformations for @xmath22 will be identities on objects . from the definition of the unit @xmath371 in @xmath296 , one has that for @xmath261 the coproduct inclusion @xmath372 is an isomorphism . thus we have an isomorphism @xmath373 . given distributive @xmath374 and @xmath304 , @xmath375 and @xmath261 , we define the hom maps of @xmath376 for @xmath377 , as the unique isomorphism such that for all @xmath378 where @xmath164 and @xmath132 , @xmath379 and @xmath380 , the diagram @xmath381{(\opeof\limits_{ij})x(x_{ij-1},x_{ij})}="c " [ d][r(.5)]{\gamma(e{\comp}f)x(a , b)}="e " , [ r]{\ope\limits_i\gamma(f)x(x_{i-1},x_i)}="b " [ d]{\gamma(e)\gamma(f)x(a , b)}="d " ) " a":"b"^-{\ope\limits_i\opc\limits_j}:"d"^-{c_{x_{i\bullet}}}:"e"^-{\gamma_2 } " a":"c"_-{\opc\limits_{ij}}:"e"_-{c_{x_{ij}}}}\ ] ] in @xmath196 commutes , where @xmath382 and @xmath383 for @xmath384 . we have selected the notation so as to match up with the development of @xcite section(4 ) , and the proof of the first part of proposition(4.1 ) of _ loc . _ interpretted in the present context gives    [ prop : gamma - psfunctor ] the coherences @xmath385 just defined make @xmath22 into a pseudo - functor . given a monad @xmath101 on @xmath18 over @xmath24 , and a set @xmath82 , one obtains by restriction a monad @xmath386 on the category @xmath387 of @xmath16-graphs with fixed object set @xmath82 . let us write @xmath388 for the functor labelled as @xmath22 in @xcite . then for a given distributive multitensor @xmath122 , our present @xmath22 and @xmath388 are related by the formula @xmath389 where the @xmath33 on the right hand side of this equation indicates a singleton . in other words we have just given the `` many - objects version '' of the theory presented in @xcite section(4 ) . having just established the machinery to convert distributive multitensors on @xmath16 to monads on @xmath18 over @xmath24 , we shall now relate the enriched categories to the algebras . this involves two things : seeing that the normalised monads constructed from distributive multitensors are path - like , and understanding the relationship between @xmath22 and the construction of section([sec : monad->mult ] ) of multitensors from monads . [ lem : gamma - path - like ] let @xmath16 and @xmath196 have coproducts and @xmath304 preserve coproducts in each variable . then @xmath390 is path - like . the condition that @xmath194 is path - like can be expressed more 2-categorically . @xmath391 { 1}="tr " [ d ] { \ca gv_{\bullet}}="mr " [ l ] { m\ca gv_{\bullet}}="ml " [ d ] { } = " bl " [ r ] { w}="br " " tl"(:"ml"_{p}:"mr"^-{*},:"tr":"mr"^{(a , x , b)}:"br"^{t } ) " tl " [ d(.5)r(.35 ) ] : @{=>}[r(.3)]^{\lambda } } } } [ r(4)d(.1 ) ] * { \xybox{\xygraph{!{0;(1.5,0):(0,.67 ) : : } { * /(a , x , b)}="tl " [ r ] { 1}="tr " [ d ] { \ca gv_{\bullet}}="mr " [ l ] { m\ca gv_{\bullet}}="ml " [ d ] { mv}="bl " [ r ] { w}="br " " tl"(:"ml"_{p}(:"bl"_{m\varepsilon}:"br"_-{e},:"mr"^-{*}),:"tr":"mr"^{(a , x , b)}:"br"^{\gamma{e } } ) " tl " [ d(.5)r(.35 ) ] : @{=>}[r(.3)]^{\lambda } " ml " [ d(.5)r(.35 ) ] : @{=>}[r(.3)]^{\gamma_e}}}}}\ ] ] writing @xmath392 for 2-cell part of the comma object , @xmath101 is path - like iff the 2-cell on the left exhibits @xmath393 as a colimit . to see this recall from ( [ ssec : gamma ] ) that the set of components of @xmath394 may be regarded as the set of sequences of objects of @xmath6 starting at @xmath29 and finishing at @xmath30 , that each of these components has a terminal object , and that @xmath395 is terminal in the component corresponding to the sequence @xmath250 the situation for a given @xmath122 is depicted on the right in the previous display , and by definition this composite 2-cell is a colimit . thus it suffices to show that the component of the 2-cell @xmath322 at @xmath396 is invertible for all sequences @xmath250 from @xmath6 . the component of @xmath322 at a general @xmath397 is the coproduct inclusion @xmath398 ^ -{c_w } & & { \coprod\limits_{(z_0, ... ,z_m ) } \ope\limits_i y(z_{i-1},z_i)}}\ ] ] corresponding to the sequence @xmath399 where @xmath400 . in the case of @xmath401 , @xmath402 , and for summands corresponding to sequences @xmath183 different from @xmath287 , we will have @xmath403 for some @xmath46 . by the distributivity of @xmath122 those summands will be @xmath151 , whence @xmath404 will be invertible .    given a distributive multitensor @xmath124 note that one can apply @xmath22 to it and then @xmath405 to the result . one has @xmath406 where the @xmath407 in the sum are elements of @xmath154 and @xmath408 and @xmath409 . unless the sequence @xmath410 is just an in - order list @xmath411 of the elements of @xmath154 , at least one of the homs @xmath412 must be @xmath151 making that summand @xmath151 by the distributivity of @xmath122 . thus the coproduct inclusion @xmath413 is invertible . moreover using the explicit description of the multitensor @xmath414 one may verify that this isomorphism is compatible with the units and substitutions , and so we have    [ lem : gamma - bar ] if @xmath124 is a distributive multitensor on a category @xmath16 with coproducts , then one has an isomorphism @xmath415 of multitensors . together with lemma([lem : gamma - path - like ] ) and proposition([prop : pl - alg<->cat ] ) this implies    [ cor : gamma - alg - ecat ] if @xmath124 is a distributive multitensor on a category @xmath16 with coproducts , then one has @xmath302 commuting with the forgetful functors into @xmath18 . we now describe the sense in which @xmath22 and @xmath405 are adjoint . first let us note that equation([eq : tbar ] ) defining the construction @xmath405 in section([ssec : defnmonad ] ) may be seen as providing functors @xmath416 for all @xmath417 in @xmath99 . we have abused notation slightly by denoting by @xmath18 ( resp . @xmath258 ) the category of @xmath16-enriched graphs together with its forgetful functor into @xmath24 . in ( [ ssec : defnmonad ] ) we considered only the case @xmath418 and when @xmath419 is part of a monad , but equation([eq : tbar ] ) obviously makes sense in this more general context . in order to relate this with @xmath22 we make    [ def : ndist ] let @xmath16 and @xmath196 have coproducts . a functor @xmath420 over @xmath24 is _ distributive _ when @xmath421 preserves coproducts in each variable . we denote by @xmath422 the full subcategory of @xmath423 consisting of the distributive functors from @xmath18 to @xmath258 . [ prop : pl - adjoint - char ] let @xmath16 and @xmath196 be categories with coproducts . then we have an adjunction @xmath424 { \ndist(v , w)}="nd " " d":@<1ex>"nd"^-{\gamma_{v , w}}|{}="t":@<1ex>"d"^-{\overline{(-)}_{v , w}}|{}="b " " t":@{}"b"|-{\perp}}\ ] ] whose unit is invertible . a distributive @xmath420 is in the image of @xmath425 iff it is path - like .    by lemma([lem : gamma - bar ] ) applying @xmath22 does indeed produce a distributive functor , so @xmath425 is well - defined and one has an isomorphism of @xmath426 with the identity . for @xmath127 and @xmath261 one has @xmath427 induced by the hom - maps of @xmath263 , giving @xmath428 natural in @xmath101 , and so by @xcite lemma(2.6 ) to establish the adjunction it suffices to show that @xmath429 is inverted by @xmath405 . to this end note that when @xmath430 for @xmath431 , the above summands are non - initial iff the sequence @xmath432 is the sequence @xmath433 , by the distributivity of @xmath101 . the characterisation of path - likeness now follows too , since this condition on a given @xmath101 is by definition the same as the invertibility of @xmath434 .    an immediate consequence of proposition([prop : pl - adjoint - char ] ) and proposition([prop : gammae - basic])([geb1 ] ) below is the following result . a direct proof is also quite straight forward and is left as an exercise . [ cor : pl->copr - pres ] let @xmath16 and @xmath196 be categories with coproducts . if @xmath420 over @xmath24 is distributive and path - like , then it preserves coproducts . [ [ ssec : intro - reexpress ] ]    let us now regard @xmath22 as a pseudo - functor @xmath435 that is to say , we take for granted the inclusion of @xmath297 in @xmath58 . in this section we shall give a systematic account of the categorical properties that @xmath22 preserves . the machinery we are developing gives an elegant inductive description of the monads @xmath436 for strict @xmath0-categories , provides explanations of some of their key properties , and gives a shorter account of the central result of @xcite on the equivalence between @xmath0-multitensors and @xmath145-operads . let @xmath392 be a regular cardinal . an object @xmath346 in a category @xmath16 is _ connected _ when the representable functor @xmath437 preserves coproducts , and @xmath346 is _ @xmath392-presentable _ when @xmath437 preserves @xmath392-filtered colimits . the object @xmath346 is said to be _ small _ when it is @xmath392-presentable for some regular cardinal @xmath392 . a category @xmath16 is _ extensive _ when it has coproducts and for all families @xmath438 of objects of @xmath16 , the functor @xmath439 is an equivalence of categories . a more elementary characterisation is that @xmath16 is extensive iff it has coproducts , pullbacks along coproduct coprojections and given a family of commutative squares @xmath440 ^ -{c_i } \ar[d]_{f_i } & x \ar[d]^{f } \\ { y_i } \ar[r]_-{d_i } & y}\ ] ] where @xmath441 such that the @xmath442 form a coproduct cocone , the @xmath443 form a coproduct cocone iff these squares are all pullbacks . it follows that coproducts are disjoint and the initial object of @xmath16 is strict . another sufficient condition for extensivity is provided by    [ lem : easy - ext ] if a category @xmath16 has disjoint coproducts and a strict initial object , and every @xmath444 is a coproduct of connected objects , then @xmath16 is extensive . the proof is left as an easy exercise . note this condition is not necessary : there are many extensive categories whose objects do nt decompose into coproducts of connected objects , for example , take the topos of sheaves on a space which is not locally connected . a category is _ lextensive _ when it is extensive and has finite limits . there are many examples of lextensive categories : grothendieck toposes , the category of algebras of any higher operad and the category of topological spaces and continuous maps are all lextensive .    denoting the terminal object of a lextensive category @xmath16 by @xmath33 , the representable @xmath445 has a left exact left adjoint @xmath446 which sends a set @xmath82 to the copower @xmath447 . this functor enables one to express coproduct decompositions of objects of @xmath16 , _ internal to @xmath16 _ because to give a map @xmath448 in @xmath16 is the same thing as giving an @xmath34-indexed coproduct decomposition of @xmath2 . the lextensive categories in which every object decomposes into a sum of connected objects are characterised by the following well - known result . [ prop : lext - decompose - charn ] for a lextensive category @xmath16 the following statements are equivalent :    1 . every @xmath444 can be expressed as a coproduct of connected objects.[copcon ] 2 . the functor @xmath449 has a left adjoint.[pi0 ]    the most common instance of this is when @xmath16 is a grothendieck topos . the toposes @xmath16 satisfying the equivalent conditions of proposition([prop : lext - decompose - charn ] ) are said to be _ locally connected_. this terminology is reasonable since for a topological space @xmath2 , one has that @xmath2 is locally connected as a space iff its associated topos of sheaves is locally connected in this sense . a set @xmath450 of objects of @xmath16 is a _ strong generator _ when for all maps @xmath48 , if @xmath451 is bijective for all @xmath452 then @xmath76 is an isomorphism . a locally small category @xmath16 is _ locally @xmath392-presentable _ when it is cocomplete and has a strong generator consisting of small objects . finally recall that a functor is _ accessible _ when it preserves @xmath392-filtered colimits for some regular cardinal @xmath392 . the theory of locally presentable categories is one of the high points of classical category theory , and this notion admits many alternative characterisations @xcite @xcite @xcite . for instance locally presentable categories are exactly those categories which are the @xmath24-valued models for a limit sketch . grothendieck toposes are locally presentable because each covering sieve in a grothendieck topology on a category @xmath453 gives rise to a cone in @xmath454 , and a sheaf is exactly a functor @xmath455 which sends these cones to limit cones in @xmath24 . that is to say a grothendieck topos can be seen as the models of a limit sketch which one obtains in an obvious way from any site which presents it . just as locally presentable categories generalise grothendieck toposes , the following notion generalises locally connected grothendieck toposes . [ def : lcc ] a locally small category @xmath16 is _ locally c - presentable _ when it is cocomplete and has a strong generator consisting of small connected objects . just as locally presentable categories have many alternative characterisations we have the following result for locally c - presentable categories . its proof is obtained by applying the general results of @xcite in the case of the doctrine for @xmath392-small connected categories , which is `` sound '' ( see @xcite ) , and proposition([prop : lext - decompose - charn ] ) . [ thm : conn - gabulm ] for a locally small category @xmath16 the following statements are equivalent .    1 . @xmath16 is locally c - presentable.[lc1 ] 2 . @xmath16 is cocomplete and has a small dense subcategory consisting of small connected objects.[lc2 ] 3 . @xmath16 is a full subcategory of a presheaf category for which the inclusion is accessible , coproduct preserving and has a left adjoint.[lc4 ] 4 . @xmath16 is the category of models for a limit sketch whose distingished cones are connected.[lc5 ] 5 . @xmath16 is locally presentable and every object of @xmath16 is a coproduct of connected objects.[lc6 ] 6 . @xmath16 is locally presentable , extensive and the functor @xmath456 has a left adjoint.[lc7 ]    [ ex : lc - groth - toposes ] by theorem([thm : conn - gabulm])([lc7 ] ) a grothendieck topos is locally connected in the usual sense iff its underlying category is locally c - presentable . just as with locally presentable categories , locally c - presentable categories are closed under many basic categorical constructions . for instance from theorem([thm : conn - gabulm])([lc6 ] ) , one sees immediately that the slices of a locally c - presentable category are locally c - presentable from the corresponding result for locally presentable categories . another instance of this principle is the following result . [ thm : acc - monad ] if @xmath16 is locally c - presentable and @xmath101 is an accessible coproduct preserving monad on @xmath16 , then @xmath102 is locally c - presentable . first we recall that colimits in @xmath102 can be constructed explicitly using colimits in @xmath16 and the accessibility of @xmath101 ( see for instance @xcite for a discussion of this ) . by definition we have a regular cardinal @xmath392 such that @xmath101 preserves @xmath392-filtered colimits and @xmath16 is locally @xmath392-presentable . defining @xmath457 to be the full subcategory of @xmath16 consisting of the @xmath392-presentable and connected objects , @xmath458 is a monad with arities in the sense of @xcite . one has a canonical isomorphism @xmath459 ^ -{v^t(i,1 ) } \ar[d]_{u } \save \pos?="d " \restore & { \psh { \theta}_t } \ar[d]^{\res_j } \save \pos?="c " \restore \\ v \ar[r]_-{v(i_0,1 ) } & { \psh { \theta}_0 } \pos " d";"c " * * @ { } ; ? * { \iso}}\ ] ] in the notation of @xcite . thus @xmath460 is accessible since @xmath461 creates colimits , and @xmath101 and @xmath462 are accessible . by the nerve theorem of @xcite @xmath460 is also fully faithful , it has a left adjoint since @xmath102 is cocomplete given by left extending @xmath46 along the yoneda embedding , and so we have exhibited @xmath102 as conforming to theorem([thm : conn - gabulm])([lc4 ] ) . [ ex : lc - noperad - algebras ] an @xmath0-operad for @xmath463 in the sense of @xcite , gives a finitary coproduct preserving monad on the category @xmath8 of @xmath0-globular sets , and its algebras are just the algebras of the monad . thus the category of algebras of any @xmath0-operad is locally c - presentable by theorem([thm : acc - monad ] ) .      at the object level , to apply @xmath22 is to apply @xmath51 , so we shall now collect together many of the categorical properties that @xmath51 preserves . for @xmath16 with an initial object @xmath151 , we saw in section([ssec : nmonad->monmonad ] ) how to construct coproducts in @xmath18 explicitly . from this explicit construction , it is clear that the connected components of a @xmath16-graph @xmath2 may be described as follows . objects @xmath29 and @xmath30 of @xmath2 are in the same connected component iff there exists a sequence @xmath129 of objects of @xmath2 such that for @xmath156 the hom @xmath464 is non - initial . moreover @xmath2 is clearly the coproduct of its connected components , coproducts are disjoint and the initial object of @xmath18 , whose @xmath24 of objects is empty , is strict . thus by lemma([lem : easy - ext ] ) we obtain    [ prop : gv - ext ] if @xmath16 has an initial object then @xmath18 is extensive and every object of @xmath18 is a coproduct of connected objects . given finite limits in @xmath16 it is straight forward to construct finite limits in @xmath18 directly . the terminal @xmath16-graph has one object and its only hom is the terminal object of @xmath16 . given maps @xmath465 and @xmath466 in @xmath18 their pullback @xmath180 can be constructed as follows . objects are pairs @xmath467 where @xmath29 is an object of @xmath1 and @xmath231 is an object of @xmath346 such that @xmath468 . the hom @xmath469 is obtained as the pullback of @xmath470 ^ -{f } & { b(fa_1,fa_2 ) } & { c(c_1,c_2 ) } \ar[l]_-{g}}\ ] ] in @xmath16 . thus one has    [ prop : gv - lext ] if @xmath16 has finite limits then so does @xmath18 . if in addition @xmath16 has an initial object , then @xmath18 is lextensive and every object of @xmath16 is a coproduct of connected objects .    as for cocompleteness one has the following result due to betti , carboni , street and walters . [ prop : gv - cocomp]@xcite if @xmath16 is cocomplete then so is @xmath18 and @xmath174 is cocontinuous . we now turn to local c - presentability . first we require a general lemma which produces a dense subcategory of @xmath18 from one in @xmath16 in a canonical way . [ lem : gv - dense ] let @xmath450 be a full subcategory of @xmath16 and suppose that @xmath16 has an initial object . define an associated full subcategory @xmath471 of @xmath18 as follows :    * @xmath472 . * @xmath473 .    if @xmath450 is dense then so is @xmath471 . for a regular cardinal @xmath392 , if the objects of @xmath450 are @xmath392-presentable then so are those of @xmath471 .    given functions @xmath474 natural in @xmath475 , we must show that there is a unique @xmath48 such that @xmath476 . the object map of @xmath76 is forced to be @xmath477 , and naturality with respect to the maps @xmath478 ^ -{0 } & ( d ) & 0 \ar[l]_-{1}}\ ] ] ensures that the functions @xmath479 amount to @xmath477 together with functions @xmath480 natural in @xmath452 for all @xmath261 . by the adjointness @xmath238 these maps are in turn in bijection with maps @xmath481 natural in @xmath452 for all @xmath261 , and so by the density of @xmath450 one has unique @xmath482 in @xmath16 such that@xmath483 . thus @xmath477 and the @xmath482 together form the object and hom maps of the unique desired map @xmath76 .    since any colimit in @xmath18 is preserved by @xmath175 @xcite , one can easily check directly that @xmath26 is @xmath392-presentable for all @xmath392 . let @xmath452 be @xmath392-presentable . one has natural isomorphisms @xmath484 exhibiting @xmath485 as a coproduct of functors that preserve @xmath392-filtered colimits , and thus is itself @xmath392-filtered colimit preserving . [ cor : gv - lc ] if @xmath16 is locally presentable then @xmath18 is locally c - presentable . immediate from theorem([thm : conn - gabulm])([lc2 ] ) , lemma([lem : gv - dense ] ) and proposition([prop : gv - cocomp ] ) .    in @xcite kelly and lack proved that if @xmath16 is locally presentable then so is @xmath18 by an argument almost identical to that given here . the only difference is that in their version of lemma([lem : gv - dense ] ) , their @xmath471 differs from ours only in that they use @xmath486 where we use @xmath26 , and they instead prove that @xmath471 is a strong generator given that @xmath450 is . we have given the above proof because the present form of lemma([lem : gv - dense ] ) is more useful to us in section([ssec : gamma - pres12-lra ] ) . next we shall see that @xmath51 preserves toposes . first a lemma of independent interest . [ lem : abstract - wreath ] let @xmath453 be a category and @xmath487 be a lextensive category . consider the category @xmath488 constructed from @xmath453 as follows . there is an injective on objects fully faithful functor @xmath489 and @xmath488 has an additional object @xmath26 not in the image of @xmath490 . moreover for each @xmath491 one has maps @xmath492 and for all @xmath493 one has the equations @xmath494 and @xmath495 . then @xmath496 $ ] is equivalent to the full subcategory of @xmath497 $ ] consisting of those @xmath2 such that @xmath6 is a copower of @xmath33 , the terminal object of @xmath487 . let us write @xmath498 for the full subcategory of @xmath497 $ ] described in the statement of the lemma . we shall describe the functors @xmath499 \rightarrow \ca f } & & { ( -)^{- } : \ca f \rightarrow \ca g[\op { \c},\ca e ] } \end{array}\ ] ] which provide the desired equivalence directly . given @xmath500 $ ] define @xmath501 and @xmath502 in the obvious way this definition is functorial in @xmath2 and @xmath346 . conversely given @xmath503 choose a set @xmath504 such that @xmath505 . such a set is determined uniquely up to isomorphism . then define the homs of @xmath506 via the pullbacks @xmath507 _ { } \save \pos?(.3)="lpb " \restore \ar[r]^- { } \save \pos?(.3)="tpb " \restore & { yc_+ } \ar[d]^{(y\sigma , y\tau ) } \save \pos?(.3)="rpb " \restore \\ { 1 } \ar[r]_-{(a , b ) } \save \pos?(.3)="bpb " \restore & { y0{\times}y0 } \pos " rpb " ; " lpb " * * @ { } ; ? ! { " bpb";"tpb"}="cpb " * * @ { } ; ? * * @{- } ; " tpb " ; " cpb " * * @ { } ; ? * * @{-}}}\ ] ] in @xmath487 for all @xmath508 . note that since @xmath487 is lextensive and hence distributive , @xmath509 is itself the coproduct of copies of @xmath33 indexed by such pairs @xmath47 . the natural isomorphisms @xmath510 come from extensivity . [ cor : gv - topos ]    1 . if @xmath16 is a presheaf topos then so is @xmath18.[g6 ] 2 . if @xmath16 is a grothendieck topos then @xmath18 is a locally connected grothendieck topos.[g7 ]    applying lemma([lem : abstract - wreath ] ) in the case where @xmath453 is small and @xmath511 one obtains the formula @xmath512 and thus ( [ g6 ] ) . since a grothendieck topos is a left exact localisation of a presheaf category , the 2-functoriality of @xmath51 together with ( [ g6 ] ) , corollary([cor : gv - lc ] ) and example([ex : lc - groth - toposes ] ) , implies that to establish ( [ g7 ] ) it suffices to show that @xmath51 preserves left exact functors between categories with finite limits . this follows immediately from the explicit description of finite limits in @xmath18 given in the proof of proposition([prop : gv - lext ] ) . the construction @xmath513 in the previous proof is easily discovered by thinking about why , as pointed out in section([ssec : enriched - graphs ] ) , applying @xmath51 successively to the empty category does one produce the categories of @xmath0-globular sets for @xmath40 . the construction @xmath513 is just the general construction on categories , which when applied successively to the empty category produces the categories @xmath514 for @xmath40 , presheaves on which are by definition @xmath0-globular sets . first we note that by the explicit description of coproducts , the terminal object and pullbacks of enriched graphs , and the formula of corollary([cor : explicit - gamma ] ) , one has    [ prop : gammae - basic ] let @xmath16 and @xmath196 have coproducts and @xmath304 be distributive .    1 . @xmath301 preserves coproducts.[geb1 ] 2 . if @xmath122 preserves the terminal object then so does @xmath301 . if @xmath122 preserves pullbacks then so does @xmath301 . the precise conditions under which @xmath22 preserves and reflects cartesian natural transformations are identified by proposition([prop : gamma - cart ] ) . first we require a lemma which generalises lemma(7.4 ) of @xcite , whose proof follows easily from the explicit description of pullbacks in @xmath18 discussed in ( [ ssec : gamma - pres ] ) . [ lem : pb - hom ] suppose that @xmath16 has pullbacks . given a commutative square ( i ) @xmath515_{f } \save \pos?="d " \restore \ar[r]^-{h } & x \ar[d]^{g } \save \pos?="c " \restore \\ y \ar[r]_-{k } & z \pos " d";"c " * * @ { } ; ? * { \textnormal{i } } } } } ; ( 1,0)*-!l{\xybox{\xymatrix{{w(a , b ) } \ar[d]_{f_{a , b } } \save \pos?="d " \restore \ar[r]^-{h_{a , b } } & { x(ha , hb ) } \ar[d]^{g_{ha , hb } } \save \pos?="c " \restore \\ { y(a , b ) } \ar[r]_-{k_{a , b } } & { z(ha , hb ) } \pos " d";"c " * * @ { } ; ? * { \textnormal{ii } } } } } * * @ { } ? * { } \end{xy}}\ ] ] in @xmath18 such that @xmath477 and @xmath516 are identities , one has for each @xmath517 commuting squares ( ii ) as in the previous display . the square ( i ) is a pullback iff for all @xmath517 , the square ( ii ) is a pullback in @xmath16 . [ prop : gamma - cart ] let @xmath16 have coproducts , @xmath196 be extensive , and @xmath122 and @xmath307 be distributive . then @xmath518 is cartesian iff @xmath519 is cartesian . let @xmath48 be in @xmath18 , @xmath261 and @xmath432 be a sequence of objects of @xmath2 such that @xmath251 and @xmath252 . for each such @xmath520 it suffices by lemma([lem : pb - hom ] ) , to show that the square ( ii ) in the commutative diagram @xmath521 { \gamma{e}x(a , b)}="i12 " [ d ] { \gamma{e}y(fa , fb)}="i22 " [ l ] { \ope\limits_iy(fx_{},fx_i)}="i21 " " i11 " [ ul ] { \gamma{f}x(a , b)}="o11 "   " i12 " [ ur ] { \opf\limits_ix(x_{},x_i)}="o12 " " i21 " [ dl ] { \opf\limits_iy(fx_{},fx_i)}="o21 " " i22 " [ dr ] { \gamma{f}y(fa , fb)}="o22 " " i11":"i12"^-{c_{x_i}}:"i22"^{\gamma{e}(f)_{a , b}}|{}="l3 " " i11":"i21"_{\ope\limits_if}|{}="l2":"i22"_-{c_{x_i } } " o11":"o12"^-{c_{x_i}}:"o22"^{\gamma{f}(f)_{a , b}}|{}="l4 " " o11":"o21"_{\opf\limits_if}|{}="l1":"o22"_-{c_{x_i } } " i11":"o11"^{\phi } " i21":"o21"^{\phi } " i12":"o12"^{\gamma\phi } " i22":"o22"^{\gamma\phi } " l1":@{}"l2"|(.4)*{(iii)}:@{}"l3"|*{(i)}:@{}"l4"|(.7)*{(ii)}}\ ] ] is a pullback . the square ( i ) and the largest square are pullbacks since @xmath196 is extensive and the @xmath231-maps are coproduct inclusions , and ( iii ) is a pullback since @xmath522 is cartesian . thus for all such @xmath432 the composite of ( i ) and ( ii ) is a pullback . the result follows since the @xmath523 form coproduct cocones and @xmath196 is extensive . conversely suppose that we have @xmath524 in @xmath16 for @xmath156 . then by the isomorphisms @xmath525 and @xmath526 , and their naturality with respect to @xmath522 one may identify the naturality square on the left @xmath527 ^ -{\phi_{x_i } } \ar[d]_{\ope\limits_if_i } & { \opf\limits_i } \ar[d]^{\opf\limits_if_i } \\ { \ope\limits_iy_i } \ar[r]_-{\phi_{y_i } } & { \opf\limits_iy_i } } } } ; ( 1,0)*-!l{\xybox{\xymatrix{{\gamma(e)(x_1, ... ,x_n)(0,n ) } \ar[r]^-{\gamma(\phi)_{0,n } } \ar[d]_{\gamma(e)(f_1, ... ,f_n)_{0,n } } & { \gamma(f)(x_1, ... ,x_n)(0,n ) } \ar[d]_{\gamma(f)(f_1, ... ,f_n)_{0,n } } \\ { \gamma(e)(y_1, ... ,y_n)(0,n ) } \ar[r]_-{\gamma(\phi)_{0,n } } & { \gamma(f)(y_1, ... ,y_n)(0,n ) } } } } * * @ { } ? * { } \end{xy}}\ ] ] with that on the right in the previous display , which is cartesian by lemma([lem : pb - hom ] ) and since @xmath528 is cartesian , and so @xmath522 is indeed cartesian . @xmath22 s compatibility with the bicategory structure of @xmath296 is expressed in    [ prop : gamma - cart2 ]    1 . let @xmath304 and @xmath374 be distributive and @xmath196 be extensive . if @xmath122 and @xmath292 preserve pullbacks then so does @xmath309.[cart2 - 1 ] 2 . let @xmath529 and @xmath530 be distributive and pullback preserving , and @xmath196 be extensive . if @xmath531 and @xmath532 are cartesian then so is @xmath533.[cart2 - 2 ]    ( [ cart2 - 1 ] ) . suppose that for @xmath156 we have pullback squares @xmath534_{h_i } \save \pos?(.3)="lpb " \restore \ar[r]^-{f_i } \save \pos?(.3)="tpb " \restore & { b_i } \ar[d]^{k_i } \save \pos?(.3)="rpb " \restore \\ { d_i } \ar[r]_-{g_i } \save \pos?(.3)="bpb " \restore & { c_i } \pos " rpb " ; " lpb " * * @ { } ; ? ! { " bpb";"tpb"}="cpb " * * @ { } ; ? * * @{- } ; " tpb " ; " cpb " * * @ { } ; ? * * @{-}}}\ ] ] in @xmath220 . then for each partition @xmath535 of @xmath0 we have a commutative diagram @xmath536 { ( \opeof\limits_i)b_i}="i12 " [ d ] { ( \opeof\limits_i)d_i}="i22 " [ l ] { ( \opeof\limits_i)c_i}="i21 " " i11 " [ ul ] { \ope\limits_i\opf\limits_ja_{ij}}="o11 "   " i12 " [ ur ] { \ope\limits_i\opf\limits_jb_{ij}}="o12 " " i21 " [ dl ] { \ope\limits_i\opf\limits_jc_{ij}}="o21 " " i22 " [ dr ] { \ope\limits_i\opf\limits_jd_{ij}}="o22 " " i11":"i12"^-{(\opeof\limits_i)f_i}:"i22"^{(\opeof\limits_i)k_i}|{}="l3 " " i11":"i21"_{(\opeof\limits_i)h_i}|{}="l2":"i22"_-{(\opeof\limits_i)g_i } " o11":"o12"^-{(\ope\limits_i\opf\limits_j)f_{ij}}:"o22"^{(\ope\limits_i\opf\limits_j)k_{ij}}|{}="l4 " " o11":"o21"_{(\ope\limits_i\opf\limits_j)h_{ij}}|{}="l1":"o22"_-{(\ope\limits_i\opf\limits_j)g_{ij } } " o11":"i11"^{c_{n_i } } " o21":"i21"^{c_{n_i } } " o12":"i12"^{c_{n_i } } " o22":"i22"^{c_{n_i } } " l1":@{}"l2"|(.35)*{(i)}:@{}"l3"|*{(ii)}:@{}"l4"|(.7)*{(iii)}}\ ] ] in @xmath196 in which the @xmath231-maps are coproduct inclusions , the coproducts being indexed over the set of all such partitions . for the outer square @xmath164 and @xmath132 . we must show that ( ii ) is a pullback . the squares ( i ) and ( iii ) are pullbacks because @xmath196 is extensive . the large square is a pullback because @xmath122 and @xmath292 preserve pullbacks . thus the composite of ( i ) and ( ii ) is a pullback , and so the result follows by the extensivity of @xmath196 . ( [ cart2 - 2 ] ) . given @xmath537 for @xmath156 and @xmath535 we have a commutative diagram @xmath536 { ( \opeof\limits_i)b_i}="i12 " [ d ] { ( \opeofpr\limits_i)b_i}="i22 " [ l ] { ( \opeofpr\limits_i)a_i}="i21 " " i11 " [ ul ] { \ope\limits_i\opf\limits_ja_{ij}}="o11 "   " i12 " [ ur ] { \ope\limits_i\opf\limits_jb_{ij}}="o12 " " i21 " [ dl ] { \opepr\limits_i\opfpr\limits_ja_{ij}}="o21 " " i22 " [ dr ] { \opepr\limits_i\opfpr\limits_jb_{ij}}="o22 " " i11":"i12"^-{(\opeof\limits_i)f_i}:"i22"^{\psi{\comp}\phi } " i11":"i21"_{\psi{\comp}\phi}:"i22"_-{(\opeofpr\limits_i)f_i } " o11":"o12"^-{(\ope\limits_i\opf\limits_j)f_{ij}}:"o22"^{\psi\phi } " o11":"o21"_{\psi\phi}:"o22"_-{(\opepr\limits_i\opfpr\limits_j)f_{ij } } " o11":"i11"^{c_{n_i } } " o21":"i21"^{c_{n_i } } " o12":"i12"^{c_{n_i } } " o22":"i22"^{c_{n_i}}}\ ] ] in @xmath196 , to which we apply a similar argument as in ( [ cart2 - 1 ] ) to demonstrate that the inner square is cartesian . let @xmath16 be lextensive . then by proposition([prop : gamma - cart2 ] ) the monoidal structure of @xmath316 restricts to pullback preserving @xmath538 and cartesian transformations between them . a multitensor @xmath124 on @xmath16 is _ cartesian _ when @xmath122 preserves pullbacks and @xmath125 and @xmath126 are cartesian . by slicing over @xmath122 one obtains a monoidal category @xmath539 , whose objects are cartesian transformations @xmath540 . to give @xmath541 a monoid structure is to give @xmath1 the structure of a cartesian multitensor such that @xmath542 is a cartesian multitensor morphism . such an @xmath541 is called an _ @xmath122-multitensor_. the category of @xmath122-multitensors is denoted @xmath543 . [ ex : non - sig - operads ] let us denote by @xmath544 the multitensor on @xmath24 given by finite products . by the lextensivity of @xmath24 , @xmath544 is an lra multitensor and thus cartesian . a @xmath544-multitensor is the same thing as a non - symmetric operad in @xmath24 . for given a cartesian multitensor map @xmath545 , one obtains the underlying sequence @xmath546 of sets of the corresponding operad as @xmath547 . one uses the cartesianness of the naturality squares corresponding to the maps @xmath548 to recover @xmath122 from the @xmath549 . similarly the multitensor structure of @xmath122 corresponds to the unit and substitution maps making the @xmath549 into an operad . given a cartesian normalised monad @xmath101 on @xmath18 , one obtains a monoidal category @xmath550 , whose objects are cartesian transformations @xmath551 over @xmath24 . explicitly , to say that a general collection , which is a cartesian transformation @xmath551 , is over @xmath24 , is to say that the components of @xmath542 are identities on objects maps of @xmath16-graphs . the tensor product of @xmath550 is obtained via composition and the monad structure of @xmath101 , and a monoid structure on @xmath541 is a cartesian monad structure on @xmath1 such that @xmath542 is a cartesian monad morphism . such an @xmath541 is called a _ @xmath101-operad over @xmath24_. the category of @xmath101-operads over @xmath24 is denoted as @xmath552 . by the results of this subsection @xmath22 induces a strong monoidal functor @xmath553 and we shall now see that this functor is a monoidal equivalence . applying this equivalence to the monoids in the respective monoidal categories gives the promised general equivalence between multitensors and operads over @xmath24 in corollary([cor : mult - nop - equiv ] ) below . [ lem : transfer - dpl ] let @xmath16 be a lextensive category and @xmath101 be a cartesian monad on @xmath18 over @xmath24 . let @xmath551 be a collection over @xmath24 .    1 . if @xmath101 is distributive then so is @xmath1.[tdpl1 ] 2 . if @xmath101 is path - like then so is @xmath1.[tdpl2 ]    ( [ tdpl1 ] ) : given an @xmath0-tuple @xmath554 of objects of @xmath16 and a coproduct cocone @xmath555 where @xmath156 , we must show that the hom - maps @xmath556 form a coproduct cocone . for @xmath557 we have a pullback square @xmath558 { a(x_1, ... ,x_i, ... ,x_n)(0,n)}="tr " [ d ] { t(x_1, ... ,x_i, ... ,x_n)(0,n)}="br " [ l ] { t(x_1, ... ,x_{ij}, ... ,x_n)(0,n)}="bl " " tl"(:"tr"^-{a(x_1, ... ,c_j, ... ,x_n)_{0,n}}:"br"^{\alpha},:"bl"_{\alpha}:"br"_-{t(x_1, ... ,c_j, ... ,x_n)_{0,n } } ) " tl " [ r(.1)d(.4 ) ] { \xybox{\xygraph{!{0;(0.2,0 ) : } ( : @{-}[u ] , : @{-}[l])}}}}\ ] ] and by the distributivity of @xmath101 and lemma([lem : pb - hom ] ) , the @xmath559 form a coproduct cocone , and thus so do the @xmath560 by the extensivity of @xmath16 . + ( [ tdpl2 ] ) : given @xmath127 , @xmath261 and a sequence @xmath129 of objects of @xmath2 such that @xmath251 and @xmath252 , we have the map @xmath561 and we must show that these maps , where the @xmath562 range over all sequences from @xmath29 to @xmath30 , form a coproduct cocone . by the path - likeness of @xmath101 we know that the maps @xmath257 form a coproduct cocone , so we can use the cartesianness of @xmath542 , lemma([lem : pb - hom ] ) and the extensivity of @xmath16 to conclude as in ( [ tdpl1 ] ) . [ prop : coll - equiv ] let @xmath16 be lextensive and @xmath563 be a distributive cartesian multitensor on @xmath16 . then @xmath564 is a monoidal equivalence @xmath565 . the functor @xmath564 is the result of applying the functor @xmath566 of proposition([prop : pl - adjoint - char ] ) over @xmath122 . thus by proposition([prop : pl - adjoint - char ] ) , proposition([prop : gamma - cart ] ) and lemma([lem : transfer - dpl ] ) , @xmath564 is an equivalence . [ cor : mult - nop - equiv ] let @xmath16 be lextensive and @xmath563 be a distributive cartesian multitensor on @xmath16 . then applying @xmath564 gives @xmath567 . local right adjoint monads , especially defined on presheaf categories , are fundamental to higher category theory . indeed a deeper understanding of such monads is the key to understanding the relationship between the operadic and homotopical approaches to the subject @xcite . we will now understand the conditions under which @xmath22 preserves local right adjoints . first we require two lemmas . [ lem : partial - adjoint ] let @xmath568 be a functor , @xmath16 be cocomplete , @xmath220 be a small dense full subcategory of @xmath196 , and @xmath569 be a partial left adjoint to @xmath318 , that is to say , one has isomorphisms @xmath570 natural in @xmath571 and @xmath444 . defining @xmath572 as the left kan extension of @xmath212 along the inclusion @xmath573 , one has @xmath574 .    denoting by @xmath575 the canonical forgetful functor for @xmath576 and recalling that @xmath577 , one obtains the desired natural isomorphism as follows @xmath578(lp,\textnormal{const}(x ) ) } & { \iso } & { \lim_{f{\in}i / y } v(l(\textnormal{dom}(f)),x ) } \\ & { \iso } & { \lim_f w(\textnormal{dom}(f),rx ) } & { \iso } &   { \ca b(y , rx ) } \end{array}\ ] ] for all @xmath444 . [ lem : lra - dense ] let @xmath208 be a functor , @xmath16 be cocomplete and @xmath196 have a small dense subcategory @xmath220 . then @xmath101 is a local right adjoint iff every @xmath579 with @xmath580 admits a generic factorisation . if in addition @xmath16 has a terminal object denoted @xmath33 , then generic factorisations in the case @xmath581 suffice .    for the first statement ( @xmath582 ) is true by definition so it suffices to prove the converse . the given generic factorisations provide a partial left adjoint @xmath583 to @xmath584 where @xmath34 is the inclusion of @xmath220 . now @xmath585 is a small dense subcategory of @xmath586 , and so by the previous lemma @xmath212 extends to a genuine left adjoint to @xmath587 . in the case where @xmath16 has @xmath33 one requires only generic factorisations in the case @xmath581 by the results of @xcite section(2 ) . the analogous result for presheaf categories , with the representables forming the chosen small dense subcategory , was discussed in @xcite section(2 ) . [ prop : gammae - lra ] let @xmath16 and @xmath196 be locally c - presentable and @xmath304 be distributive . if @xmath304 is a local right adjoint then so is @xmath301 . let @xmath450 be a small dense subcategory of @xmath196 consisting of small connected objects . by lemma([lem : lra - dense ] ) and lemma([lem : gv - dense ] ) it suffices to exhibit generic factorisations of maps @xmath588 where @xmath225 is either @xmath26 or @xmath589 for some @xmath452 . in the case where @xmath225 is @xmath26 the first arrow in the composite @xmath478 & { \gamma{e}0 } \ar[r]^-{\gamma{e}t } & { \gamma{e}1}}\ ] ] is generic because @xmath26 is the initial @xmath196-graph with one object ( and @xmath590 here is the unique map ) . in the case where @xmath591 , to give @xmath76 is to give a map @xmath592 in @xmath16 , by corollary([cor : explicit - gamma ] ) since @xmath348 is connected . since @xmath122 is a local right adjoint , @xmath549 is too and so one can generically factor @xmath593 to obtain @xmath594 ^ -{g'_f } & { \ope\limits_iz_i } \ar[r]^-{\ope\limits_it } & { e_n1}}\ ] ] from which we obtain the generic factorisation @xmath595 ^ -{g_f } & { \gamma{e}z } \ar[r]^-{\gamma{e}t } & { \gamma{e}1}}\ ] ] where @xmath596 , the object map of @xmath358 is given by @xmath166 and @xmath597 , and the hom map of @xmath358 is @xmath598 composed with the coproduct inclusion . first note that while it is a very different thing for @xmath304 to preserve coproducts compared with preserving coproducts in each variable , the situation is simpler for @xmath392-filtered colimits , where @xmath392 is any regular cardinal . note that @xmath599 preserves @xmath392-filtered colimits in each variable iff @xmath292 preserves @xmath392-filtered colimits . for given a connected category @xmath346 , the colimit of a functor @xmath600 constant at say @xmath2 is of course @xmath2 , and since @xmath392-filtered colimits are connected , one can prove @xmath601 by keeping all but the variable of interest constant . for the converse it is sufficient to prove that @xmath292 preserves colimits of chains of length less than @xmath392 , and this follows by a straight forward transfinite induction . since @xmath111 is a sum of @xmath602 s , from the connectedness of @xmath392-filtered colimits it is clear that @xmath122 preserves @xmath392-filtered colimits iff each @xmath603 does , and so we have proved    [ lem : multi - lambda ] for @xmath304 the following statements are equivalent for any regular cardinal @xmath392 .    1 . @xmath122 preserves @xmath392-filtered colimits in each variable . @xmath122 preserves @xmath392-filtered colimits . @xmath603 preserves @xmath392-filtered colimits for all @xmath40 . as already mentioned , colimits in @xmath18 for a cocomplete @xmath16 were calculated in @xcite . let us spell out transfinite composition in @xmath18 . given an ordinal @xmath392 and a @xmath392-chain @xmath604 in @xmath18 with colimit @xmath2 , one may consider the induced @xmath392-chain @xmath605 in @xmath24 . this will have colimit @xmath606 because @xmath392-filtered colimits and products commute in @xmath24 and @xmath607 is cocontinuous . for @xmath377 let us denote by @xmath608 the full subcategory of the category of elements of ( [ chain2 ] ) , consisting of those elements which are sent to @xmath47 by the universal cocone . we shall call this the _ @xmath47-component _ of the chain ( [ chain1 ] ) . now @xmath608 is of course no longer a chain , but one may easily verify that it is @xmath392-filtered . by the explicit description of colimits in @xmath24 , the @xmath608 are just the connected components of the category of elements of ( [ chain2 ] ) . to pairs @xmath609 which are elements of @xmath608 , one may associate the corresponding hom @xmath610 , and in this way build a functor @xmath611 . the hom @xmath31 is the colimit of this functor . [ prop : gamma - accessible ] let @xmath16 and @xmath196 be cocomplete , @xmath304 be distributive and @xmath392 be a regular cardinal . if @xmath122 preserves @xmath392-filtered colimits in each variable then @xmath301 preserves @xmath392-filtered colimits . it suffices to show @xmath301 preserves colimits of @xmath392-chains . consider the chain ( [ chain1 ] ) in @xmath18 . for all @xmath40 one has an @xmath145-ary version of ( [ chain2 ] ) , that is involving @xmath145-fold instead of binary cartesian products in @xmath24 . these of course also commute with @xmath392-filtered colimits . similarly one obtains a @xmath392-filtered category @xmath612 and a functor @xmath613 applying @xmath301 does nothing at the object level . let us write @xmath614 for the @xmath47-component of the chain obtained by applying @xmath301 to ( [ chain1 ] ) , and @xmath615 for the corresponding functor into @xmath196 . from the explicit description of @xmath22 s effect on homs of corollary([cor : explicit - gamma ] ) , one sees that @xmath615 is the coproduct of the composites @xmath616 ^ -{f_{x_0, ... ,x_n } } & { v^n } \ar[r]^-{e_n } & w}.\ ] ] over all sequences @xmath129 starting at @xmath29 and finishing at @xmath30 . by lemma([lem : multi - lambda ] ) the colimits of the @xmath617 are preserved by the @xmath549 , and so by the explicit description of colimits of @xmath392-chains in @xmath258 , the colimit of ( [ chain1 ] ) is indeed preserved by @xmath301 . let us recall the construction @xmath618 . given a monad @xmath147 on @xmath16 a category with products , one has a multitensor @xmath619 defined by @xmath620 and the unit and substitution is induced in the obvious way from @xmath149 and @xmath150 . when @xmath16 is lextensive , @xmath101 is a local right adjoint , and @xmath149 and @xmath150 are cartesian , it follows that @xmath619 is a local right adjoint its unit and multiplication are also cartesian . when @xmath101 preserves coproducts and the cartesian product for @xmath16 is distributive , then @xmath619 is a distributive multitensor . if in addition finite limits and filtered colimits commute in @xmath16 ( which happens when , for example @xmath16 is locally finitely presentable ) , then @xmath619 is finitary . moreover by proposition(2.8 ) of @xcite one has @xmath621 where the enrichment on the right hand side is with respect to cartesian products . thus one can consider the following inductively - defined sequence of monads    * put @xmath622 equal to the identity monad on @xmath24 . * given a monad @xmath9 on @xmath623 , define the monad @xmath624 on @xmath625 . recalling that @xmath623 is the category of @xmath0-globular sets . @xcite[thm : enhopi - main - theorem ] for @xmath40 , @xmath626 is the strict @xmath0-category monad on @xmath0-globular sets . this monad is coproduct preserving , finitary and local right adjoint .    by ( [ eq : tcross - cat ] ) and corollary([cor : gamma - alg - ecat ] ) , one has @xmath627 and so by definition @xmath628 is the category of strict @xmath0-categories and strict @xmath0-functors between them . by the remarks at the beginning of this section and corollary([cor : gv - topos ] ) @xmath629 will produce a distributive , finitary , local right adjoint multitensor on a presheaf category when it is fed a coproduct preserving , finitary , local right adjoint monad on a presheaf category . by corollary([cor : gv - topos ] ) , proposition([prop : gammae - basic ] ) , proposition([prop : gamma - accessible ] ) , proposition([prop : gammae - lra ] ) and proposition([prop : gamma - cart2 ] ) , @xmath22 will produce a coproduct preserving , finitary , local right adjoint monad on a presheaf category when it is fed a distributive , finitary , local right adjoint multitensor on a presheaf category . thus the monads @xmath626 are indeed coproduct preserving , finitary and local right adjoint for all @xmath40 .    the objects of @xmath630 @xmath0-operads over @xmath24  were in @xcite @xcite called `` normalised '' @xmath0-operads . many @xmath0-categorical structures of interest , such as weak @xmath0-categories , can be defined as algebras of @xmath0-operads over @xmath24 . objects of @xmath631 are called @xmath0-multitensors . these are a nice class of lax monoidal structures on the category of @xmath0-globular sets . by corollary([cor : mult - nop - equiv ] ) and theorem([thm : enhopi - main - theorem ] ) one obtains    @xcite[cor : nnp1op - nmult ] for all @xmath40 , applying @xmath22 gives @xmath632 . that is to say , @xmath22 exhibits @xmath145-operads over @xmath24 and @xmath0-multitensors as the same thing , and under this correspondence , the algebras of the operad correspond to the categories enriched in the associated multitensor by corollary([cor : gamma - alg - ecat ] ) . up to this point @xmath22 has been our notation for the process @xmath633 and @xmath405 has been our notation for the reverse construction . for the most complete analysis of these constructions one must acknowledge that they are the object maps of 2-functors in two important ways . this 2-functoriality together with the formal theory of monads @xcite gives a satisfying explanation of how it is that monad distributive laws arise naturally in this subject ( see @xcite ) . as the lax - algebras of a 2-monad @xmath114 ( see section([ssec : lmc ] ) ) , lax monoidal categories form a 2-category @xmath234 . see @xcite for a complete description of the 2-category of lax algebras for an arbitrary 2-monad . explicitly a lax monoidal functor between lax monoidal categories @xmath634 and @xmath635 consists of a functor @xmath636 , and maps @xmath637 natural in the @xmath177 such that @xmath638 { f_1hx}="r " [ dl ] { he_1x}="b " " l"(:"r"^-{u_{hx}}:"b"^{\psi_x},:"b"_{hu_x } ) } } } [ r(5 ) ] { \xybox{\xygraph{!{0;(2,0):(0,.5 ) : : } { \opf\limits_i\opf\limits_jhx_{ij}}="tl " [ r ] { \opf\limits_ih\ope\limits_jx_{ij}}="tm " [ r ] { h\ope\limits_i\ope\limits_jx_{ij}}="tr " [ l(.5)d ] { h\ope\limits_{ij}x_{ij}}="br " [ l ] { \opf\limits_{ij}hx_{ij}}="bl " " tl " ( : @<1ex>"tm"^-{\opf\limits_i\psi}:@<1ex>"tr"^-{\psi\ope\limits_j}:"br"^{h\sigma},:"bl"_{\sigma{h}}:@<1ex>"br"_-{\psi})}}}}\ ] ] commute for all @xmath2 and @xmath639 in @xmath16 . a monoidal natural transformation between lax monoidal functors @xmath640 consists of a natural transformation @xmath641 such that @xmath642 { h\ope\limits_ix_i}="tr " [ d ] { k\ope\limits_ix_i}="br " [ l ] { \opf\limits_ikx_i}="bl " " tl " ( : @<1ex>"tr"^-{\psi}:"br"^{\phi\ope\limits_i},:"bl"_{\opf\limits_i\phi}:@<1ex>"br"_-{\kappa})}\ ] ] commutes for all @xmath177 . [ def:2cat - distmult ] the 2-category @xmath20 of distributive multitensors , is defined to be the full sub-2-category of @xmath234 consisting of the @xmath634 such that @xmath16 has coproducts and @xmath122 is distributive . for any 2-category @xmath222 recall the 2-category @xmath643 from @xcite of monads in @xmath222 . another way to describe this very canonical object is that it is the 2-category of lax algebras of the identity monad on @xmath222 . explicitly the 2-category @xmath644 has as objects pairs @xmath645 where @xmath16 is a category and @xmath101 is a monad on @xmath16 . an arrow @xmath646 is a pair consisting of a functor @xmath636 and a natural transformation @xmath647 satisfying the obvious 2 axioms : these are just the `` unary '' analogues of the axioms for a lax monoidal functor written out above . for example , any lax monoidal functor @xmath648 as above determines a monad functor @xmath649 . a monad transformation between monad functors @xmath650 consists of a natural transformation @xmath641 satisfying the obvious axiom . for example a monoidal natural transformation @xmath522 as above is a monad transformation @xmath651 . in fact as we are interested in monads over @xmath24 , we shall work not with @xmath644 but rather with @xmath652 . an object @xmath645 of this latter 2-category is a category @xmath16 equipped with a functor into @xmath24 , together with a monad @xmath101 on @xmath16 which `` acts fibre - wise '' with respect to this functor . that is @xmath101 s object map does nt affect the underlying object set , similarly for the arrow map of @xmath101 , and the components of @xmath101 s unit and multiplication are identities on objects in the obvious sense . an arrow @xmath646 of @xmath652 is a pair @xmath648 as in the case of @xmath644 , with the added condition that @xmath653 s components are the identities on objects , and similarly the 2-cells of @xmath652 come with an extra identity - on - object condition . we shall now exhibit the 2-functor @xmath19 which on objects is given by @xmath654 . let @xmath655 be a lax monoidal functor between distributive lax monoidal categories . then for @xmath127 and @xmath261 , we define the hom map @xmath656 to be the composite of @xmath657 { \coprod\limits_{a = x_0, ... ,x_n = b } h\ope\limits_ix(x_{i-1},x_i)}="m " " l":@<1ex>"m"^-{\coprod \psi}}\ ] ] and @xmath658 s coproduct preservation obstruction map . it follows easily from the definitions that @xmath659 as defined here satisfies the axioms of a monad functor . moreover given a monoidal natural transformation @xmath660 , it also follows easily from the definitions that @xmath661 is a monad transformation . it is also straight - forward to verify that these assignments are 2-functorial .      by proposition([prop : pl - adjoint - char ] ) and corollary([cor : pl->copr - pres ] ) we understand objects of the image of @xmath22 and we collect this information in    [ prop : image - gamma - objects ] for @xmath16 a category with coproducts , a monad @xmath662 over @xmath24 is in the image of @xmath22 iff @xmath101 is distributive and path - like . moreover any such @xmath101 automatically preserves coproducts . one recovers the distributive multitensor @xmath122 such that @xmath663 as @xmath664 . since the construction @xmath405 is itself obviously 2-functorial , the arrows and 2-cells in the image of @xmath22 may also be easily characterised . [ prop : image - gamma-1 - 2-cells ]    1 . let @xmath634 and @xmath635 be distributive lax monoidal categories . a monad functor of the form @xmath665 is in the image of @xmath22 iff @xmath666 for some @xmath667.[gamma - char1 ] 2 . let @xmath668 be lax monoidal functors between distributive lax monoidal categories . a monad transformation @xmath669 is in the image of @xmath22 iff it is of the form @xmath670.[gamma - char2 ]    by definition monad functors and transformations in the image of @xmath22 have the stated properties , so we must prove the converse . given @xmath671 such that the components of @xmath653 are the identities on objects , one recovers for @xmath672 from @xmath16 , the corresponding lax monoidal functor coherence map as the hom map from @xmath26 to @xmath0 of the component @xmath673 . that is to say , we apply @xmath405 to the appropriate monad functors to prove ( [ gamma - char1 ] ) , and we do the same to the appropriate monad transformations to obtain ( [ gamma - char2 ] ) .    [ def : plmnd ] we denote by @xmath674 the following 2-category . its objects are monads @xmath675 over @xmath24 such that @xmath16 has coproducts and @xmath101 is distributive and path - like . its arrows are arrows @xmath676 of @xmath652 , and its 2-cells are 2-cells @xmath677 of @xmath652 .    thus from the proof of proposition([prop : image - gamma-1 - 2-cells ] ) we have    [ cor:2eq - mult - mon ] @xmath22 and @xmath405 provide a 2-equivalence @xmath678 . lax algebras of a 2-monad organise naturally into _ two _ different 2-categories depending on whether one takes lax or oplax algebra morphisms . so in particular one has the 2-category @xmath679 of lax monoidal categories , _ _ op__lax - monoidal functors between them and monoidal natural transformations between those . the coherence @xmath653 for an oplax @xmath655 goes in the other direction , and so its components look like this : @xmath680 the reader should easily be able to write down explicitly the two coherence axioms that this data must satisfy , as well as the condition that must be satisfied by a monoidal natural transformation between oplax monoidal functors . similarly there is a dual version @xmath681 of the 2-category @xmath643 of monads in a given 2-category @xmath222 discussed above @xcite . an arrow @xmath646 of @xmath682 consists of a functor @xmath636 and a natural transformation @xmath683 satisfying the two obvious axioms . an arrow of @xmath682 is called a monad opfunctor . as before @xmath684 differs from @xmath652 in that all the categories involved come with a functor into @xmath24 , and all the functors and natural transformations involved are compatible with these forgetful functors . we now describe the dual version of the 2-functoriality of @xmath22 discussed in sections([ssec : gamma-2-functor ] ) and ( [ ssec : gamma - image ] ) . when defining the one - cell map of @xmath22 in section([ssec : gamma-2-functor ] ) we were helped by the fact that the coproduct preservation obstruction went the right way : see the definition of the monad functor @xmath685 above . this time however we will not be so lucky , and for this reason we must restrict ourselves in the following definition to coproduct preserving oplax monoidal functors . [ def : opdistmult ] the 2-category @xmath686 is defined to be the locally full sub-2-category of @xmath679 consisting of the distributive lax monoidal categories , and the oplax monoidal functors @xmath648 such that @xmath658 preserves coproducts . we denote by @xmath687 the following 2-category . its objects are monads @xmath675 over @xmath24 such that @xmath16 has coproducts and @xmath101 is distributive and path - like . its arrows are arrows @xmath676 of @xmath684 such that @xmath658 preserves coproducts . its 2-cells are 2-cells @xmath677 of @xmath684 . we now define @xmath688 with object map @xmath654 as before , and the rest of its definition is obtained by modifying the earlier definition of @xmath22 in what should now be the obvious way . the proof of the following result is obtained by a similar such modification of the proof of corollary([cor:2eq - mult - mon ] ) . [ cor:2eq - mult - mon - dual ] @xmath22 and @xmath405 provide a 2-equivalence @xmath689      as explained in @xcite the assignment @xmath690 is in fact the object map of a strict 3-functor . just exploiting 2-functoriality here and corollaries([cor:2eq - mult - mon ] ) and ( [ cor:2eq - mult - mon - dual ] ) one immediately obtains    [ thm : monmon - distlaw ] @xmath22 and @xmath405 provide two 2-equivalences of 2-categories :    1 . @xmath691.[mdl1 ] 2 . @xmath692.[mdl2 ]    the meaning of this result is understood by understanding what the objects of the 2-categories involved are , that is to say , what monads are in each of the 2-categories @xmath20 , @xmath686 , @xmath674 and @xmath687 . a very beautiful observation of @xcite is that to give a monad on @xmath645 in @xmath643 is to give another monad @xmath225 on @xmath16 , together with a distributive law @xmath693 . similarly to give a monad on @xmath645 in @xmath681 is to give another monad @xmath225 on @xmath16 , together with a distributive law @xmath694 in the other direction . thus the 2-categories @xmath695 and @xmath696 really have the same objects : such an object being a pair of monads on the same category and a distributive law between them . thus both @xmath697 and @xmath698 are 2-categories whose objects are monad distributive laws between monads defined on categories of enriched graphs , with some extra conditions . on the other hand a monad in the 2-category @xmath234 of lax monoidal categories and lax monoidal functors is also a well - known thing , and such things are usually called _ monoidal monads_. similarly an opmonoidal monad @xmath101 on a monoidal category @xmath16 , that is to say a monad on @xmath16 in @xmath679 , comes with the extra data of coherence maps @xmath699 that are compatible with the monad structure . if for instance @xmath700 is just cartesian product , then the product obstruction maps for @xmath101 endow it with an opmonoidal structure in a unique way .    by definition the objects of @xmath701 are monoidal monads defined on distributive lax monoidal categories , and the objects of @xmath702 are coproduct preserving opmonoidal monads defined on distributive lax monoidal categories . thus the meaning of theorem([thm : monmon - distlaw ] ) is that it exhibits these kinds of monoidal and opmonoidal monads as being equivalent to certain kinds of distributive laws . we shall spell this out precisely in corollaries([cor : monmon - distlaw ] ) and ( [ cor : opmonmon - distlaw ] ) below . let @xmath634 be a lax monoidal category and @xmath101 be a monad on @xmath16 . in section([ssec : laxalg - const1 ] ) we saw that when @xmath101 is a monoidal monad , that is to say one has coherence maps @xmath703 making the underlying endofunctor of @xmath101 a lax monoidal functor and the unit and multiplication monoidal natural transformations , then one has another multitensor on @xmath16 given on objects by @xmath704 , and with unit and substitution given by the composites @xmath705 { tx}="m " [ r(1.5 ) ] { te_1x}="r " " l":"m"^-{\eta}:"r"^-{tu } } } } [ r(5)d(.1 ) ] { \xybox{\xygraph{!{0;(2,0 ) : } { t\ope\limits_it\ope\limits_j x_{ij}}="l " [ r ] { t^2\ope\limits_i\ope\limits_j x_{ij}}="m " [ r ] { t\ope\limits_{ij}x_{ij}}="r " " l":@<1ex>"m"^-{t{\tau}e}:@<1ex>"r"^-{\mu\sigma}}}}}\ ] ] in particular if @xmath122 is distributive and @xmath101 preserves coproducts , then this new multitensor @xmath706 is also distributive . if instead @xmath101 has the structure of an opmonoidal monad , with the coherences @xmath707 going in the other direction , then in the same way one can construct a new multitensor @xmath708 on @xmath16 which on objects is defined by given by @xmath709 . once again if @xmath122 is distributive and @xmath101 coproduct preserving , then @xmath708 is a distributive multitensor . in particular when @xmath122 is cartesian product , @xmath708 is the multitensor @xmath619 of section([ssec : induction ] ) .    with regards to monoidal monads , unpacking what theorem([thm : monmon - distlaw])([mdl1 ] ) says at the object level gives    [ cor : monmon - distlaw ] let @xmath634 be a distributive lax monoidal category and @xmath101 be a monad on @xmath16 . to give maps @xmath710 making @xmath101 into a monoidal monad on @xmath634 , is the same as giving a monad distributive law @xmath711 whose components are the identities on objects . in the case where @xmath101 preserves coproducts one may readily verify that @xmath712 as monads , and so by corollary([cor : gamma - alg - ecat ] ) one understands what the algebras of this composite monad @xmath713 are . [ cor : algebras - gtgammae ] if in the situation of corollary([cor : monmon - distlaw ] ) @xmath101 also preserves coproducts , then @xmath714 .    similarly , one can unpack what theorem([thm : monmon - distlaw])([mdl2 ] ) says at the object level , witness @xmath715 and use corollary([cor : gamma - alg - ecat ] ) to conclude    [ cor : opmonmon - distlaw ] let @xmath634 be a distributive lax monoidal category and @xmath101 be a coproduct preserving monad on @xmath16 . to give maps @xmath716 making @xmath101 into an opmonoidal monad on @xmath634 , is the same as giving a monad distributive law @xmath717 whose components are the identities on objects . moreover @xmath718 . [ ex : cheng1 ] from the inductive description of @xmath436 of section([ssec : induction ] ) and corollary([cor : opmonmon - distlaw ] ) one obtains a distibutive law @xmath719 for all @xmath0 , between monads on @xmath623 , and the composite monad @xmath720 . thus we have recaptured the decomposition of @xcite of the strict @xmath0-category monad into a `` distributive series of monads '' . pursuing the idea of the previous example , we shall now begin to recover and in some senses generalise cheng s analysis and description @xcite of the trimble definition of weak @xmath0-category .    from @xcite example(2.6 ) non - symmetric operads in the usual sense can be regarded as multitensors . here we shall identify a non - symmetric operad @xmath721 in a braided monoidal category @xmath16 , with the multitensor @xmath722 it generates . recall that one object @xmath122-categories for @xmath122 a non - symmetric operad are precisely algebras of the operad @xmath122 in the usual sense . if @xmath700 is cartesian product , then the projections @xmath723 are the components of a cartesian multitensor map @xmath724 . conversely such a cartesian multitensor map exhibits @xmath122 as an operad via @xmath725 for all @xmath40 . let @xmath16 be a distributive category and @xmath101 a coproduct preserving monad on @xmath16 . let us denote by @xmath726 a non - symmetric operad in @xmath102 . the `` @xmath429 '' is meant to denote the @xmath101-algebra actions , that is @xmath727 is the @xmath101-algebra structure , and so @xmath122 denotes the underlying operad in @xmath16 . since @xmath728 preserves products it is the underlying functor of a strong monoidal functor @xmath729 between lax monoidal categories . since the composites @xmath730 { te_n \times \prod\limits_i tx_i}="m " [ r ] { e_n \times \prod\limits_itx_i}="r " " l":@<1ex>"m"^-{\textnormal{prod . obstn.}}:@<1ex>"r"^-{\varepsilon_n \times \id}}\ ] ] form the components of an opmonoidal structure for the monad @xmath101 , we find ourselves in the situation of corollary([cor : opmonmon - distlaw ] ) and so obtain    [ prop : monad - trimble ] let @xmath16 be a distributive category , @xmath101 a coproduct preserving monad on @xmath16 and @xmath726 a non - symmetric operad in @xmath102 . then one has a distributive law @xmath731 between monads on @xmath18 , and isomorphisms @xmath732 of categories over @xmath18 . this result has an operadic counterpart . [ prop : operad - trimble ] let @xmath16 be a lextensive category , @xmath101 a cartesian and coproduct preserving monad on @xmath16 , @xmath733 a @xmath101-operad and @xmath726 a non - symmetric operad in @xmath734 . then the monad @xmath735 , whose algebras by proposition([prop : monad - trimble ] ) are @xmath726-categories , has a canonical structure of a @xmath736-operad .    with @xmath737 we must exhibit a cartesian monad map @xmath738 . we have the cartesian multitensor map @xmath739 which exhibits the multitensor @xmath726 as a non - symmetric operad , thus @xmath740 is also a cartesian multitensor map , and since @xmath16 is lextensive @xmath22 sends this to a cartesian monad morphism . the required cartesian monad map is thus @xmath741 . recall the path - space functor @xmath742 discussed in section([ssec : enriched - graphs ] ) . to say that a non - symmetric topological operad @xmath1 acts on @xmath180 is to say that @xmath180 factors as @xmath94 { \enrich a}="m " [ r ] { \ca g(\top)}="r " " l":"m"^-{p_a}:"r"^-{u^a}}\ ] ] the main example to keep in mind is the version of the little intervals operad recalled in @xcite definition(1.1 ) . as this @xmath1 is a contractible non - symmetric operad , @xmath1-categories may be regarded as a model of @xmath1-infinity spaces . since @xmath180 is a right adjoint , @xmath743 is also a right adjoint by the dubuc adjoint triangle theorem . a product preserving functor @xmath744 into a distributive category , may be regarded as the underlying functor of a strong monoidal functor @xmath745 between lax monoidal categories . applying @xmath22 to this gives us a monad functor @xmath746 with underlying functor @xmath747 , which amounts to giving a lifting @xmath748 as indicated in the commutative diagram @xmath749 { \enrich a}="tm " [ r ] { \enrich { qa}}="tr " [ d ] { \ca gv}="br " [ l ] { \ca g(\top)}="bl " " tl"(:"tm"^-{p_a}(:"tr"^-{\overline{q}}:"br"^{u^{qa}},:"bl":"br"_-{\ca g(q)}),:"bl"_{p})}\ ] ] and so we have produced another product preserving functor @xmath750 where @xmath751 and @xmath752 . the functor @xmath748 is product preserving since @xmath753 is and @xmath754 creates products . the assignment @xmath755 in the case where @xmath1 is as described in @xcite definition(1.1 ) , is the inductive process lying at the heart of the trimble definition . in this definition one begins with the connected components functor @xmath756 and defines the category @xmath757 of `` trimble 0-categories '' to be @xmath24 . the induction is given by @xmath758 and so this definition constructs not only a notion of weak @xmath0-category but the product preserving @xmath759 s to be regarded as assigning the fundamental @xmath0-groupoid to a space . applying proposition([prop : monad - trimble ] ) to this situation produces the monad on @xmath0-globular sets whose category of algebras is @xmath760 as well as its decomposition into an iterative series of monads witnessed in @xcite section(4.2 ) . applying proposition([prop : operad - trimble ] ) and the inductive description of @xmath436 of section([ssec : induction ] ) exhibits these monads as @xmath0-operads . applied to the normalised @xmath13-operad for gray categories @xcite , the results of the section([sec : reexpress ] ) produce a lax monoidal structure @xmath122 on the category of @xmath15-globular sets whose enriched categories are exactly gray categories . for this example it turns out that @xmath123 is @xmath761 , and so providing a lift of @xmath122 in the sense of definition([def : lift ] ) amounts to the construction of a tensor product of @xmath15-categories whose enriched categories are gray categories , that is to say , an abstractly constructed gray tensor product . by the main result of this section theorem([thm : lift - mult ] ) , _ every _ @xmath0-multitensor has a lift which is unique given certain properties . while the proof of theorem([thm : lift - mult ] ) is fairly abstract , and the uniqueness has the practical effect that in the examples we never have to unpack an explicit description of the lifted multitensors provided by the theorem , we provide such an unpacking in section([ssec : explicit - lifting ] ) anyway . this enables us to give natural conditions when the construction of the lifted multitensor is simpler . doing all this requires manipulating some of the transfinite constructions that arise in monad theory , and we give a self - contained review of these in the appendix .      in appendix [ sec : dubuc ] we recall an explicit description , for a given monad morphism @xmath762 between accessible monads on a locally presentable category @xmath16 , of the left adjoint @xmath763 to the canonical forgetful functor @xmath764 induced by @xmath522 . the key point about @xmath763 is that it is constructed via a transfinite process involving only _ connected _ colimits in @xmath16 . the importance of this is underscored by    [ lem : concol - pathlike ] let @xmath16 be a category with an initial object , @xmath196 be a cocomplete category , @xmath210 be a small connected category and @xmath765\ ] ] be a functor . suppose that @xmath292 sends objects of @xmath210 to normalised functors , and arrows of @xmath210 to natural transformations whose components are identities on objects . * then the colimit @xmath766 of @xmath292 may be chosen to be normalised.[cpl1 ]    given such a choice of @xmath767 :    * if @xmath768 is path - like for all @xmath557 , then @xmath767 is also path - like.[cpl2 ] * if @xmath768 is distributive for all @xmath557 , then @xmath767 is also distributive.[clp3 ]    colimits in @xmath769 $ ] are computed componentwise from colimits in @xmath258 and so for @xmath127 we must describe a universal cocone with components @xmath770 we demand that the @xmath771 are identities on objects . this is possible since the @xmath772 form the constant diagram on @xmath6 by the hypotheses on @xmath292 . for @xmath261 we choose an arbitrary colimit cocone @xmath773 in @xmath196 . one may easily verify directly that since @xmath210 is connected , the @xmath771 do indeed define a univeral cocone for all @xmath2 in order to establish ( 1 ) . since the properties of path - likeness and distributivity involve only colimits at the level of the homs as does the construction of @xmath767 just given , ( 2 ) and ( 3 ) follow immediately since colimits commute with colimits in general . with these preliminaries in hand we are now ready to present the monad version of the multitensor lifting theorem , and then the lifting theorem itself . [ lem : mnd - lift - mult ] let @xmath16 be a locally presentable category , @xmath318 be a coproduct preserving monad on @xmath16 , @xmath225 be an accessible and normalised monad on @xmath18 , and @xmath774 be a monad morphism whose components are identities on objects . denote by @xmath101 the monad induced by @xmath775 . * one may choose @xmath763 so that @xmath101 becomes normalised . given such a choice of @xmath763 :    * if @xmath225 is path - like then @xmath101 is path - like . * if @xmath225 is distributive then @xmath101 is distributive . let @xmath392 be the regular cardinal such that @xmath225 preserves @xmath392-filtered colimits . to verify that @xmath776 is a normalised monad one must verify : ( i ) @xmath101 is normalised , and ( ii ) the components of the unit @xmath777 are identities on objects . since @xmath778 is a retraction of @xmath779 , it will then follow that the components of @xmath778 are also identities on objects . but @xmath101 is normalised iff @xmath780 is normalised , and @xmath781 , so for ( i ) it suffices to show that one can choose @xmath763 making @xmath782 normalised . this follows by a transfinite induction using the explicit description of @xmath782 of section([sec : dubuc ] ) and lemma([lem : concol - pathlike ] ) .    for the initial step note that @xmath783 can be chosen to be normalised , because @xmath784 is a componentwise - identity on objects natural transformation between normalised functors , since the monads @xmath225 and @xmath785 are normalised . thus the coequaliser defining @xmath783 is a connected colimit involving only normalised functors and componentwise - identity on objects natural transformations , and so @xmath783 can be taken to be normalised by lemma([lem : concol - pathlike ] ) . for the inductive steps the argument is basically the same : at each stage one is taking connected colimits of normalised functors and componentwise identity on objects natural transformations , so that by lemma([lem : concol - pathlike ] ) one stays within the subcategory of @xmath786 $ ] consisting of such functors and natural transformations . moreover using lemma([lem : concol - pathlike ] ) @xmath101 will be path - like if @xmath225 is .    as for ( ii ) it suffices to prove that the components of @xmath787 are identities on objects . writing @xmath788 for transfinite composite constructed as part of the definition of @xmath763 ( note that @xmath789 by definition ) recall from the end of section([sec : dubuc ] ) that one has a commutative square @xmath790 ^ -{\ca g\rho } \ar[d]_{{\phi}u^m } & { u^{\ca gr } } \ar[d]^{\ca g(u^r)\eta^t } \\ { su^{\ca gr } } \ar[r]_-{q } & { u^s\phi_{!}}}\ ] ] where @xmath791 is the 2-cell datum for @xmath318 s eilenberg - moore object , which we recall is preserved by @xmath51 . now @xmath791 is componentwise the identity objects since @xmath792 is and @xmath791 is a retraction of it , @xmath793 is the identity on objects by definition , and @xmath794 is by construction , so the result follows . recall from definition([def : lift ] ) that a _ lift _ of @xmath124 is a normal multitensor @xmath142 on @xmath141 together with an isomorphism @xmath143 which commutes with the forgetful functors into @xmath144 . when in addition @xmath305 is distributive , we say that it is a _ distributive lift _ of @xmath122 . recall from @xcite that for any category @xmath16 , the functor @xmath795 which sends a monad @xmath101 on @xmath16 to the forgetful functor @xmath103 , is fully - faithful . [ thm : lift - mult ] let @xmath124 be a distributive multitensor on @xmath16 a locally presentable category , and let @xmath122 be accessible in each variable . then @xmath122 has a distributive lift @xmath305 , which is unique up to isomorphism . write @xmath796 for the distributive multitensor on @xmath16 whose unary part is @xmath123 and whose non - unary parts are constant at @xmath151 . there is an obvious inclusion @xmath797 of multitensors and one clearly has @xmath798 applying lemma([lem : mnd - lift - mult ] ) with @xmath799 , @xmath800 and @xmath801 one produces a path - like , normalised and distributive monad @xmath101 on @xmath802 , because @xmath225 is accessible by proposition([prop : gamma - accessible ] ) . thus one has a distributive multitensor @xmath173 on @xmath141 . applying proposition([prop : pl - alg<->cat ] ) to @xmath173 , and corollary([cor : gamma - alg - ecat ] ) to @xmath122 , gives @xmath803 in view of the monadicity of @xmath804 . that is to say , @xmath173 is a distributive lift of @xmath122 . as for uniqueness suppose that @xmath142 is a distributive lift of @xmath122 . then by corollary([cor : gamma - alg - ecat ] ) and proposition([prop : pl - adjoint - char ] ) , @xmath805 is a distributive monad on @xmath144 and one has @xmath806 commuting with the forgetful functors into @xmath144 . by the fully - faithfulness of @xmath807 recalled above , one has an isomorphism @xmath808 of monads , and thus by proposition([prop : pl - adjoint - char ] ) an isomorphism @xmath809 of multitensors .    applying this result to any normalised @xmath145-operad @xmath1 , exhibits its algebras as categories enriched in the algebras of some @xmath0-operad . the @xmath0-operad is @xmath810 , and tensor product over which one enriches is @xmath811 . in cases where we already know what our tensor product ought to be , the uniqueness part of theorem([thm : lift - mult ] ) ensures that it is . an instance of this is    [ ex : gray ] in @xcite the normalised 3-operad @xmath23 whose algebras are gray categories was constructed . as we have already seen , @xmath812 is a lax monoidal structure on @xmath813 whose enriched categories are gray categories , and @xmath814 is the operad for strict 2-categories . note that the usual gray tensor product is symmetric monoidal closed and thus distributive . thus by theorem([thm : lift - mult ] ) @xmath815 is the gray tensor product . in other words , the general methods of this paper have succeeded in producing the gray tensor product of @xmath15-categories from the operad @xmath23 . more generally given a distributive tensor product @xmath700 on the category of algebras of an @xmath0-operad @xmath109 , and a normalised @xmath145-operad @xmath1 whose algebras are the categories enriched in @xmath109-algebras , theorem([thm : lift - mult ] ) exhibits @xmath700 as the more generally constructed @xmath811 . [ ex : crans ] in @xcite sjoerd crans explicitly constructed a tensor product on the category of gray - categories . this explicit construction was extremely complicated . it is possible to exhibit the crans tensor product as an instance of our general theory , by rewriting his explicit constructions as the construction of the 4-operad whose algebras are teisi in his sense . the multitensor @xmath122 associated to this 4-operad has @xmath123 equal to the 3-operad for gray categories . thus theorem([thm : lift - mult ] ) constructs a lax tensor product of gray categories whose enriched categories are teisi . since the tensor product explicitly constructed by crans is distributive , the uniqueness of part of theorem([thm : lift - mult ] ) ensures that it is indeed @xmath305 , since teisi are categories enriched in the crans tensor product by definition . honestly writing the details of the 4-operad of example([ex : crans ] ) is a formidable task and we have omitted this here . in the end though , such details will not be important , because such a tensor product ( or more properly a biclosed version thereof ) will only be really useful once it is given a conceptual definition . let @xmath16 be a symmetric monoidal model category which satisfies the conditions of @xcite or the monoid axiom of @xcite . in this case the category of pruned @xmath0-operads of @xcite can be equipped with a monoidal model structure @xcite . so we can speak of cofibrant @xmath0-operads in @xmath16 .    for @xmath273 let us fix a particular cofibrant and contractible @xmath33-operad @xmath1 . the algebras of @xmath816 can be called _ @xmath25-categories enriched in @xmath16_. up to homotopy the choice of @xmath816 is not important . so we can speak of _ the _ category of @xmath25-categories . for @xmath334 we denote by @xmath817 a cofibrant contractible @xmath15-operad in @xmath16 . let @xmath818 be the corresponding multitensor on @xmath18 . one can always choose @xmath817 in such a way that its unary part is @xmath816 . as in @xcite for an arbitary multitensor @xmath122 , one object @xmath122-categories are called @xmath122-monoids . similarly one object @xmath25-categories are called @xmath25-monoids . [ thm : a - infinity - app ]    1 . there is a distributive lift @xmath819 of @xmath109 to the category of @xmath25-categories . 2 .   @xmath819 restricts to give a multitensor @xmath346 on the category of @xmath25-monoids . the category of @xmath346-monoids is equivalent to the category of algebras of @xmath820 and therefore is quillen equivalent to the category of the algebras of the little squares operad . the first statement is a direct consequence of theorem([thm : lift - mult ] ) . the second statement follows from the fact that @xmath819 is the cartesian product on the object level . the last statement follows from the theorem(8.6 ) of @xcite . applying this result to the case @xmath821 with its folklore model structure one recovers    [ cor : js - braided ] [ joyal - street ] the category of braided monoidal categories is equivalent to the category of monoidal categories equipped with multiplication . the previous corollary is proved by joyal and street @xcite by a direct application of a `` categorified '' eckmann - hilton argument . the following analogous result for @xmath15-categories appears to be new . [ cor : coh - bm2c ] the category of braided monoidal @xmath15-categories is equivalent to the category of gray - monoids with multiplication . apply theorem([thm : a - infinity - app ] ) with @xmath822 equipped with the gray tensor product and lack s folklore model structure for 2-categories @xcite . thus theorem([thm : a - infinity - app ] ) should be considered as an @xmath823-generalisation of the above corollaries . we believe it sheds some light on the problem of defining the tensor product of @xmath25-algebras initiated by @xcite . as explained in the introduction , the negative result of @xcite shows that there is no hope to get an `` honest '' tensor product of such algebras . thus the multitensor @xmath346 constructed in theorem([thm : a - infinity - app ] ) is genuinely lax , and exhibits laxity as a way around the aforementioned negative result . in future work we will generalise this theorem to arbitrary dimensions . let us now instantiate the constructions of section([sec : dubuc ] ) to produce a more explicit description of the lifted multitensor @xmath305 . beyond mere instantiation this task amounts to reformulating everything in terms of hom maps which live in @xmath16 , because in our case the colimits being formed in @xmath18 at each stage of the construction are connected colimits diagrams whose morphisms are all identity on objects . moreover these fixed object sets are of the form @xmath154 for @xmath40 . +   + * notation*. we shall be manipulating sequences of data and so we describe here some notation that will be convenient . a sequence @xmath824 from some set will be denoted more tersely as @xmath825 leaving the length unmentioned . similarly a sequence of sequences @xmath826 of elements from some set will be denoted @xmath827  the variable @xmath46 ranges over @xmath164 and the variable @xmath134 ranges over @xmath132 . triply - nested sequences look like this @xmath828 , and so on . these conventions are more or less implicit already in the notation we have been using all along for multitensors . see especially section([ssec : lmc ] ) and @xcite . we denote by @xmath829 the ordinary sequence obtained from the @xmath244-tuply nested sequence @xmath830 by concatenation . in particular given a sequence @xmath825 , the set of @xmath827 such that @xmath831 is just the set of partitions of the original sequence into doubly - nested sequences , and will play an important role below . this is because to give the substitution maps for a multitensor @xmath122 on @xmath16 , is to give maps @xmath832 for all @xmath833 and @xmath834 from @xmath16 such that @xmath835 . +   + the monad map @xmath762 is taken as @xmath836 where @xmath837 is the inclusion of the unary part of the multitensor @xmath122 . note the notational abuse we regard write @xmath123 for the multitensor on @xmath16 obtained from @xmath122 by ignoring ( ie setting to constant at @xmath151 ) the non - unary parts , but also as the monad on @xmath16  and so @xmath838 as monads . the role of @xmath104 in @xmath839 is played by sequences @xmath840 of @xmath123-algebras regarded as objects of @xmath841 as in section([ssec : defnmonad ] ) . the transfinite induction produces for each ordinal @xmath227 and each sequence of @xmath123-algebras as above of length @xmath0 , morphisms @xmath842 in @xmath18 which are identities on objects , and thus we shall now evolve this notation so that it only records what s going on in the hom between @xmath26 and @xmath0 . by the definition of @xmath225 we have the equation on the left @xmath843 and the equation on the right is a definition . because of these definitions and that of @xmath225 we have the equation @xmath844 the data for the hom maps of the @xmath845 thus consists of morphisms @xmath846 in @xmath16 whenever one has @xmath847 as sequences of @xmath123-algebras . to summarise , the output of the transfinite process we are going to describe is , for each ordinal @xmath227 , the following data . for each sequence @xmath840 of @xmath123-algebras , one has an object @xmath848 and morphisms @xmath849 of @xmath16 where @xmath847 . +   + * initial step*. first we put @xmath850 , @xmath851 , and then form the coequaliser @xmath852 { \ope\limits_ix_i}="m " [ r ] { \opeone\limits_i(x_i , x_i)}="r " " l":@<2ex>"m"^-{\sigma } " l":"m"_-{\ope\limits_ix_i}:@<1ex>"r"^-{q^{(0)}_{(x_i , x_i)}}}\ ] ] in @xmath16 to define @xmath853 . put @xmath854 and @xmath855 . +   + * inductive step*. assuming that @xmath845 , @xmath856 and @xmath857 are given , we have maps @xmath858 { \ope\limits_i\opempone\limits_{jk}}="r " " l":"r"^-{\ope\limits_iv^{(m ) } } } } } & & { \xybox{\xygraph{!{0;(2,0 ) : } { \ope\limits_i\ope\limits_j\opem\limits_k}="l " [ r ] { \ope\limits_{ij}\opem\limits_k}="m " [ r ] { \ope\limits_{ij}\opempone\limits_k}="r " " l":"m"^-{{\sigma}\opem\limits_k}:"r"^-{q^{(m ) } } } } } \end{array}\ ] ] and these are used to provide the parallel maps in the coequaliser @xmath859 { \coprod\limits_{\con(x_{ij},x_{ij})=(x_i , x_i ) } \ope\limits_i\opempone\limits_j(x_{ij},x_{ij})}="m " [ d ] { \opemptwo\limits_i ( x_i , x_i)}="r " " l":@<-2ex>"m " " l":@<2ex>"m":"r"^-{(v^{(m{+}1)}_{(x_{ij},x_{ij})})}}\ ] ] which defines the @xmath860 , the commutative diagram @xmath861 { e_1\opempone\limits_i ( x_i , x_i)}="il " [ r(4 ) ] { \coprod\limits_{\con(x_{ij},x_{ij})=(x_i , x_i ) } \ope\limits_i\opempone\limits_j(x_{ij},x_{ij})}="ir " [ ru ] { \opemptwo\limits_i ( x_i , x_i)}="r " " l":"il"_-{u}:"ir":"r"^(.35){v^{(m{+}1)}_{(x_i , x_i)}}:@{<-}"l"_-{q^{(m{+}1)}_{(x_i , x_i)}}}\ ] ] in which the unlabelled map is the evident coproduct inclusion defines @xmath862 , and @xmath863 . +   + * limit step*. define @xmath864 as the colimit of the sequence given by the objects @xmath865 and morphisms @xmath866 for @xmath867 , and @xmath868 for the component of the universal cocone at @xmath869 . @xmath870 { \colsum\limits_{\con(x_{ij},x_{ij})=(x_i , x_i ) } \ope\limits_i\oper\limits_j(x_{ij},x_{ij})}="tm " [ r ] { \colim_{r{<}m } \oper\limits_i ( x_i , x_i)}="tr " [ d ] { \opem\limits_i ( x_i , x_i)}="br " [ l ] { \coprod\limits_{\con(x_{ij},x_{ij})=(x_i , x_i ) } \ope\limits_i\opem\limits_j(x_{ij},x_{ij})}="bm " [ l ] { \coprod\limits_{\con(x_{ijk},x_{ijk})=(x_i , x_i ) } \ope\limits_i\ope\limits_j\opem\limits_k(x_{ijk},x_{ijk})}="bl " " tl":@<1ex>"tm"^-{\sigma^{(<{m})}}:@<1ex>"tr"^-{v^{(<{m } ) } } " tl":@<-1ex>"tm"_-{(ev)^{(<{m})}}:@<-1ex>@{<-}"tr"_-{u^{(<{m } ) } } " bl":"bm"_-{\mu}:@{<-}"br"_-{uc } " tl":"bl"^{o_{m,2 } } " tm":"bm"^{o_{m,1 } } " tr":@{=}"br"}\ ] ] as before we write @xmath871 and @xmath872 for the obstruction maps , and @xmath231 denotes the evident coproduct injection . the maps @xmath873 , @xmath874 , @xmath875 and @xmath876 are by definition induced by @xmath877 , @xmath878 , @xmath879 and @xmath880 for @xmath867 respectively . define @xmath845 as the coequaliser of @xmath881 and @xmath882 , @xmath883 and @xmath884 . +   + instantiating corollary([cor : explicit - phi - shreik ] ) to the present situation gives    [ cor : lifted - obj ] let @xmath16 be a locally presentable category , @xmath392 a regular cardinal , and @xmath122 a distributive @xmath392-accessible multitensor on @xmath16 . then for any ordinal @xmath227 with @xmath885 one may take @xmath886 where the action @xmath887 is given as the composite @xmath888 { \opempone\limits_i ( x_i , x_i)}="m " [ r ] { \opem\limits_i ( x_i , x_i)}="r " " l":"m"^-{v^{(m)}}:"r"^-{(q^{(m)})^{-1}}}\ ] ] as an explicit description of the object map of the lifted multitensor @xmath305 on @xmath141 .    in corollaries ( [ cor : phi - shreik - simple ] ) and ( [ cor : vexp - simple ] ) , in which the initial data is a monad map @xmath762 between monads on a category @xmath16 together with an algebra @xmath104 for @xmath114 , we noted the simplification of our constructions when @xmath225 and @xmath889 preserve the coequaliser @xmath890 { sx}="m " [ r ] { q_1x}="r " " l":@<-1ex>"m"_-{sx } " l":@<1ex>"m"^-{\mu^ss(\phi)}:"r"^-{q_0}}\ ] ] in @xmath16 , which is part of the first step of the inductive construction of @xmath763 . in the present situation the role of @xmath16 is played by the category @xmath18 , the role of @xmath225 is played by @xmath891 , and the role of @xmath104 played by a given sequence @xmath840 of @xmath123-algebras , and so the role of the coequaliser ( [ eq : monad - coeq ] ) is now played by the coequaliser @xmath892 { \gamma e(x_i)}="m " [ r ] { q_1}="r " " l":@<-1ex>"m"_-{sx } " l":@<1ex>"m"^-{\mu^ss(\phi)}:"r"^-{q^{(0)}}}\ ] ] in @xmath18 . here we have denoted by @xmath893 the @xmath16-graph with objects @xmath154 and homs given by @xmath894 taking the hom of ( [ eq : monad - coeq2 ] ) between @xmath26 and @xmath0 gives the coequaliser @xmath895 { \ope\limits_ix_i}="m " [ r ] { \opeone\limits_i(x_i , x_i)}="r " " l":"m"_-{\ope\limits_ix_i } " l":@<2ex>"m"^-{\sigma}:@<1ex>"r"^-{q^{(0)}}}\ ] ] in @xmath16 which is part of the first step of the explicit inductive construction of @xmath305 . we shall refer to ( [ eq : mult - coeq ] ) as the _ basic coequaliser associated to the sequence @xmath840 _ of @xmath123-algebras . note that all coequalisers under discussion here are reflexive coequalisers , with the common section for the basic coequalisers given by the maps @xmath896 . the basic result which expresses why reflexive coequalisers are nice , is the @xmath897-lemma , which we record here for the reader s convenience . a proof can be found in @xcite . [ lem:3by3 ] * @xmath897-lemma*. given a diagram @xmath898 ^ -{f_1 } \ar@<-1ex>[r]_{g_1 } \ar@<1ex>[d]^{b_1 } \ar@<-1ex>[d]_{a_1 } & b \ar[r]^-{h_1 } \ar@<1ex>[d]^{b_2 } \ar@<-1ex>[d]_{a_2 } & c \ar@<1ex>[d]^{b_3 } \ar@<-1ex>[d]_{a_3 } \\ d \ar@<1ex>[r]^-{f_2 } \ar@<-1ex>[r]_-{g_2 } & e \ar[r]^-{h_2 } & f \ar[d]^{c } \\ & & h}\ ] ] in a category such that : ( 1 ) the two top rows and the right - most column are coequalisers , ( 2 ) @xmath899 and @xmath900 have a common section , ( 3 ) @xmath901 and @xmath902 have a common section , ( 3 ) @xmath903 , ( 4 ) @xmath904 , ( 5 ) @xmath905 and ( 6 ) @xmath906 ; then @xmath907 is a coequaliser of @xmath903 and @xmath904 .    if @xmath908 is a functor which preserves connected colimits of a certain type , then it also preserves these colimits in each variable separately , because for a connected colimit , a cocone involving only identity arrows is a universal cocone . the most basic corollary of the @xmath897-lemma says that the converse of this is true for reflexive coequalisers . [ cor:3by3 ] let @xmath908 be a functor . if @xmath292 preserves reflexive coequalisers in each variable separately then @xmath292 preserves reflexive coequalisers .    and this can be proved by induction on @xmath0 using the @xmath897-lemma in much the same way as @xcite lemma(1 ) . the most well - known instance of this is    [ cor:3by3 - 2]@xcite let @xmath239 be a biclosed monoidal category . then the @xmath0-fold tensor product of reflexive coequalisers in @xmath239 is again a reflexive coequaliser .    in particular note that by corollary([cor:3by3 ] ) a multitensor @xmath122 preserves ( some class of ) reflexive coequalisers iff it preserves them in each variable separately . returning to our basic coequalisers an immediate consequence of the explicit description of @xmath891 and corollary([cor:3by3 ] ) is    [ lem : reformulate - simplifying - conditions ] let @xmath122 be a distributive multitensor on @xmath16 a cocomplete category , and @xmath840 a sequence of @xmath123-algebras . if @xmath122 preserves the basic coequalisers associated to all the subsequences of @xmath840 , then for all @xmath909 , @xmath910 preserves the coequaliser ( [ eq : monad - coeq2 ] ) .    and applying this lemma and corollary([cor : phi - shreik - simple ] ) gives    [ cor : lifted - obj - simple ] let @xmath16 be a locally presentable category , @xmath392 a regular cardinal , @xmath122 a distributive @xmath392-accessible multitensor on @xmath16 and @xmath840 a sequence of @xmath123-algebras . if @xmath122 preserves the basic coequalisers associated to all the subsequences of @xmath840 , then one may take @xmath911 where the action @xmath29 is defined as the unique map such that @xmath912 . note in particular that when the sequence @xmath840 of @xmath123-algebras is of length @xmath332 or @xmath273 , the associated basic coequaliser is absolute . in the @xmath332 case the basic coequaliser is constant at @xmath913 , and when @xmath273 the basic coequaliser may be taken to be the canonical presentation of the given @xmath123-algebra . thus in these cases it follows from corollary([cor : lifted - obj - simple ] ) that @xmath914 and @xmath915 . reformulating the explicit description of the unit in corollary([cor : vexp - simple ] ) one recovers the fact from our explicit descriptions , that the unit of @xmath305 is the identity , which was of course true by construction .    to complete the task of giving a completely explicit description of the multitensor @xmath305 we now turn to unpacking its substitution . so we assume that @xmath122 is a distributive @xmath392-accessible multitensor on @xmath16 a locally presentable category , and fix an ordinal @xmath227 so that @xmath885 , so that @xmath305 may be constructed as @xmath916 as in corollary([cor : lifted - obj ] ) . by transfinite induction on @xmath917 we shall generate the following data : @xmath918 and @xmath919 whenever @xmath847 , such that @xmath920 { \operpone\limits_{ij}\opem\limits_k}="tr " [ d ] { \opem\limits_{ijk}}="br " [ l ] { \ope\limits_i\opem\limits_{jk}}="bl " " tl":"tr"^-{v^{(r)}e^{(m)}}:"br"^-{\sigma^{(r{+}1)}}:@{<-}"bl"^-{(q^{(m)})^{-1}v^{(m)}}:@{<-}"tl"^-{\ope\limits_i\sigma^{(r)}}}\ ] ] commutes . +   + * initial step*. define @xmath921 to be the identity and @xmath922 as the unique map such that @xmath923 by the universal property of the coequaliser @xmath853 . +   + * inductive step*. define @xmath924 as the unique map such that @xmath925 using the universal property of @xmath926 as a coequaliser . +   + * limit step*. when @xmath917 is a limit ordinal define @xmath927 as induced by the @xmath928 for @xmath929 and the universal property of @xmath930 as the colimit of the sequence of the @xmath931 for @xmath929 . then define @xmath932 as the unique map such that @xmath933 using the universal property of @xmath879 as a coequaliser . +   + the fact that the transfinite construction just specified was obtained from that for corollary([cor : induced - monad - very - explicit ] ) , by taking @xmath934 and looking at the homs , means that by corollaries ( [ cor : induced - monad - very - explicit ] ) and ( [ cor : vexp - simple ] ) one has    [ cor : induced - substitution - very - explicit ] let @xmath16 be a locally presentable category , @xmath392 a regular cardinal , @xmath122 a distributive @xmath392-accessible multitensor on @xmath16 and @xmath840 a sequence of @xmath123-algebras . then one has @xmath935 as an explicit description of the substitution of @xmath305 . if moreover @xmath122 preserves the basic coequalisers of all the subsequences of @xmath840 , then one may take @xmath936 as the explicit description of the substitution .      recall @xcite @xcite that when @xmath222 has eilenberg - moore objects , the one and 2-cells of the 2-category @xmath643 admit another description . given monads @xmath645 and @xmath937 in @xmath222 , and writing @xmath103 and @xmath938 for the one - cell data of their respective eilenberg - moore objects , to give a monad functor @xmath939 , is to give @xmath658 and @xmath940 such that @xmath941 . this follows immediately from the universal property of eilenberg - moore objects . similarly to give a monad 2-cell @xmath942 is to give @xmath943 and @xmath944 commuting with @xmath728 and @xmath945 . note that eilenberg - moore objects in @xmath58 are computed as in @xmath99 , and we shall soon apply these observations to the case @xmath946 . suppose we have a lax monoidal functor @xmath655 , that @xmath16 and @xmath196 are locally presentable , and that @xmath122 and @xmath292 are accessible . then we obtain a commutative diagram @xmath947 { \ca gv^{e_1}}="tm " [ r ] { \ca gv}="tr " [ d ] { \ca gw}="br " [ l ] { \ca gw^{f_1}}="bm " [ l ] { \enrich f}="bl " " tl":"tm":"tr " " bl":"bm":"br " " tl":"bl " " tm":"bm " " tr":"br"}\ ] ] of forgetful functors in @xmath58 . applying the previous paragraph to the left - most square gives a monad morphism @xmath948 , and then applying @xmath405 to this gives the lax monoidal functor @xmath949 between the induced lifted multitensors . arguing similarly for monoidal transformations and monad 2-cells , one finds that the assignment @xmath950 is 2-functorial . let @xmath951 be an @xmath0-multitensor . in terms of the previous paragraph , this is the special case @xmath952 , @xmath953 , @xmath954 . the lifted multitensor corresponding to @xmath955 is just cartesian product for strict @xmath0-categories . one has a component of @xmath956 for each sequence @xmath957 of strict @xmath0-categories , and since @xmath958 as a right adjoint preserves products , this component may be regarded as a map @xmath959 of @xmath123-algebras . if in particular @xmath123 is itself @xmath9 and @xmath960 , then these components of @xmath956 give a canonical comparison from the lifted tensor product @xmath305 of @xmath0-categories to the cartesian product . for instance , when @xmath122 is the multitensor corresponding to the 3-operad for gray categories , then @xmath956 gives the well - known comparison map from the gray tensor product of 2-categories to the cartesian product , which we recall is actually a componentwise biequivalence . returning to the general situation , it is routine to unpack the assignment @xmath961 as in section([ssec : explicit - lifting ] ) and so obtain the following 1-cell counterpart of corollary([cor : lifted - obj - simple ] ) . [ cor : free - lift-1cell ] let @xmath655 be a lax monoidal functor such that @xmath16 and @xmath196 are locally presentable , and @xmath122 and @xmath292 are accessible . let @xmath554 be a sequence of objects of @xmath16 . then the component of @xmath962 at the sequence @xmath963 of free @xmath123-algebras is just @xmath964 . let @xmath16 be a category and @xmath965 a class of maps in @xmath16 . denote by @xmath966 the class of maps in @xmath16 that have the right lifting property with respect to all the maps in @xmath965 . that is to say , @xmath48 is in @xmath966 iff for every @xmath967 in @xmath965 , @xmath542 and @xmath968 such that the outside of @xmath969 { x}="tr " [ d ] { y}="br " [ l ] { b}="bl " " tl " ( : " tr"^-{\alpha}:"br"^{f},:"bl"_{i}(:"br"_-{\beta},:@{.>}"tr"|{\gamma}))}\ ] ] commutes , then there is a @xmath970 as indicated such that @xmath971 and @xmath972 . an @xmath973 is called a _ trivial @xmath965-fibration_. the basic facts about @xmath966 that we shall use are summarised in    [ lem : basic - tf ] let @xmath16 be a category , @xmath965 a class of maps in @xmath16 , @xmath210 a set and @xmath974 a family of maps in @xmath16 .    1 . @xmath966 is closed under composition and retracts.[tfib1 ] 2 . if @xmath16 has products and each of the @xmath975 is a trivial @xmath965-fibration , then @xmath976 is also a trivial @xmath965-fibration.[tfib2 ] 3 . the pullback of a trivial @xmath965-fibration along any map is a trivial @xmath965-fibration.[tfib3 ] 4 . if @xmath16 is extensive and @xmath977 is a trivial @xmath965-fibration , then each of the @xmath975 is a trivial fibration.[tfib4 ] 5 . if @xmath16 is extensive , the codomains of maps in @xmath965 are connected and each of the @xmath975 is a trivial @xmath965-fibration , then @xmath977 is a trivial @xmath965-fibration.[tfib5 ]    ( [ tfib1])-([tfib3 ] ) is standard . if @xmath16 is extensive then the squares @xmath978 { \coprod_jx_j}="tr " [ d ] { \coprod_jy_j}="br " [ l ] { y_j}="bl " " tl " ( : " tr":"br"^{\coprod_jf_j},:"bl"_{f_j}:"br")}\ ] ] whose horizontal arrows are the coproduct injections are pullbacks , and so ( [ tfib4 ] ) follows by the pullback stability of trivial @xmath965-fibrations . as for ( [ tfib5 ] ) note that for @xmath967 in @xmath965 , the connectedness of @xmath109 ensures that any square as indicated on the left @xmath979 { \coprod_jx_j}="tr " [ d ] { \coprod_jy_j}="br " [ l ] { b}="bl " " tl " ( : " tr":"br"^{\coprod_jf_j},:"bl"_{i}:"br " ) } } } [ r(4 ) ] { \xybox{\xygraph{!{0;(1.5,0):(0,.666 ) : : } { s}="tl " [ r ] { x_j}="tr " [ d ] { y_j}="br " [ l ] { b}="bl " " tl " ( : " tr":"br"^{f_j},:"bl"_{i}:"br")}}}}\ ] ] factors through a unique component as indicated on the right , enabling one to induce the desired filler . let @xmath980 be functors and @xmath965 be a class of maps in @xmath16 . a natural transformation @xmath981 is a _ trivial @xmath965-fibration _ when its components are trivial @xmath965-fibrations .    note that since trivial @xmath965-fibrations in @xmath16 are pullback stable , this reduces , in the case where @xmath196 has a terminal object @xmath33 and @xmath522 is cartesian , to the map @xmath982 being a trivial @xmath965-fibration .    given a category @xmath16 with an initial object , and a class of maps @xmath965 in @xmath16 , we denote by @xmath983 the class of maps in @xmath18 containing the maps @xmath984 where @xmath985 . the proof of the following lemma is trivial . [ lem : ind - tf ] let @xmath16 be a category with an initial object and @xmath965 a class of maps in @xmath16 . then @xmath48 is a trivial @xmath983-fibration iff it is surjective on objects and all its hom maps are trivial @xmath965-fibrations . in particular starting with @xmath986 the category of globular sets and @xmath987 the empty class of maps , one generates a sequence of classes of maps @xmath988 of globular sets by induction on @xmath0 by the formula @xmath989 since @xmath990 may be identified with @xmath991 , and moreover one has inclusions @xmath992 . more explicitly , the set @xmath988 consists of @xmath145 maps : for @xmath993 one has the inclusion @xmath994 , where @xmath244 here denotes the representable globular set , that is the `` @xmath244-globe '' , and @xmath995 is the @xmath244-globe with its unique @xmath244-cell removed . one defines @xmath996 to be the union of the @xmath988 s . note that by definition @xmath997 . there is another version of the induction just described to produce , for each @xmath40 , a class @xmath998 of maps of @xmath999 . the set @xmath1000 consists of the functions @xmath1001 so @xmath1002 is the class of bijective functions . for @xmath40 , @xmath1003 . as maps of globular sets , the class @xmath1004 consists of all the maps of @xmath1005 together with the unique map @xmath1006 . let @xmath1007 . an @xmath0-operad @xmath1008 is _ contractible _ when it is a trivial @xmath998-fibration . an @xmath0-multitensor @xmath1009 is _ contractible _ when it is a trivial @xmath998-fibration .    by the preceeding two lemmas , an @xmath145-operad @xmath1010 over @xmath24 is contractible iff the hom maps of @xmath1011 are trivial @xmath1004-fibrations .      as one would expect an @xmath145-operad over @xmath24 is contractible iff its associated @xmath0-multitensor is contractible . this fact has quite a general explanation . recall the 2-functoriality of @xmath22 described in section([ssec : gamma-2-functor ] ) and that of the lifting described in section([ssec : functoriality - lifting ] ) . [ prop : contractible ] let @xmath655 be a lax monoidal functor between distributive lax monoidal categories , and @xmath965 a class of maps in @xmath196 . suppose that @xmath196 is extensive , @xmath658 preserves coproducts and the codomains of maps in @xmath965 are connected . then the following statements are equivalent    * @xmath653 is a trivial @xmath965-fibration . * @xmath1012 is a trivial @xmath983-fibration . and moreover when in addition @xmath16 and @xmath196 are locally presentable and @xmath122 and @xmath292 are accessible , these conditions are also equivalent to    * the components of @xmath1013 at sequences @xmath1014 of free @xmath123-algebras are trivial @xmath965-fibrations .    for each @xmath127 the component @xmath1015 is the identity on objects and for @xmath261 , the corresponding hom map is obtained as the composite of @xmath1016 and the canonical isomorphism that witnesses the fact that @xmath658 preserves coproducts . in particular note that for any sequence @xmath112 of objects of @xmath16 , regarded as @xmath16-graph in the usual way , one has @xmath1017 thus @xmath1018 follows from lemmas([lem : basic - tf ] ) and ( [ lem : ind - tf ] ) . @xmath1019 follows immediately from corollary([cor : free - lift-1cell ] ) . [ cor : contractible ] let @xmath1007 , @xmath1010 be an @xmath1020-operad over @xmath24 and @xmath951 be the corresponding @xmath0-multitensor . tfsae :    1 . @xmath1010 is contractible . @xmath951 is contractible . the components of @xmath1021 of section([ssec : functoriality - lifting])([eq : mult->prod ] ) are trivial @xmath998-fibrations of @xmath0-globular sets , when the @xmath840 are free strict @xmath0-categories .    by induction one may easily establish that the codomains of the maps in any of the classes : @xmath988 , @xmath998 , @xmath996 are connected so that proposition([prop : contractible ] ) may be applied . [ ex : gray - contractible ] applying this last result to the 3-operad @xmath23 for gray - categories , the contractibility of @xmath23 is a consequence of the fact that the canonical 2-functors from the gray to the cartesian tensor product are identity - on - object biequivalences .      continuing the discussion from section([ssec : tci ] ) , we now explain why the operads which describe trimble @xmath0-categories " are contractible . this result appears in @xcite as theorem(4.8 ) and exhibits trimble @xmath0-categories as weak @xmath0-categories in the batanin sense . let us denote by @xmath1022 the set of inclusions @xmath1023 of the @xmath0-sphere into the @xmath0 disk for @xmath40 . as we remarked in section([ssec : enriched - graphs ] ) these may all be obtained by successively applying the reduced suspension functor @xmath126 to the inclusion of the empty space into the point . as we recalled in section([ssec : tci ] ) , a basic ingredient of the trimble definition is a version of the little intervals operad which acts on the path spaces of any space . a key property of this operad is that it is contractible  a topological operad @xmath1 being contractible when for each @xmath0 the unique map @xmath1024 is in @xmath1025 . this is equivalent to saying that the cartesian multitensor map @xmath1026 is a trivial @xmath1022-fibration . a useful fact about the class trivial @xmath1022-fibrations is that it gets along with the construction of path - spaces in the sense of    [ lem : tf - path - spaces ] if @xmath48 is a trivial @xmath1022-fibration then so is @xmath50 for all @xmath1027 .    to give a commutative square as on the left in @xmath1028 { x(a , b)}="tr " [ d ] { y(fa , fb)}="br " [ l ] { d^n}="bl " " tl"(:"tr":"br",:"bl":"br " ) } } } & & & { \xybox{\xygraph{!{0;(2,0):(0,.5 ) : : } { s^n}="tl " [ r ] { ( a , x , b)}="tr " [ d ] { ( fa , y , fb)}="br " [ l ] { d^{n{+}1}}="bl " " tl"(:"tr":"br",:"bl":"br " ) } } } \end{array}\ ] ] is the same as giving a commutative square in @xmath37 as on the right in the previous display , by @xmath43 . the square on the right admits a diagonal filler @xmath1029 since @xmath76 is a trivial @xmath1022-fibration , and thus so does the square on the left . we shall write @xmath1030 for the forgetful functor for each @xmath0 . the relationship between trivial fibrations of spaces and of globular sets is expressed in    [ prop : pres - tf ] if @xmath48 is a trivial @xmath1022-fibration then @xmath1031 is a trivial @xmath1032-fibration . we proceed by induction on @xmath0 . having the right lifting property with respect to the inclusions @xmath1033 ensures that @xmath76 surjective and injective on path components , and thus is inverted by @xmath1034 . for the inductive step we assume that @xmath1035 sends trivial @xmath1022-fibrations to trivial @xmath998-fibrations and suppose that @xmath76 is a trivial @xmath1022-fibration . then so are all the maps it induces between path spaces by lemma([lem : tf - path - spaces ] ) . but from the inductive definition of @xmath1036 recalled in section([ssec : tci ] ) , we have @xmath1037 and so @xmath1038 is a morphism of @xmath11-globular sets which is surjective on objects ( as argued already in the @xmath332 case ) and whose hom maps are trivial @xmath998-fibrations by induction . thus the result follows by lemma([lem : ind - tf ] ) .    in section([ssec : tci ] ) we exhibited @xmath760 as the algebras of an @xmath0-operad by a straight - forward application of two abstract results propositions([prop : monad - trimble ] ) and ( [ prop : operad - trimble ] ) . we now provide a third such result relating to contractibility . [ prop : tf - formal ] given the data and hypotheses of proposition([prop : operad - trimble ] ) : @xmath16 is a lextensive category , @xmath101 a cartesian and coproduct preserving monad on @xmath16 , @xmath733 a @xmath101-operad and @xmath726 a non - symmetric operad in @xmath734 . suppose furthermore that a class @xmath965 of maps of @xmath16 is given , and that the non - symmetric operad @xmath1039 and the @xmath101-operad @xmath653 are trivial @xmath965 fibrations . then the @xmath736-operad @xmath1040 of proposition([prop : operad - trimble ] ) is a trivial @xmath1041-fibration .    by definition this monad morphism may be written as the composite @xmath1042 . since @xmath653 is a trivial @xmath965-fibration so is @xmath1043 by lemma([lem : basic - tf ] ) , and thus @xmath1044 is a trivial @xmath1041-fibration by proposition([prop : contractible ] ) . since @xmath542 is a trivial @xmath965-fibration , @xmath1045 is a trivial @xmath1041-fibration again by proposition([prop : contractible ] ) , and so the result follows since trivial fibrations compose .    starting with a contractible topological operad @xmath1 which acts on path spaces , proposition([prop : pres - tf ] ) ensures that @xmath1046 will be a contractible non - symmetric operad of @xmath0-globular sets . then proposition([prop : tf - formal ] ) may be applied to give , by induction on @xmath0 , the contractibility of the @xmath0-operad defining trimble @xmath0-categories . we would like to acknowledge clemens berger , richard garner , andr joyal , steve lack , joachim kock , jean - louis loday , paul - andr mellis , ross street and dima tamarkin for interesting discussions on the substance of this paper . we would also like to acknowledge the centre de recerca matemtica in barcelona for the generous hospitality and stimulating environment provided during the thematic year 2007 - 2008 on homotopy structures in geometry and algebra . the first author would like to acknowledge the financial support on different stages of this project of the scott russell johnson memorial foundation , the australian research council ( grant no.dp0558372 ) and luniversit paris 13 . the second author would like to acknowledge the support of the anr grant no . the third author would like to acknowledge the laboratory pps ( preuves programmes systmes ) in paris , the max planck institute in bonn and the institut des hautes tudes scientifique in bures sur yvette for the excellent working conditions he enjoyed during this project , as well as macquarie university for the hospitality he enjoyed in august of 2008 . 10    j.  adamek , f.  borceux , s.  lack , and j.  rosicky . a classification of accessible categories . , 175:730 , 2002 . j.  adamek and j.  rosicky . . number 189 in london math soc . lecture notes . cambridge university press , 1994 .    m.  barr and c.  wells . . number  12 in tac reprints . theory and applications of categories , 2005 . m.  batanin . monoidal globular categories as a natural environment for the theory of weak @xmath0-categories . , 136:39103 , 1998 . m.  batanin . symmetrisation of @xmath0-operads and compactification of real configuration spaces . , 211:684725 , 2007 . m.  batanin , c.  berger , and d - 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groupoide non - strictes via des ensembles multi - simpliciaux . , 16:5199 , 1999 .    m.  weber . generic morphisms , parametric representations , and weakly cartesian monads . , 13:191234 , 2004 . m.  weber . familial 2-functors and parametric right adjoints . , 18:665732 , 2007 . m.  weber . yoneda structures from 2-toposes . , 15:259323 , 2007 . m.  weber . free products of higher operad algebras . arxiv:0909.4722 , 2009 . in these appendices we review some of the transfinite constructions in monad theory that we used in section([sec : lift - mult ] ) . an earlier reference for these matters is @xcite . however due to the technical nature of this material , and our need for its details when we come to making our constructions explicit , we feel that it is appropriate to give a rather thorough account of this background . let @xmath101 be a monad on a category @xmath16 that has filtered colimits and coequalisers and let @xmath1047 { ( b , b)}="r " " l":@<-1ex>"r"_-{g } " l":@<1ex>"r"^-{f}}\ ] ] be morphisms in @xmath102 . we shall now construct morphisms @xmath1048 starting with @xmath1049 by transfinite induction on @xmath0 , such that for @xmath0 large enough @xmath1050 is the coequaliser of @xmath76 and @xmath345 in @xmath102 when @xmath101 is accessible . the initial stages of this construction are described in the following diagram . @xmath1051 { t^2b}="tb " [ r ] { t^2q_1}="t0 " [ r ] { t^2q_2}="t1 " [ r ] { t^2q_3}="t2 " [ r ] { t^2q_4}="t3 " [ r ] { ... } = " t4 " " ta " [ d ] { ta}="ma " [ r ] { tb}="mb " [ r ] { tq_1}="m0 " [ r ] { tq_2}="m1 " [ r ] { tq_3}="m2 " [ r ] { tq_4}="m3 " [ r ] { ... } = " m4 " " ma " [ d ] { a}="ba " [ r ] { b}="bb " [ r ] { q_1}="b0 " [ r ] { q_2}="b1 " [ r ] { q_3}="b2 " [ r ] { q_4}="b3 " [ r ] { ... } = " b4 " " ta":@<-1ex>"tb"_-{t^2 g } " ta":@<1ex>"tb"^-{t^2f}:"t0"^-{t^2q_0}:"t1"^-{t^2q_1}:"t2"^-{t^2q_2}:"t3"^-{t^2q_3}:"t4 " " ma":@<-1ex>"mb"_-{tg } " ma":@<1ex>"mb"^-{tf}:"m0"^-{tq_0}:"m1"^-{tq_1}:"m2"^-{tq_2}:"m3"^-{tq_3}:"m4 " " ba":@<-1ex>"bb"_-{g } " ba":@<1ex>"bb"^-{f}:"b0"_-{q_0}:"b1"_-{q_1}:"b2"_-{q_2}:"b3"_-{q_3}:"b4 " " ta":@<-1ex>"ma"_-{\mu}:@<-1ex>"ba"_-{a } " ta":@<1ex>"ma"^-{ta}:@<1ex>@{<-}"ba"^-{\eta } " tb":@<-1ex>"mb"_-{\mu}:@<-1ex>"bb"_-{b } " tb":@<1ex>"mb"^-{tb}:@<1ex>@{<-}"bb"^-{\eta } " t0":"m0"^-{\mu}:@{<-}"b0"^-{\eta } " t1":"m1"^-{\mu}:@{<-}"b1"^-{\eta } " t2":"m2"^-{\mu}:@{<-}"b2"^-{\eta } " t3":"m3"^-{\mu}:@{<-}"b3"^-{\eta } " tb":"m0"^-{tv_0 } " mb":"b0"^-{v_0 } " t0":"m1"^-{tv_1 } " m0":"b1"^-{v_1 } " t1":"m2"^-{tv_2 } " m1":"b2"^-{v_2 } " t2":"m3"^-{tv_3 } " m2":"b3"^-{v_3}}\ ] ] * initial step*. define @xmath1052 to be the identity , @xmath1053 to be the coequaliser of @xmath76 and @xmath345 , @xmath1054 and @xmath1055 . note also that @xmath1056 . +   + * inductive step*. assuming that @xmath1057 , @xmath1058 and @xmath1059 are given , we define @xmath1060 to be the coequaliser of @xmath1061 and @xmath1062 , @xmath1063 and @xmath1064 . one may easily verify that @xmath1065 , and that @xmath1066 could equally well have been defined as the coequaliser of @xmath1067 and @xmath1068 . +   + * limit step*. define @xmath1069 as the colimit of the sequence given by the objects @xmath1070 and morphisms @xmath1071 for @xmath1072 , and @xmath1050 for the component of the universal cocone at @xmath1073 . @xmath1074 { \colim_{m{<}n } tq_m}="tm " [ r ] { \colim_{m{<}n } q_m}="tr " [ d ] { q_n}="br " [ l ] { tq_n}="bm " [ l ] { t^2q_n}="bl " " tl":@<1ex>"tm"^-{\mu_{<{n}}}:@<1ex>"tr"^-{v_{<{n } } } " tl":@<-1ex>"tm"_-{(tv)_{<{n}}}:@<-1ex>@{<-}"tr"_-{\eta_{<{n } } } " bl":"bm"_-{\mu}:@{<-}"br"_-{\eta } " tl":"bl"_{o_{n,2 } } " tm":"bm"^{o_{n,1 } } " tr":@{=}"br"}\ ] ] we write @xmath1075 and @xmath1076 for the obstruction maps measuring the extent to which @xmath101 and @xmath1077 preserve the colimit defining @xmath1069 . we write @xmath1078 , @xmath1079 , @xmath1080 and @xmath1081 for the maps induced by the @xmath1082 , @xmath1083 , @xmath1084 and @xmath1085 for @xmath1072 respectively . the equations @xmath1086 follow easily from the definitions . define @xmath1057 as the coequaliser of @xmath1087 and @xmath1088 , @xmath1089 and @xmath1090 . +   + * stabilisation*. we say that the sequence _ stabilises at @xmath0 _ when @xmath1058 and @xmath1091 are isomorphisms . in the case @xmath332 one may easily show that stabilisation is equivalent to just @xmath1053 being an isomorphism , which is the same as saying that @xmath1092 . [ stable - limit ] if @xmath0 is a limit ordinal and @xmath1075 and @xmath1076 are invertible , then the sequence stabilises at @xmath0 . let us write @xmath1093 , @xmath1094 for the colimit cocones . first we contemplate the diagram @xmath1074 { \colim_{m{<}n } tq_m}="tm " [ r ] { \colim_{m{<}n } q_m}="tr " [ d ] { q_n}="br " [ l ] { tq_n}="bm " [ l ] { t^2q_n}="bl " [ d ] { t^2q_{n{+}1}}="bbl " [ r ] { tq_{n{+}1}}="bbm " [ r ] { q_{n{+}1}}="bbr " [ d ] { q_{n{+}2}}="bbbr " " tl":@<1ex>"tm"^-{\mu_{<{n}}}:@<1ex>"tr"^-{v_{<{n } } } " tl":@<-1ex>"tm"_-{(tv)_{<{n}}}:@<-1ex>@{<-}"tr"_-{\eta_{<{n } } } " bl":"bm"_-{\mu}:@{<-}"br"_-{\eta } " bbl":"bbm"_-{\mu}:@{<-}"bbr"_-{\eta } " tl":"bl"_{o_{n,2 } } " tm":"bm"^{o_{n,1 } } " tr":@{=}"br " " bl":"bbl"_{t^2q_n } " bm":"bbm"^{tq_n } " br":"bbr"^{q_n}:"bbbr"^{q_{n{+}1 } } " bl":"bbm"_-{tv_n}:"bbbr"_-{v_{n{+}1 } } " bm":"bbr"_-{v_n}}\ ] ] and in general one has @xmath1095 to prove this note that from the definitions of @xmath1071 and @xmath1058 and the naturality of the @xmath1096 in @xmath227 , one may show easily that @xmath1097 , and from this last equation and all the definitions it is easy to show that @xmath1098 for all @xmath1072 from which ( [ eq : succ - limit ] ) follows . suppose that @xmath1075 and @xmath1076 are isomorphisms . then define @xmath1099 as the unique map such that @xmath1100 . it follows easily that @xmath1101 . from ( [ eq : succ - limit ] ) and the invertibility of @xmath1076 it follows easily that @xmath1102 and so there is a unique @xmath1103 such that @xmath1104 , from which it follows easily that @xmath1105 . [ really - stable ] if the sequence stabilises at @xmath0 then it stabilises at any @xmath1106 , and moreover one has an isomorphism of sequences between the given sequence @xmath1107 and the following one : @xmath1108 { ... } = " p2 " [ r ] { q_n}="p3 " [ r ] { q_n}="p4 " [ r ] { ... } = " p5 " " p1":"p2"^-{q_0}:"p3":"p4"^-{\id}:"p5"^-{\id}}\ ] ]    we show for @xmath1106 that @xmath1071 and @xmath1109 are isomorphisms , and provide the component isomorphisms @xmath1110 of the required isomorphism of sequences , by transfinite induction on @xmath227 . we define @xmath1111 to be the identity when @xmath1112 . in the initial step @xmath1113 , @xmath1071 and @xmath1109 are isomorphisms by hypothesis and we define @xmath1114 . in the inductive step when @xmath1106 is a non - limit ordinal , we must show that @xmath1115 is an isomorphism and define @xmath1116 . the key point is that @xmath1117 because with this equation in hand one defines @xmath1118 as the unique morphism satisfying @xmath1119 using the universal property of @xmath1120 , and then it is routine to verify that @xmath1121 . so for the inductive step it remains to verify ( [ eq : key ] ) . but we have @xmath1122 and so ( [ eq : key ] ) follows since @xmath1071 is an isomorphism . in the case where @xmath227 is a limit ordinal , we have stabilisation at @xmath1123 established whenever @xmath1124 by the induction hypothesis . thus the colimit defining @xmath1070 is absolute ( ie preserved by all functors ) since its defining sequence from the position @xmath0 onwards consists only of isomorphisms . thus @xmath1071 and @xmath1109 are isomorphisms by lemma([stable - limit ] ) . by induction , the previously constructed @xmath1125 s provide a cocone on the defining diagram of @xmath1070 with vertex @xmath1069 , thus one induces the isomorphism @xmath1111 compatible with the earlier @xmath1125 s and defines @xmath1126 . [ coeq - when - stable ] if the sequence stabilises at @xmath0 then @xmath1127 is a @xmath101-algebra and @xmath1128 is the coequaliser of @xmath76 and @xmath345 in @xmath102 . the unit law for @xmath1127 is immediate from the definition of @xmath1058 and the associative law is the commutativity of the outside of the diagram on the left @xmath1129 { tq_n}="tr " [ d ] { q_{n{+}1}}="mr " [ d ] { q_n}="br " [ l ] { q_{n{+}1}}="bm " [ l ] { tq_n}="bl " [ u ] { tq_{n{+}1}}="ml " " bm " [ u(.75 ) ] { q_{n{+}2}}="mm " [ u(.75 ) ] { tq_{n{+}1}}="tm " " tl":"tr"^-{\mu}:"mr"^-{v_n}:"br"^-{q_n^{-1}}:@{<-}"bm"^-{q_n^{-1}}:@{<-}"bl"^-{v_n}:@{<-}"ml"^-{tq_n^{-1}}:@{<-}"tl"^-{tv_n } " tr":"tm"^-{tq_n}:"mm"_-{v_{n{+}1}}:"bm"_-{q_{n{+}1}^{-1 } } " ml":"mm"_-{v_{n{+}1}}:"mr"^-{q_{n{+}1}^{-1 } } } } } [ r(5)u(.1 ) ] { \xybox{\xygraph{{tb}="tl " [ r(2 ) ] { tq_n}="tr " [ d ] { q_{n{+}1}}="m " [ d ] { q_n}="br " [ l(2 ) ] { b}="bl " " tl":"tr"^-{tq_{{<}n}}:"m"^-{v_n}:"br"^-{q_n^{-1}}:@{<-}"bl"^-{q_{{<}n}}(:@{<-}"tl"^-{b},:"m"^-{q_{{<}n{+}1 } } ) " m":@{}"tl"|{(i)})}}}}\ ] ] the regions of which evidently commute . the commutativity of the outside diagram on the right exhibits @xmath1130 as a @xmath101-algebra map , and this follows immediately from the commutativity of the region labelled ( i ) .    the equational form of ( i ) says @xmath1131 and we now proceed to prove this by transfinite induction on @xmath0 . the case @xmath332 is just the statement @xmath1055 . the inductive step comes out of the calculation @xmath1132 and since @xmath1133 . the case where @xmath0 is a limit ordinal is the commutativity of the outside of @xmath1134 { b}="b " [ r(3 ) ] { q_{n{+}1}}="qnp1",[dr ] { tb}="tb2 " [ r(3 ) ] { \col \ , tq_m}="ctq " [ ur ] { tq_n}="tqn " ) " tb"(:"b"^-{b}:"qnp1"^-{q_{{<}n{+}1}},:"tb2"_-{\id}|-{}="mla":@/_{1pc}/"ctq"_-{q'_{0,n}}|-{}="mma":"tqn"_-{o_{n,1}}|-{}="mra":"qnp1"_-{v_n } " b":@{}"ctq"|*{tb}="m " ) " m " ( : @{}"b"|(.6)*{q_n}="ptl",:@{}"qnp1"|(.6)*{tq_n}="ptr",:@{}"mra"|(.6)*{\col \ , tq_m}="pmr",:@{}"mma"|(.6)*{\col \ , t^2q_m}="pb",:@{}"mla"|(.6)*{t^2b}="pml " ) " b":"ptl"_(.6){q_{{<}n}}:"ptr"^-{\eta}:"qnp1"^(.4){v_n } " tb":"pml"^-{\eta}:"m"^-{tb}(:"ptr"^(.4){tq_{{<}n}},:"pmr"^-{q'_{0,n } } ) " pml":"pb"^-{q''_{0,n}}:"pmr"^-{(tv)_{{<}n}}:"ptr"^-{o_{n,1 } } " pml":"tb2"^-{\mu } " pb":"ctq"^-{\mu_{{<}n}}}\ ] ] the regions of which evidently commute . thus @xmath1130 is indeed a @xmath101-algebra map .    to see that @xmath1130 is a coequaliser let @xmath1135 such that @xmath1136 . for each ordinal @xmath227 we construct @xmath1137 such that @xmath1138 for all @xmath227 by transfinite induction on @xmath227 . when @xmath1073 we define @xmath1139 and @xmath1140 as unique such that @xmath1141 . the equation @xmath1142 is easily verified . for the inductive step we note that the commutativity of @xmath1143 { tq_n}="tl " [ r(2 ) ] { tc}="tr " [ dr ] { c}="mr " [ dl ] { tc}="br " [ l(2 ) ] { tq_{n{+}1}}="bl " [ ur ] { t^2c}="m " " ml":"tl"^-{\mu}:"tr"^-{th_n}:"mr"^-{c}:@{<-}"br"^-{c}:@{<-}"bl"^-{th_{n{+}1}}:@{<-}"ml"^-{tv_n } " m"(:@{<-}"ml"_-{t^2h_n},:"tr"^-{\mu},:"br"^-{tc})}\ ] ] and the universal property of @xmath1144 ensures there is a unique @xmath1145 such that @xmath1146 . when @xmath227 is a limit ordinal it follows from all the definitions that @xmath1147 for all @xmath1148 , and so @xmath1149 and so by the universal property of @xmath1144 there is a unique @xmath1150 such that @xmath1138 . the sequence of @xmath1151 s just constructed is clearly unique such that @xmath1139 and @xmath1138 . it follows immediately that @xmath1152 is a @xmath101-algebra map , and that @xmath1153 . conversely given @xmath1154 such that @xmath1155 , one constructs @xmath1156 as @xmath1157 , and it follows easily that @xmath1158 , @xmath1159 and @xmath1160 whence @xmath1161 and so @xmath1162 .    from these results we recover the usual theorem on the construction of coequalisers of algebras of accessible monads . [ thm : coeq - talg ] let @xmath16 be a category with filtered colimits and coequalisers , @xmath101 be a monad on @xmath16 and @xmath1047 { ( b , b)}="r " " l":@<-1ex>"r"_-{g } " l":@<1ex>"r"^-{f}}\ ] ] be morphisms in @xmath102 . if @xmath101 is @xmath392-accessible for some regular cardinal @xmath392 , then @xmath1050 as constructed above is the coequaliser of @xmath76 and @xmath345 in @xmath102 for any ordinal @xmath0 such that @xmath1163 . take the smallest such ordinal @xmath0  it is necessarily a limit ordinal , and @xmath101 and @xmath1077 by hypothesis preserve the defining colimit of @xmath1069 . thus by lemmas([stable - limit ] ) and ( [ coeq - when - stable ] ) the result follows in this case , and in general by lemmas([really - stable ] ) and ( [ coeq - when - stable ] ) . finally we mention the well - known special case when the above transfinite construction is particularly simple , that will be worth remembering . [ prop : simple - coeq ] let @xmath16 be a category with filtered colimits and coequalisers , @xmath101 be a monad on @xmath16 and @xmath1047 { ( b , b)}="r " " l":@<-1ex>"r"_-{g } " l":@<1ex>"r"^-{f}}\ ] ] be morphisms in @xmath102 . if @xmath101 and @xmath1077 preserve the coequaliser of @xmath76 and @xmath345 in @xmath16 , then the sequence @xmath1164 stabilises at @xmath33 . denoting by @xmath1165 the unique map such that @xmath1166 , @xmath1167 is the coequaliser of @xmath76 and @xmath345 in @xmath102 . refer to the diagram in @xmath16 above that describes the first few steps of the construction of @xmath1164 . since @xmath1053 and @xmath1168 are epimorphisms , the @xmath101-algebra axioms for @xmath1169 follow from those for @xmath1170 , and @xmath1053 is a @xmath101-algebra map by definition . thus @xmath287 is the coequaliser in @xmath16 of @xmath1171 and @xmath1172 , and since @xmath1168 is an epimorphism it is also the coequaliser of @xmath1173 and @xmath1174 , but so is @xmath1066 , and so @xmath1175 is the canonical isomorphism between them . to see that @xmath1176 is also invertible , apply the same argument with the composite @xmath1177 in place of @xmath1053 . the result now follows by lemma([coeq - when - stable ] ) . suppose that @xmath16 is locally presentable , @xmath1178 and @xmath1179 are monads on @xmath16 , and @xmath1180 is a morphism of monads . then one has the obvious forgetful functor @xmath1181 and when @xmath225 is accessible , @xmath804 has a left adjoint which we denote as @xmath763 . the general fact responsible for the existence of @xmath763 , and which in fact gives a formula for it in terms of coequalisers in @xmath734 , is the dubuc adjoint triangle theorem @xcite : for an algebra @xmath1182 of @xmath114 , one has the reflexive coequaliser @xmath1183 { ( sx,\mu^s_x)}="m " [ r ] { \phi_!(x , x)}="r " " l":@<2ex>"m"^-{\mu^s_xs(\phi_x)}:"l"|-{s\eta^m_x}:@<-2ex>"m"_-{sx}:"r"^-{q_{(x , x)}}}\ ] ] in @xmath734 . putting this together with section([sec : coequalisers ] ) an explicit description of the composite @xmath782 is given as follows . we construct morphisms @xmath1184 @xmath1185 starting with @xmath1186 by transfinite induction on @xmath0 . +   + * initial step*. define @xmath1052 to be the identity , @xmath1053 to be the coequaliser of @xmath1187 and @xmath1188 , @xmath1054 and @xmath1055 . note also that @xmath1189 . +   + * inductive step*. assuming that @xmath1057 , @xmath1058 and @xmath1059 are given , we define @xmath1060 to be the coequaliser of @xmath1190 and @xmath1191 , @xmath1192 and @xmath1064 . +   + * limit step*. define @xmath1193 as the colimit of the sequence given by the objects @xmath1194 and morphisms @xmath1071 for @xmath1072 , and @xmath1050 for the component of the universal cocone at @xmath1073 . @xmath1195 { \colim_{m{<}n } sq_m}="tm " [ r ] { \colim_{m{<}n } q_m}="tr " [ d ] { q_n}="br " [ l ] { sq_n}="bm " [ l ] { s^2q_n}="bl " " tl":@<1ex>"tm"^-{\mu_{<{n}}}:@<1ex>"tr"^-{v_{<{n } } } " tl":@<-1ex>"tm"_-{(sv)_{<{n}}}:@<-1ex>@{<-}"tr"_-{\eta_{<{n } } } " bl":"bm"_-{\mu}:@{<-}"br"_-{\eta } " tl":"bl"_{o_{n,2 } } " tm":"bm"^{o_{n,1 } } " tr":@{=}"br"}\ ] ] we write @xmath1075 and @xmath1076 for the obstruction maps measuring the extent to which @xmath225 and @xmath889 preserve the colimit defining @xmath1193 . we write @xmath1196 , @xmath1197 , @xmath1080 and @xmath1198 for the maps induced by the @xmath1199 , @xmath1200 , @xmath1084 and @xmath1201 for @xmath1072 respectively . define @xmath1057 as the coequaliser of @xmath1087 and @xmath1202 , @xmath1203 and @xmath1090 . +   + instantiating theorem([thm : coeq - talg ] ) to the present situation gives    [ cor : explicit - phi - shreik ] suppose that @xmath16 is a locally presentable category , @xmath114 and @xmath225 are monads on @xmath16 , @xmath1180 is a morphism of monads , and @xmath104 is an @xmath114-algebra . if moreover @xmath225 is @xmath392-accessible for some regular cardinal @xmath392 , then for any ordinal @xmath0 such that @xmath1163 one may take @xmath1204 as an explicit definition of @xmath1205 and the associated coequalising map in @xmath734 coming from the dubuc adjoint triangle theorem . [ cor : phi - shreik - simple ] suppose that under the hypotheses of corollary([cor : explicit - phi - shreik ] ) that @xmath225 and @xmath889 preserve the coequaliser of @xmath1206 and @xmath1188 in @xmath16 . then the sequence @xmath1164 stabilises at @xmath33 , and writing @xmath1207 for the unique map such that @xmath1208 , one may take @xmath1209 as an explicit definition of @xmath1205 and the associated coequalising map in @xmath734 .    here is a degenerate situation in which corollary([cor : phi - shreik - simple ] ) applies . since @xmath1210 we have @xmath1211 , but another way to view this isomorphism as arising is to apply the corollary in the case where @xmath104 is a free @xmath114-algebra , say @xmath1212 , for in this case one has the dotted arrows in @xmath1213 { smz}="m " [ r ] { sz}="r " " l":@<1.5ex>"m"^-{(\mu^sm)(s{\phi}m ) } " l":@<-1.5ex>"m"^-{s\mu^m } " l":@{<.}@<-3.5ex>"m"_-{sm\eta^m } " m":"r"^-{\mu^ss(\phi ) } " m":@{<.}@<-2ex>"r"_-{s\eta^m}}\ ] ] exhibiting @xmath1214 as a split coequaliser , and thus absolute . let us denote by @xmath776 the monad on @xmath839 induced by the adjunction @xmath775 . while a completely explicit description of this monad is unnecessary for the proof of theorem([thm : lift - mult ] ) , we will require such a description in section([ssec : explicit - lifting ] ) when we wish to give an explicit description of the `` lifted '' multitensors that this theorem provides for us . let @xmath104 be in @xmath839 , suppose @xmath225 is @xmath392-accessible and fix an ordinal @xmath0 such that @xmath1163 . then by corollary([cor : explicit - phi - shreik ] ) one may take @xmath1215 as the definition of the endofunctor @xmath101 . note that @xmath1216 is just a more refined notation for @xmath1205 . referring to the diagram @xmath1217 { mx}="tm " [ r ] { x}="tr " [ d ] { q_n}="br " [ l ] { sx}="bm " [ l ] { smx}="bl " " tl":@<-1ex>"tm"_-{mx } " tl":@<1ex>"tm"^-{\mu^m_x}:"tr"^-{x } " bl":@<-1ex>"bm"_-{sx } " bl":@<1ex>"bm"^-{\mu^s_xs(\phi_x)}:"br"_-{q_{{<}n } } " tl":"bl"_{\phi_{mx } } " tm":"bm"^{\phi_x } " tr":@{.>}"br"^{\eta^t_{(x , x)}}}\ ] ] one may define the underlying map in @xmath16 of @xmath1218 as the unique map making the square on the right commute . this makes sense since the top row is a coequaliser in @xmath16 . via the evident @xmath114-algebra structures on each of the objects in this diagram , one may in fact interpret the whole diagram in @xmath839 with the top row now being the canonical presentation coequaliser for @xmath104 , and this is why @xmath1218 is an @xmath114-algebra map . the proof that @xmath1218 possesses the universal property of the unit of @xmath775 is straight forward and left to the reader . as for @xmath1219 , it is induced from the following situation in @xmath734 : @xmath1220 { ( sq_n,\mu^s)}="m " [ r(4 ) ] { ( q_n(q_n , a\phi),a(q_n , a\phi))}="r " [ dl ] { ( q_n(x , x),a(x , x))}="b " " l":@<-1ex>"m"_-{\mu^ss(\phi ) } " l":@<1ex>"m"^-{s(a(x , x)\phi)}(:"r"^-{(q_{{<}n})_{(q_n , a\phi)}}:@{.>}"b"^(.35){\mu^t_{(x , x)}},:"b"_{a(x , x)})}\ ] ] since by definition @xmath1219 underlies an @xmath225-algebra map , to finish the proof that our definition really does describe the multiplication of @xmath101 , it suffices by the universal property of @xmath777 to show that @xmath1221 is the identity , and this is easily achieved using the defining diagrams of @xmath778 and @xmath777 together . the data of @xmath101 is still not quite explicit enough for our purposes . what remains to be done is to describe @xmath777 and ( especially ) @xmath778 in terms of the transfinite data that gives @xmath1193 . so we shall for each ordinal @xmath227 provide @xmath1222 and @xmath1223 in @xmath16 such that @xmath1224 , by transfinite induction on @xmath227 . +   + * initial step*. define @xmath1225 to be the identity , and @xmath1226 and @xmath1227 as the unique morphisms such that @xmath1228 by the universal properties of @xmath107 and @xmath1053 ( as the evident coequalisers ) respectively . +   + * inductive step*. define @xmath1229 and @xmath1230 as the unique map satisfying @xmath1231 using the universal property of @xmath1144 as a coequaliser . +   + * limit step*. when @xmath227 is a limit ordinal define @xmath1232 and @xmath1233 as the maps induced by the @xmath1234 and @xmath1235 for @xmath1236 and the universal property of @xmath1194 as the colimit of the sequence of the @xmath1237 for @xmath1236 . then define @xmath1238 and @xmath1230 as the unique map satisfying @xmath1231 using the universal property of @xmath1144 as a coequaliser . +   + the fact that the induction just given was obtained by unpacking the descriptions of @xmath777 and @xmath778 of the previous paragraph in terms of the transfinite construction of the endofunctor @xmath101 ( ie the @xmath1194 ) , is expressed by    [ cor : induced - monad - very - explicit ] suppose that @xmath16 is a locally presentable category , @xmath114 and @xmath225 are monads on @xmath16 , @xmath1180 is a morphism of monads , and @xmath104 is an @xmath114-algebra . if moreover @xmath225 is @xmath392-accessible for some regular cardinal @xmath392 , then for any ordinal @xmath0 such that @xmath1163 one may take @xmath1239 as constructed above as an explicit description underlying endofunctor , unit and multiplication of the monad generated by the adjunction @xmath775 . [ cor : vexp - simple ] under the hypotheses of corollary([cor : induced - monad - very - explicit ] ) , if for @xmath1240 , @xmath225 and @xmath889 preserve the coequaliser of @xmath1206 and @xmath1188 in @xmath16 , then one may take @xmath1241 with @xmath287 as constructed in corollary([cor : phi - shreik - simple ] ) .
one of the open problems in higher category theory is the systematic construction of the higher dimensional analogues of the gray tensor product . in this paper 
 we continue the work of @xcite to adapt the machinery of globular operads @xcite to this task . 
 the resulting theory includes the gray tensor product of 2-categories and the crans tensor product @xcite of gray categories . 
 moreover much of the previous work on the globular approach to higher category theory is simplified by our new foundations , and we illustrate this by giving an expedited account of many aspects of cheng s analysis @xcite of trimble s definition of weak @xmath0-category . by way of application 
 we obtain an `` ekmann - hilton '' result for braided monoidal 2-categories , and give the construction of a tensor product of @xmath1-infinity algebras . 
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