arXiv:1511.00824v3 [math.CT] 30 Dec 2016

CENTRAL REFLECTIONS AND NILPOTENCY IN EXACT
MAL’TSEV CATEGORIES
CLEMENS BERGER AND DOMINIQUE BOURN
Abstract. We study nilpotency in the context of exact Mal’tsev categories taking central extensions as the primitive notion. This yields a nilpotency tower
which is analysed from the perspective of Goodwillie’s functor calculus.
We show in particular that the reflection into the subcategory of n-nilpotent
objects is the universal endofunctor of degree n if and only if every n-nilpotent
object is n-folded. In the special context of a semi-abelian category, an object is
n-folded precisely when its Higgins commutator of length n + 1 vanishes.

Contents
Introduction
1. Central extensions and regular pushouts
2. Affine objects and nilpotency
3. Affine morphisms and central reflections
4. Aspects of nilpotency
5. Quadratic identity functors
6. Identity functors with bounded degree
Acknowledgements
References

1
6
17
23
32
36
46
67
67

Introduction
This text investigates nilpotency in the context of exact Mal’tsev categories. Our
purpose is twofold: basic phenomena of nilpotency are treated through universal
properties rather than through commutator calculus, emphasising the fundamental
role played by central extensions; nilpotency is then linked to an algebraic form of
Goodwillie’s functor calculus [29]. This leads to a global understanding of nilpotency
in terms of functors with bounded degree.
A Mal’tsev category is a finitely complete category in which reflexive relations are
equivalence relations [17, 19]. Important examples of exact Mal’tsev categories are
Date: November 25, 2016.
1991 Mathematics Subject Classification. 17B30, 18D35, 18E10, 18G50, 20F18.
Key words and phrases. Nilpotency, Mal’tsev category, central extension, Goodwillie functor
calculus.
1

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CLEMENS BERGER AND DOMINIQUE BOURN

Mal’tsev varieties [53] and semi-abelian categories [47]. The simplicial objects of an
exact Mal’tsev category are “internal” Kan complexes (cf. [17, 62]).
Nilpotency is classically understood via the vanishing of iterated commutators: in
a Mal’tsev variety by means of so-called Smith commutators [60, 27], in a semi-abelian
category by means of so-called Huq commutators [43, 25]. The first aim of this text
is to promote another point of view which seems more intrinsic to us and is based on
the notion of central extension, by which we mean a regular epimorphism with central
kernel relation. The n-nilpotent objects are defined to be those which can be linked
to a terminal object by a chain of n consecutive central extensions. This notion of
nilpotency is equivalent to the two aforementioned notions in their respective contexts
(cf. Proposition 2.14). In particular, we get the usual notions of n-nilpotent group,
n-nilpotent Lie algebra and n-nilpotent loop [15]. A category is called n-nilpotent if
all its objects are n-nilpotent.
For any exact Mal’tsev category with binary sums, the full subcategory spanned
by the n-nilpotent objects is a reflective Birkhoff subcategory (cf. Theorem 2.12).
This generalises the analogous known results for Mal’tsev varieties [60, 27] and semiabelian categories [25]. We denote the reflection into the subcategory of n-nilpotent
n
objects by I n and the unit of the adjunction at an object X by ηX
: X ։ I n (X).
Since an n-nilpotent object is a fortiori (n + 1)-nilpotent, the different reflections
assemble into the following nilpotency tower
X ❄PP
qq⑧⑧q⑧ ❄❄❄PPPPP
q
q
q ⑧
❄❄ PPP
qqq ⑧⑧⑧
❄❄ n PPP n+1
q
2
q
PPPηX
⑧
η
❄❄ηX
1
X
qq  ⑧⑧ ηX
q
PP' '

❄ 
q
⑧

q

xqx o q 1
2
n
o
o
o
o
o
o
o
⋆
I (X)
I (X)
I (X)
I n+1 (X) o o
in which the successive quotient maps I n+1 (X) ։ I n (X) are central extensions.
Pointed categories with binary sums will be ubiquitous throughout the text; we call
them σ-pointed for short. Among σ-pointed exact Mal’tsev categories we characterise
the n-nilpotent ones as those for which the comparison maps
θX,Y : X + Y ։ X × Y
are (n − 1)-fold central extensions (cf. Theorem 4.3). The nilpotency class of a σpointed exact Mal’tsev category measures thus the discrepancy between binary sum
and binary product. If n = 1, binary sum and binary product coincide, and all objects
are abelian group objects. A σ-pointed exact Mal’tsev category is 1-nilpotent if and
1
: X ։ I 1 (X)
only if it is an abelian category (cf. Corollary 4.4). The unit ηX
of the first Birkhoff reflection is abelianisation (cf. Proposition 4.2). Moreover, the
successive kernels of the nilpotency tower are abelian group objects as well. This
situation is reminiscent of what happens in Goodwillie’s functor calculus [29] where
“infinite loop spaces” play the role of abelian group objects. The second aim of our
study of nilpotency was to get a deeper understanding of this analogy.
Goodwillie’s notions [29] of cross-effect and degree of a functor translate well into
our algebraic setting: for each (n + 1)-tuple (X1 , . . . , Xn+1 ) of objects of a σ-pointed

CENTRAL REFLECTIONS AND NILPOTENCY IN EXACT MAL’TSEV CATEGORIES

3

category and each based endofunctor F we define an (n + 1)-cube ΞF
X1 ,...,Xn+1 consisting of the images F (Xi1 + · · · + Xik ) for all subsequences of (X1 , . . . , Xn+1 ) together
with the obvious contraction maps. We say that a functor F is of degree ≤ n if these
(n + 1)-cubes are limit-cubes for all choices of (X1 , . . . , Xn+1 ).
F
F
We denote by θX
: F (X1 + · · · + Xn+1 ) → PX
the comparison map
1 ,...,Xn+1
1 ,...,Xn+1
towards the limit of the punctured (n + 1)-cube so that F is of degree ≤ n if and only
F
F
if θX
is invertible for each choice of (n + 1)-tuple. The kernel of θX
1 ,...,Xn+1
1 ,...,Xn+1
F
is a (n + 1)-st cross-effect of F , denoted crn+1 (X1 , . . . , Xn+1 ).
A based endofunctor F is linear, i.e. of degree ≤ 1, if and only if F takes binary sums to binary products. In a semi-abelian category, the second cross-effects
cr2F (X, Y ) measure thus the failure of linearity of F . If F is the identity functor, we
drop F from the notation so that cr2 (X, Y ) denotes the kernel of the comparison map
θX,Y : X + Y → X × Y . This kernel is often denoted X ⋄ Y and called the co-smash
product of X and Y (cf. [16] and Remarks 3.10 and 6.2).
An endofunctor of a semi-abelian (or homological [4]) category is of degree ≤ n if
and only if all its cross-effects of order n + 1 vanish. For functors taking values in
abelian categories, our cross-effects agree with the original cross-effects of EilenbergMac Lane [23] (cf. Remark 6.2). For functors taking values in σ-pointed categories
with pullbacks, our cross-effects agree with those of Hartl-Loiseau [38] and Hartl-Van
der Linden [39], defined as kernel intersections (cf. Definition 5.1).
A Goodwillie type characterisation of the nilpotency tower amounts to the property
that for each n, the reflection I n into the Birkhoff subcategory of n-nilpotent objects
is the universal endofunctor of degree ≤ n. In fact, every endofunctor of degree
≤ n of a σ-pointed exact Mal’tsev category takes values in n-nilpotent objects (cf.
Proposition 6.5). The reflection I n is of degree ≤ n if and only if the identity functor
of the Birkhoff subcategory of n-nilpotent objects itself is of degree ≤ n. In the
present article we have mainly been investigating this last property.
The property holds for n = 1 because the identity functor of an abelian category
is linear. However, already for n = 2, there are examples of 2-nilpotent semi-abelian
categories which are not quadratic, i.e. do not have an identity functor of degree ≤ 2
(cf. Section 6.5). We show that a σ-pointed exact Mal’tsev category is quadratic if
and only if the category is 2-nilpotent and algebraically distributive, i.e. endowed
with isomorphisms (X × Z) +Z (Y × Z) ∼
= (X + Y ) × Z for all objects X, Y, Z (cf.
Corollary 5.16). Since algebraic distributivity is preserved under Birkhoff reflection,
the subcategory of 2-nilpotent objects of an algebraically distributive exact Mal’tsev
category is always quadratic (cf. Theorem 5.18).
Algebraic distributivity is a consequence of the existence of centralisers for subobjects as shown by Gray and the second author [13]. For pointed Mal’tsev categories,
it also follows from algebraic coherence in the sense of Cigoli-Gray-Van der Linden
[20]. Our quadraticity result implies that iterated Huq commutator [X, [X, X]] and
ternary Higgins commutator [X, X, X] coincide for each object X of an algebraically
distributive semi-abelian category (cf. Corollary 5.19 and [20, Corollary 7.2]).
There is a remarkable duality for σ-pointed 2-nilpotent exact Mal’tsev categories:
algebraic distributivity amounts to algebraic codistributivity, i.e. to isomorphisms

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CLEMENS BERGER AND DOMINIQUE BOURN

(X × Y ) + Z ∼
= (X + Z) ×Z (Y + Z) for all X, Y, Z (cf. Proposition 5.15). Indeed, the
difference between 2-nilpotency and quadraticity is precisely algebraic codistributivity
(cf. Theorem 5.5). An extension of this duality to all n ≥ 2 is crucial in relating
general nilpotency to identity functors with bounded degree.
The following characterisation is very useful: The identity functor of a σ-pointed
exact Mal’tsev category E is of degree ≤ n if and only if all its objects are n-folded
(cf. Proposition 6.5). An object is n-folded (cf. Definition 6.3) if the (n + 1)-st
X
folding map δn+1
: X + · · · + X → X factors through the comparison map θX,...,X :
X + · · · + X ։ PX,...,X . In a varietal context this can be expressed in combinatorial
terms (cf. Remark 6.4). The full subcategory Fldn (E) spanned by the n-folded objects
is a reflective Birkhoff subcategory of E, and the reflection J n : E → Fldn (E) is the
universal endofunctor of degree ≤ n (cf. Theorem 6.8). Every n-folded object is
n-nilpotent (cf. Proposition 6.13) while the converse holds if and only if the other
Birkhoff reflection I n : E → Niln (E) is also of degree ≤ n.
In the context of semi-abelian categories, closely related results appear in the work
of Hartl and his coauthors [38, 39, 40], although formulated slightly differently. In a
semi-abelian category, an object X is n-folded if and only if its Higgins commutator
of length n + 1 vanishes (cf. Remark 6.4), where the latter is defined as the image
of the composite map crn+1 (X, . . . , X) → X + · · · + X → X, cf. [38, 39, 54]. The
universal n-folded quotient J n (X) may then be identified with the quotient of X by
the Higgins commutator of length n + 1 in much the same way as the universal nnilpotent quotient I n (X) may be identified with the quotient of X by the iterated
Huq commutator of length n + 1. It was Hartl’s insight that Higgins commutators
are convenient for extending the “polynomial functors” of Eilenberg-Mac Lane [23]
to a semi-abelian context. Our treatment in the broader context of exact Mal’tsev
categories follows more closely Goodwillie’s functor calculus [29].
In a σ-pointed exact Mal’cev category, abelianisation I 1 is the universal endofunctor J 1 of degree ≤ 1 (cf. Mantovani-Metere [54]). For n > 1 however, the universal
endofunctor J n of degree ≤ n is in general a proper quotient of the n-th Birkhoff
reflection I n (cf. Corollary 6.12). In order to show that even in a semi-abelian variety
the two endofunctors may disagree, we exhibit a Moufang loop of order 16 (subloop
of Cayley’s octonions) which is 2-nilpotent but not 2-folded (cf. Section 6.5). Alternatively, Mostovoy’s modified lower central series of a loop [55] yields other examples
of a similar kind provided the latter agrees with the successive Higgins commutators
of the loop (cf. [39, Example 2.15] and [61]).
We did not find a simple categorical structure that would entail the equivalence
between n-nilpotency and n-foldedness for all n. As a first step in this direction
we show that an n-nilpotent semi-abelian category has an identity functor of degree
≤ n if and only if its n-th cross-effect is multilinear (cf. Theorem 6.22). We also
show that the nilpotency tower has the desired universal property if and only if it is
homogeneous, i.e. for each n, the n-th kernel functor is of degree ≤ n (cf. Theorem
6.23). This is preserved under Birkhoff reflection (cf. Theorem 6.24). The categories
of groups and of Lie algebras have homogeneous nilpotency towers so that a group,
resp. Lie algebra is n-nilpotent if and only if it is n-folded, and the Birkhoff reflection

CENTRAL REFLECTIONS AND NILPOTENCY IN EXACT MAL’TSEV CATEGORIES

5

I n is here indeed the universal endofunctor of degree ≤ n. The category of triality
groups [22, 28, 37] also has a homogeneous nilpotency tower although it contains the
category of Moufang loops as a full coreflective subcategory, and the latter has an
inhomogeneous nilpotency tower (cf. Section 6.5).
There are several further ideas closely related to the contents of this article which
we hope to address in future work. Let us mention two of them:
The associated graded object of the nilpotency tower ⊕n≥1 K[I n (X) ։ I n−1 (X)]
is a functor in X taking values in graded abelian group objects. For the category of
groups this functor actually takes values in graded Lie rings and as such preserves
n-nilpotent objects and free objects, cf. Lazard [51]. It is likely that for a large class of
semi-abelian categories, the associated graded object of the nilpotency tower carries
a similar algebraic structure. It would be interesting to establish the relationship
between this algebraic structure and the cross-effects of the identity functor.
It follows from [17, Theorem 4.2] and [59, Theorem IV.4] that the simplicial objects
of a pointed Mal’tsev variety carry a Quillen model structure in which the weak
equivalences are the maps inducing a quasi-isomorphism on Moore complexes. Such
a model structure also exists for the simplicial objects of a semi-abelian category with
enough projectives, cf. [59, 62]. In both cases, regular epimorphisms are fibrations,
and the trivial fibrations are precisely the regular epimorphisms for which the kernel
is homotopically trivial. This implies that Goodwillie’s homotopical cross-effects [29]
agree here with our algebraic cross-effects.
Several notions of homotopical nilpotency are now available. The first is the least
n
: X• ։ I n (X• ) is a trivial fibration, the second (resp.
integer n for which the unit ηX
•
third) is the least integer n for which X• is homotopically n-folded (resp. the value
of an n-excisive approximation of the identity). The first is a lower bound for the
second, and the second is a lower bound for the third invariant. For simplicial groups
the first invariant recovers the Berstein-Ganea nilpotency for loop spaces [2], the
second the cocategory of Hovey [42], and the third the Biedermann-Dwyer nilpotency
for homotopy nilpotent groups [3]. Similar chains of inequalities have recently been
studied by Eldred [24] and Costoya-Scherer-Viruel [21].
The plan of this article is as follows.
Section 1 reviews the notions of central extension and regular pushout. At the
end an algebraic Beck-Chevalley condition for pushouts of regular epimorphisms in
an exact Mal’tsev category is established.
Section 2 presents our definition of nilpotency and studies under which conditions
the n-nilpotent objects form a reflective Birkhoff subcategory.
Section 3 investigates central reflections, the motivating example being the reflection of the category of (n + 1)-nilpotent objects into the category of n-nilpotent
objects. The unit of these central reflections is shown to be pointwise affine.
Section 4 establishes first aspects of nilpotency. The nilpotency class of a σ-pointed
exact Mal’cev category is related to universal properties of the comparison map θX,Y :
X + Y → X × Y . This leads to a new family of binary tensor products interpolating
between binary sum and binary product.

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CLEMENS BERGER AND DOMINIQUE BOURN

Section 5 studies the σ-pointed exact Mal’tsev categories with quadratic identity
functor. They are characterised among the 2-nilpotent ones as those which are algebraically distributive, resp. algebraically codistributive.
Section 6 studies the σ-pointed exact Mal’tsev categories with an identity functor
of degree ≤ n. They are characterised as those in which all objects are n-folded.
Every n-folded object is shown to be n-nilpotent. Several sufficient criteria for the
converse are given. The semi-abelian varieties of groups, Lie algebras, Moufang loops
and triality groups are discussed.
1. Central extensions and regular pushouts
In this introductory section we review the notion of central equivalence relation
and study basic properties of the associated class of central extensions, needed for
our treatment of nilpotency. By central extension we mean a regular epimorphism
with central kernel relation [8, 11, 12]. This algebraic concept of central extension
has to be distinguished from the axiomatic concept of Janelidze-Kelly [44] which is
based on a previously chosen admissible Birkhoff subcategory. Nevertheless, it is
known that with respect to the Birkhoff subcategory of abelian group objects, the
two approaches yield the same class of central extensions in any congruence modular
variety (cf. [45, 46]) as well as in any exact Mal’tsev category (cf. [11, 26, 31]).
We assume throughout that our ambient category is a Mal’tsev category, i.e. a
finitely complete category in which every reflexive relation is an equivalence relation,
cf. [4, 6, 17, 19]. Most of the material of this section is well-known to the expert, and
treated in some detail here mainly to fix notation and terminology.
One exception is Section 1.6 which establishes an “algebraic” Beck-Chevalley condition for pushouts of regular epimorphisms in exact Mal’tsev categories, dual to
the familiar Beck-Chevalley condition for pullbacks of monomorphisms in elementary
toposes. In recent and independent work, Gran-Rodelo [32] consider a weaker form
of this condition and show that it characterises regular Goursat categories.
1.1. Smith commutator of equivalence relations. –
An equivalence relation R on X will be denoted as a reflexive graph (p0 , p1 ) :
R ⇒ X with section s0 : X → R, but whenever convenient we shall consider R as a
subobject of X × X. By a fibrant map of equivalence relations (X, R) → (Y, S) we
mean a natural transformation of the underlying reflexive graphs such that the three
naturality squares are pullback squares.
A particularly important equivalence relation is the kernel relation R[f ] of a morphism f : X → Y which is part of the following diagram:
p1

R[f ] o

p0

/

/X

f

/ Y.

The discrete equivalence relation ∆X on X is the kernel relation R[1X ] of the
identity map 1X : X → X. The indiscrete equivalence relation ∇X on X is the kernel
relation R[ωX ] of the unique map ωX from X to a terminal object.

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7

Two equivalence relations R, S on the same object X are said to centralise each
other if the square
R

(s0 ,1S )
R ×O X S o
S
(1R ,sS
0)

R

p

pR
0

pS
1

# 
/X

admits a (necessarily unique) filler which makes the diagram commute, cf. [12, 57].
In set-theoretical terms such a filler amounts to the existence of a “partial Mal’tsev
operation” on X, namely (considering R ×X S as a subobject of X × X × X) a ternary
operation p : R ×X S → X such that x = p(x, y, y) and p(x, x, y) = y. We shall follow
Marino Gran and the second author in calling p : R ×X S → X a connector between
R and S, cf. [11, 12].
In a finitely cocomplete regular Mal’tsev category, there exists for each pair (R, S)
of equivalence relations on X a smallest effective equivalence relation [R, S] on X
such that R and S centralise each other in the quotient X/[R, S]. This equivalence
relation is the so-called Smith commutator of R and S, cf. [8, 12, 57, 60].
In these terms R and S centralise each other precisely when [R, S] = ∆X . The
Smith commutator is monotone in each variable and satisfies
[R, S] = [S, R] and f ([R, S]) ⊂ [f (R), f (S)]
for each regular epimorphism f : X → Y , where f (R) denotes the direct image of the
subobject R ⊂ X × X under the regular epimorphism f × f : X × X → Y × Y . The
Mal’tsev condition implies that this direct image represents an equivalence relation
on Y . Note that equality f ([R, S]) = [f (R), f (S)] holds if and only if the direct image
f ([R, S]) is an effective equivalence relation on Y , which is always the case in an exact
Mal’tsev category.
1.2. Central equivalence relations and central extensions. An equivalence relation R on X is said to be central if [R, ∇X ] = ∆X . A central extension is by
definition a regular epimorphism with central kernel relation. An n-fold central extension is the composite of n central extensions. An n-fold centrally decomposable
morphism is the composite of n morphisms with central kernel relation.
The indiscrete equivalence relation ∇X is a central equivalence relation precisely
when X admits an internal Mal’tsev operation p : X × X × X → X. In pointed
Mal’tsev categories such a Mal’tsev operation amounts to an abelian group structure
on X, cf. [4, Proposition 2.3.8]. An object X of a pointed Mal’tsev category (D, ⋆D )
is thus an abelian group object if and only if the map X → ⋆D is a central extension.
Central equivalence relations are closed under binary products and inverse image
along monomorphisms. In a regular Mal’tsev category, central equivalence relations
are closed under direct images, cf. [12, Proposition 4.2] and [4, Proposition 2.6.15].
Lemma 1.1. In a regular Mal’tsev category, an n-fold centrally decomposable morphism can be written as an n-fold central extension followed by a monomorphism.

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CLEMENS BERGER AND DOMINIQUE BOURN

Proof. It suffices to show that a monomorphism ψ followed by a central extension φ
can be rewritten as a central extension φ′ followed by a monomorphism ψ ′ . Indeed,
the kernel relation R[φψ] is central, being the restriction ψ −1 (R[φ]) of the central
equivalence relation R[φ] along the monomorphism ψ; therefore, by regularity, one
obtains φψ = ψ ′ φ′ where φ′ is quotienting by the kernel relation R[φψ].

Lemma 1.2. In a Mal’tsev category, morphisms with central kernel relation are closed
under pullback. In a regular Mal’tsev category, central extensions are closed under
pullback.
Proof. It suffices to show the first statement. In the following diagram,
p′1

R[f ′ ] o

′
/X

p′0

R(x,y)

/ Y′
y

x

p1


R[f ] o

f′

/

/ 
/X

p0

f


/Y

if the right square is a pullback, then the left square is a fibrant map of equivalence
relations. This permits to lift the connector p : R[f ] ×X ∇X → X so as to obtain a

connector p′ : R[f ′ ] ×X ′ ∇X ′ → X ′ .
f

g

Lemma 1.3. Let X → Y → Z be morphisms in a Mal’tsev category.
If gf is a morphism with central kernel relation then so is f . More generally, if
gf is n-fold centrally decomposable then so is f .
Proof. Since R[f ] ⊂ R[gf ], the commutation relation [R[gf ], ∇X ] = ∆X implies the
commutation relation [R[f ], ∇X ] = ∆X . Assume now gf = kn · · · k1 where each ki is
a morphism with central kernel relation. In the following pullback
P o

γ

/X

φ
gf

ψ


Y

g


/Z

φ is the unique map such that γφ = 1X and ψφ = f . If we denote by hi the
morphism with central kernel relation obtained by pulling back ki along g, we get f =
hn · · · h2 (h1 φ). Since φ is a monomorphism, the kernel relation R[h1 φ] = φ−1 (R[h1 ]) is
central, and hence f is the composite of n morphisms with central kernel relation. 
Proposition 1.4 (Corollary 3.3 in [8]). In a finitely cocomplete regular Mal’tsev
category, each morphism f : X → Y factors canonically as in
ηf

/ / X/[∇X , R[f ]]
❧❧
f
❧❧❧

 v❧❧❧❧❧❧
o X/R[f ]
Y o
X

ζf

CENTRAL REFLECTIONS AND NILPOTENCY IN EXACT MAL’TSEV CATEGORIES

9

where ηf is a regular epimorphism and ζf has a central kernel relation. If f is a
regular epimorphism then ζf is a central extension.
This factorisation has the following universal property. Any commutative diagram
of undotted arrows as below (with Zf = X/[∇X , R[f ]]), such that ζ ′ has a central
kernel relation, produces a unique dotted map
x

X❅
❅❅ ηf
❅❅
❅❅

/ X′
❆❆ ′
❆❆ η
❆❆
❆
z
/ Z′
⑥
⑥
⑥
⑥⑥
~⑥⑥ ζ ′
/ Y′

Zf

f

⑦
⑦⑦
⑦
 ~⑦⑦ ζf
Y

y

rendering the whole diagram commutative.
Proposition 1.5. Let n be a positive integer. In a finitely cocomplete regular Mal’tsev
category, each morphism has a universal factorisation into a regular epimorphism
followed by an n-fold centrally decomposable morphism. Each n-fold central extension
has an initial factorisation into n central extensions.
Proof. We proceed by induction, the case n = 1 being treated in Proposition 1.4.
Suppose the assertion holds up to level n − 1 with f = ζfn−1 ζfn−2 · · · ζf1 ηf1 and ηf1 a
regular epimorphism. Take the universal factorisation of ηf1 . The universality of this
new factorisation is then a straightforward consequence of the induction hypothesis
and the universal property stated in Proposition 1.4.
Starting with an n-fold central extension f , its universal factorisation through a
composite of n − 1 morphisms with central kernel relation makes the regular epimorphism ηf1 a central extension by Lemma 1.3, and therefore produces a factorisation
into n central extensions which is easily seen to be the initial one.

1.3. Regular pushouts. In a regular category, any pullback square of regular epimorphisms is also a pushout square, as follows from the pullback stability of regular
epimorphisms. In particular, a commuting square of regular epimorphisms
X

f

x


X′

f′

//Y
 y
/ / Y′

is a pushout whenever the comparison map (x, f ) : X → X ′ ×Y ′ Y to the pullback of
y along f ′ is a regular epimorphism. Such pushouts will be called regular, cf. [7]. A
regular pushout induces in particular regular epimorphisms on kernel relations which
we shall denote R(x, y) : R[f ] ։ R[f ′ ] and R(f, f ′ ) : R[x] ։ R[y].
For a regular Mal’tsev category the following more precise result holds.

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CLEMENS BERGER AND DOMINIQUE BOURN

Proposition 1.6 (cf. Proposition 3.3 in [7]). In a regular Mal’tsev category, a commuting square of regular epimorphisms like in (1.3) is a regular pushout if and only
if one of the following three equivalent conditions holds:
(a) the comparison map X → X ′ ×Y ′ Y is a regular epimorphism;
(b) the induced map R(x, y) : R[f ] → R[f ′ ] is a regular epimorphism;
(c) the induced map R(f, f ′ ) : R[x] → R[y] is a regular epimorphism.
Accordingly, central extensions are closed under regular pushouts.
Proof. We already mentioned that (a) implies (b) and (c) in any regular category. It
remains to be shown that in a regular Mal’tsev category (b) or (c) implies (a).
For this, it is useful to notice that in a regular category condition (a) holds if and
only if the composite relation R[f ] ◦ R[x] equals the kernel relation of the diagonal of
the square by Theorem 5.2 of Carboni-Kelly-Pedicchio [17]. Since this kernel relation
is given by x−1 (R[f ′ ]), and condition (b) just means that x(R[f ]) = R[f ′ ], it suffices
to establish the identity R[f ] ◦ R[x] = x−1 (x(R[f ])). In a regular Mal’tsev category,
the composition of equivalence relations is symmetric and coincides with their join.
The join R[f ] ∨ R[x] is easily identified with x−1 (x([R[f ])).
The second assertion follows from (b) resp. (c) and the closure of central kernel
relations under direct image in regular Mal’tsev categories.

Corollary 1.7. In a regular Mal’tsev category, any commutative square
f

Xo

s

x


X′ o

f′

//Y
 y
/ / Y′

s′

with a parallel pair of regular epimorphisms and a parallel pair of split epimorphisms
is a regular pushout.
Proof. The induced map R(f, f ′ ) : R[x] → R[y] is a split and hence regular epimorphism so that the pushout is regular by Proposition 1.6.

Corollary 1.8. In an exact Mal’tsev category, pushouts of regular epimorphisms
along regular epimorphims exist and are regular pushouts.
Proof. Given a pair (f, x) of regular epimorphisms with common domain, consider
the following diagram
p1

R[f ] o
x̃


So

p0

/

/X

f

y

x

q1
q0


/ ′
X
/

//Y

f′


/ / Y′

in which S denotes the direct image x(R[f ]). By exactness, this equivalence relation
on X ′ has a quotient Y ′ . The induced right square is then a regular pushout.


CENTRAL REFLECTIONS AND NILPOTENCY IN EXACT MAL’TSEV CATEGORIES

11

Remark 1.9. It follows from [6] that Corollary 1.7 characterises regular Mal’tsev categories among regular categories, while [17, Theorem 5.7] shows that Corollary 1.8
characterises exact Mal’tsev categories among regular categories.
Remark 1.10. It is worthwhile noting that in any category a commuting square of
epimorphisms in which one parallel pair admits compatible sections is automatically a
pushout square. Dually, a commuting square of monomorphisms in which one parallel
pair admits compatible retractions is automatically a pullback square.
Lemma 1.11 (cf. [11], Lemma 1.1). In a pointed regular category, a regular pushout
induces a regular epimorphism on parallel kernels
K[f ] /
K(x,y)


K[f ′ ] /

/X

f

y

x


/ X′

//Y

f′


// Y′

so that the kernel K[f ′ ] of f ′ is the image under x of the kernel K[f ] of f .
Proof. This follows from the fact that, in any pointed category, the induced map on
kernels factors as a pullback of the comparison map (x, f ) : X ։ X ′ ×Y ′ Y followed
by an isomorphism.

Proposition 1.12. In a finitely cocomplete regular Mal’tsev category, consider the
following diagram of pushouts
X

ηf

x


X′

/ / Zf

ζf

z

ηf ′


/ / Zf ′

ζf ′

//Y
 y
// Y′

in which the upper row represents the universal factorisation of the regular epimorphism f into a regular epimorphism ηf followed by a central extension ζf .
If the outer rectangle is a regular pushout then the right square as well, and the
lower row represents the universal factorisation 1.4 of its composite f ′ = ζf ′ ηf ′ .
Proof. Since on vertical kernel relations R[x] → R[z] → R[y] we get a regular epimorphism, the second one R[z] → R[y] is a regular epimorphism as well, and the right
square is a regular pushout by Proposition 1.6. Therefore, ζf ′ is a central extension
by Corollary 1.7. It remains to be shown that the lower row fulfills the universal
property of factorisation 1.4. Consider the following diagram
X

ηf

/ / Zf

ζf

//Y

z

 y
 z′′ ζ ′
ηf ′
f
// Y′
/ / Zf ′
X ′ ◆◆
qq8 8
◆◆◆
q
q
◆
qq
η ◆◆◆
& ′ qqq ζ
Z

x

12

CLEMENS BERGER AND DOMINIQUE BOURN

with central extension ζ. According to Proposition 1.4, there is a unique dotted
factorisation z ′′ making the diagram commute. Since the left square is a pushout, z ′′
factors uniquely and consistently through z ′ : Zf ′ → Z ′ , showing that the lower row
has indeed the required universal property.

Proposition 1.13. In a finitely cocomplete exact Mal’tsev category, the universal
factorisation of a regular epimorphism through an n-fold central extension is preserved
under pushouts along regular epimorphisms.
Proof. Let us consider the following diagram of pushouts
η

X
x


X′

η

/ / Zn

ζn

/ / Zn−1

 zn
 zn
′
/ / Zn′
/ / Zn−1
′
′
ζn

ζ1

...

Z1

...

 z1
Z1′

//Y

 y
// Y′
′

ζ1

in which the upper row is the universal factorisation 1.5 of a regular epimorphism
f : X → Y through an n-fold central extension. By Corollary 1.8, all pushouts are
′
regular. Therefore, the morphisms ζk′ : Zk′ → Zk−1
are central extensions for all
k. It remains to be shown that the lower row satisfies the universal property of the
factorisation 1.5 of f ′ : X ′ → Y ′ through an n-fold central extension. This follows
induction on n beginning with the case n = 1 proved in Proposition 1.12.

Proposition 1.14. Let D be an exact Mal’tsev category. Consider the following
diagram of pushouts
X
xn


X′

/ / Xn−1 fn−1/ / Xn−2

fn

xn−1


′
/ / Xn−1
′

fn

...

X1

...

 x1
X1′

xn−2


′
/ / Xn−2

′
fn−1

f1

/ / X0
x0


/ / X0′
′

f1

f0

//Y
y


/ / Y′
′

f0

in which xn : X ։ X ′ is a regular epimorphism.
If the upper row represents an n-fold central extension of the regular epimorphism
f0 : X0 ։ Y in the slice category D/Y then the lower row represents an n-fold central
extension of f0′ : X0′ ։ Y ′ in the slice category D/Y ′ .
Proof. Let us set φi = f0 · f1 · · · fi and φ′i = f0′ · f1′ · · · fi′ . Since the indiscrete equivalence relation ∇f0 on the object f0 : X0 ։ Y of the slice category D/Y is given by
R[f0 ], our assumption on the upper row translates into the conditions
[R[fi ], R[φi ]] = ∆Xi for 1 ≤ i ≤ n.
Since any of the rectangles is a regular pushout by Corollary 1.8, we get xi (R[fi ]) =
R[fi′ ] and xi (R[φi ]) = R[φ′i ], and consequently [R[fi′ ], R[φ′i ]] = ∆Xi′ for all i.


CENTRAL REFLECTIONS AND NILPOTENCY IN EXACT MAL’TSEV CATEGORIES

13

1.4. Regular pushouts in pointed Mal’tsev categories with binary sums.
In a pointed category with binary sums and binary products, each pair of objects
(X1 , X2 ) defines a canonical comparison map θX1 ,X2 : X1 + X2 → X1 × X2 , uniquely
determined by the requirement that the composite morphism
Xi /

θX1 ,X2

/ X1 + X2

/ X1 × X2

/ / Xj

is the identity (resp. the null morphism) if i = j (resp. i 6= j), where i, j ∈ {1, 2}.
Recall that θX1 ,X2 is a strong epimorphism for all objects X1 , X2 precisely when the
category is unital in the sense of the second author, and that every pointed Mal’tsev
category is unital, cf. [4, 6]. In a regular category strong and regular epimorphisms
coincide.
Note also that an exact Mal’tsev category has coequalisers for reflexive pairs, so
that an exact Mal’tsev category with binary sums has all finite colimits. In order to
shorten terminology, we call σ-pointed any pointed category with binary sums.
Later we shall need the following two examples of regular pushouts.
Proposition 1.15. For any regular epimorphism f : X ։ Y and any object Z of a
σ-pointed regular Mal’tsev category, the following square
X+Z

θX,Z

/ / X ×Z

f +Z


Y +Z

θY,Z

 f ×Z
/ / Y ×Z

is a regular pushout.
Proof. The regular epimorphism θR[f ],Z : R[f ] + Z ։ R[f ] × Z factors as below
θ

R[f ],Z
/ / R[f ] × Z = R[f × Z]
R[f ] + Z
▼▼▼
5
❧❧❧5
▼▼▼
❧❧❧
❧
▼▼▼
❧
❧❧❧
&
R[f + Z]

inducing a regular epimorphism R[f + Z] → R[f × Z] on the vertical kernel relations
of the square above. Proposition 1.6 allows us to conclude.

Corollary 1.16. For any objects X, Y, Z of a σ-pointed regular Mal’tsev category,
the following square
(X + Y ) + Z

θX+Y,Z

θX,Y +Z


(X × Y ) + Z

is a regular pushout.

θX×Y,Z

/ / (X + Y ) × Z
 θX,Y ×Z
/ / (X × Y ) × Z

14

CLEMENS BERGER AND DOMINIQUE BOURN

1.5. Central subobjects, centres and centralisers. In a pointed Mal’tsev category (D, ⋆D ), two morphisms with common codomain f : X → Z and g : Y → Z are
said to commute [9, 43] if the square
(0Y X ,1Y )
X ×O Y o
Y
φf,g

(1X ,0XY )

X

f

g

# 
/Z

admits a (necessarily unique) filler φf,g : X × Y → Z making the whole diagram
commute, where 0XY : X → ⋆D → Y denotes the zero morphism. A monomorphism
Z ֌ X which commutes with the identity 1X : X → X is called central, and the
corresponding subobject is called a central subobject of X.
Every regular epimorphism f : X ։ Y with central kernel relation R[f ] has a
central kernel K[f ]. In pointed protomodular categories, the converse is true: the
centrality of K[f ] implies the centrality of R[f ], so that central extensions are precisely
the regular epimorphisms with central kernel, cf. [33, Proposition 2.2].
Recall [4, 5] that a pointed category is protomodular precisely when the category
has pullbacks of split epimorphisms, and for each split epimorphism, section and
kernel-inclusion form a strongly epimorphic cospan. Every finitely complete protomodular category is a Mal’tsev category [4, Proposition 3.1.19]. The categories of
groups and of Lie algebras are pointed protomodular. Moreover, in both categories,
each object possesses a centre, i.e. a maximal central subobject. Central group (resp.
Lie algebra) extensions are thus precisely regular epimorphisms f : X ։ Y with
kernel K[f ] contained in the centre of X. This is of course the classical definition of
a central extension in group (resp. Lie) theory.
In these categories, there exists more generally, for each subobject N of X, a
so-called centraliser, i.e. a subobject Z(N ֌ X) of X which is maximal among
subobjects commuting with N ֌ X. The existence of centralisers has far-reaching
consequences, as shown by James Gray and the second author, cf. [13, 35, 36]. Since
they are useful for our study of nilpotency, we discuss some of them here.
Following [6], we denote by PtZ (D) the category of split epimorphisms (with chosen
section) in D over a fixed codomain Z, cf. Section 3.2. For each f : Z → Z ′ pulling
back along f defines a functor f ∗ : PtZ ′ (D) → PtZ (D) which we call pointed basechange along f . In particular, the terminal map ωZ : Z → 1D defines a functor
(ωZ )∗ : D → PtZ (D). Since in a pointed regular Mal’tsev category D, morphisms
commute if and only if their images commute, morphisms in PtZ (D) of the form
φf,f ′

/
X ×Z
⑦? Y
❍c❍❍❍❍
❍❍❍❍ ⑦s⑦⑦⑦⑦⑦
pZ ❍❍❍# ⑦⑦⑦r
Z
f

correspond bijectively to morphisms f : X → K[r] such that X −→ K[r] ֌ Y
commutes with s : Z ֌ Y in D. Therefore, if split subobjects have centralisers in D,

CENTRAL REFLECTIONS AND NILPOTENCY IN EXACT MAL’TSEV CATEGORIES

15

then for each object Z, the functor (ωZ )∗ : D → PtZ (D) : X 7→ X × Z admits a right
adjoint (ωZ )∗ : PtZ (D) → D : (r, s) 7→ K[r] ∩ Z(s).
A category with the property that for each object Z, the functor (ωZ )∗ has a right
adjoint is called algebraically cartesian closed [13]. Algebraic cartesian closedness
implies canonical isomorphisms (X × Z) +Z (Y × Z) ∼
= (X + Y ) × Z for all objects
X, Y, Z, a property we shall call algebraic distributivity, cf. Section 5.3.
1.6. An algebraic Beck-Chevalley condition. –
The dual of an elementary topos is an exact Mal’tsev category, cf. [17, Remark
5.8]. This suggests that certain diagram lemmas for elementary toposes admit a
dual version in our algebraic setting. Supporting this analogy we establish here an
“algebraic dual” of the well-known Beck-Chevalley condition. As a corollary we get a
diagram lemma which will be used several times in Section 6.
Another instance of the same phenomenon is the cogluing lemma for regular epimorphisms in exact Mal’tsev categories (cf. proof of Theorem 6.23a and Corollary
1.8) which is dual to a gluing lemma for monomorphisms in an elementary topos.
Lemma 1.17 (cf. Lemma 1.1 in [30]). Consider a commutative diagram
X

Y

x

y

/ / X′

/ X ′′


/ /Y′


/ Y ′′

in a regular category.
If the outer rectangle is a pullback and the left square is a regular pushout (1.3)
then left and right squares are pullbacks.
Proof. The whole diagram contains three comparison maps: one for the outer rectangle, denoted φ : X → Y ×Y ′′ X ′′ , one for the left and one for the right square, denoted
respectively φl : X → Y ×Y ′ X ′ and φr : X ′ → Y ′ ×Y ′′ X ′′ . We get the identity
φ = y ∗ (φr ) ◦ φl where y ∗ denotes base-change along y. Since the outer rectangle is a
pullback, φ is invertible so that φl is a section and y ∗ (φr ) a retraction.
Since the left square is a regular pushout, the comparison map φl is a regular
epimorphism and hence φl and y ∗ (φr ) are both invertible. Since y is a regular epimorphism in a regular category, base-change y ∗ is conservative so that φr is invertible
as well, i.e. both squares are pullbacks.

Proposition 1.18 (Algebraic Beck-Chevalley). Let D be an exact Mal’tsev category
with pushouts of split monomorphisms along regular epimorphisms.
Any pushout of regular epimorphisms
Ū

ḡ

u


U

g

/ / V̄
 v
//V

16

CLEMENS BERGER AND DOMINIQUE BOURN

yields a functor isomorphism ḡ! u∗ ∼
= v ∗ g! from the fibre PtU (D) to the fibre PtV (D).
Proof. We have to show that for any point (r, s) over U , the following diagram
Ū ′ ❇ ′
G ❇❇u
r̄ ✍✍✍✍
❇! !
✍✍ ✍
✍✍ ✍✍
U′
✍
F
r ✌✌ ✌
✍✍ ✍✍✍
✌
✌
✍
✌
✌✌
Ū ❆
✌✌✌✌ ḡ
❆❆
✌
✌
u ❆ ✌✌ ✌
✌
U

ḡ′

/ / V̄ ′ ′
G ❇❇v❇
r̄ ✍✍✍✍
❇! !
✍
′
✍
g
/ /V′
✍✍ ✍✍✍
✍
F
✍✍ ✍✍✍
✌✌✌✌
✌
✍
✌
r ′ ✌✌ ✌
/ / V̄
✌
❆❆
✌✌✌✌
❆
v ❆ ✌✌ ✌
✌
//V
g
′

in which (r̄, s̄) = u∗ (r, s) and g! (r, s) = (r′ , s′ ) and ḡ! (r̄, s̄) = (r̄′ , s̄′ ), has a right face
which is a downward-oriented pullback; indeed, this amounts to the required identity
v ∗ (r′ , s′ ) = (r̄′ , s̄′ ).
Since bottom face and the upward-oriented front and back faces are pushouts, the
top face is a pushout as well, which is regular by Corollary 1.8. Taking pullbacks
in top and bottom faces induces a split epimorphism U ′ ×V ′ V̄ ′ ։ U ×V V̄ through
which the left face of the cube factors as in the following commutative diagram
Ū ′
r̄


Ū

/ / U ′ ×V ′ V̄ ′

/ / U′
r



/ / U ×V V̄


//U

in which the left square is a regular pushout by Corollary 1.7. Lemma 1.17 shows
then that the right square is a pullback. Therefore, we get the following cube
U ′ ×V ′ V̄▲ ′
▲▲▲
⑥⑥⑥>
▲▲& &
⑥⑥⑥⑥⑥
⑥
⑥⑥⑥
⑥
U′
⑥⑥⑥⑥⑥
r ☎☎☎A
⑥
☎☎
~⑥⑥⑥⑥⑥
☎☎☎☎☎
☎
U ×V V̄P
ḡ
☎☎☎☎
PPP
PPP
☎☎☎☎☎
☎

P( (
☎
U
g

/ / V̄ ′ ′
G ❇❇v❇
r̄ ✎✎✎✎
❇
✎✎ ✎
// V′
✎✎ ✎✎
✎✎ ✎✎
✍ ✍F
✍✍ ✍✍✍
✎ ✎✎
r ′ ✍ ✍✍
/ V̄
✍✍
❅❅
❅
✍✍ ✍
v ❅❅ ✍✍ ✍✍
✍
//V
′

in which the downward-oriented left face and the bottom face are pullbacks. Therefore, the composite of the top face followed by the downward-oriented right face is a
pullback. Moreover, as above, the top face is a regular pushout. It follows then from
Lemma 1.17 that the downward-oriented right face is a pullback as required.

Corollary 1.19. In an exact Mal’tsev category with pushouts of split monomorphisms
along regular epimorphisms, each commuting square of natural transformations of split

CENTRAL REFLECTIONS AND NILPOTENCY IN EXACT MAL’TSEV CATEGORIES

17

epimorphisms
ŪG ′ ❇ ′
u
f¯ ✍✍ ✍ ❇❇❇
!!
✍✍ ✍✍
✍✍ ✍✍
U′
F
✍✍ ✍✍
✌
f
✌ ✌✌
✍ ✍✍
✌
✌ ✌✌
Ū
✌✌✌✌ ḡ
❆❆
✌
✌
✌✌✌✌
u ❆❆
U

ḡ′

/ / V̄ ′ ′
✍ ✍G ❇❇v❇
✍
❇! !
✍ ✍✍
g′
/ /V′
✍✍ ✍✍✍
F
✍✍ ✍✍
✌✌✌✌
✍ ✍✍
✌
✌
f ′ ✌✌ ✌
/ / V̄
✌ ✌✌
❆❆v
✌
❆❆ ✌✌✌✌✌
//V
f¯′

g

such that all horizontal arrows are regular epimorphisms, and front and back faces are
upward-oriented pushouts, induces the following upper pushout square
ḡ′

Ū✮ T✮′ ❑
✮✮ ✮✮ ❑❑❑❑(f¯,u′ )
❑❑
✮✮ ✮✮
❑❑
✮✮ ✮✮
❑% %
✮✮ ✮✮
Ū
×U U ′
f¯ ✮ ✮
✮✮ ✮✮
✁✁✁@
✮✮ ✮✮
✁✁✁✁✁
✁
✮
✮
✁✁
✁✁✁✁
Ū ✁
ḡ

/ / V̄ ′
✮✮ T✮✮ ❑❑❑
❑❑(f¯′ ,v′ )
✮✮
❑❑
f¯′ ✮ ✮
❑❑
✮✮ ✮✮
❑% %
✮
′✮
ḡ×g ✮ ✮
/ / V̄ ×V V ′
✮✮ ✮✮
✮✮ ✮✮
@
✮✮ ✮✮
✮
✮
/ / V̄

in which the kernel relation of the regular epimorphism (f¯′ , v ′ ) : V̄ ′ → V̄ ×V V ′ may
be identified with the intersection R[f¯′ ] ∩ R[v ′ ].
Proof. Taking downward-oriented pullbacks in left and right face of the first diagram
yields precisely a diagram as studied in the proof of Proposition 1.18. This implies
that the front face of the second diagram is an upward-oriented pushout. Since
the back face of the second diagram is also an upward-oriented pushout, the upper
square is a pushout as well, as asserted. The kernel relation of the comparison map
(f¯′ , v ′ ) : V̄ ′ → V̄ ×V V ′ is the intersection of the kernel relations of f¯′ and of v ′ . 
2. Affine objects and nilpotency
2.1. Affine objects.
Definition 2.1. Let C be a full subcategory of a Mal’tsev category D.
An object X of D is said to be C-affine if there exists a morphism f : X → Y in
D with central kernel relation R[f ] and with codomain Y in C.
The morphism f is called a C-nilindex for X.
We shall write Aff C (D) for the full replete subcategory of D spanned by the C-affine
objects of D. Clearly Aff C (D) contains C.
When C consists only of a terminal object 1D of D, we call the C-affine objects
simply the affine objects of D and write Aff C (D) = Aff(D). Recall that the unique
morphism X → 1D has a central kernel relation precisely when the indiscrete equivalence relation on X centralises itself, which amounts to the existence of a (necessarily
unique associative and commutative) Mal’tsev operation on X.

18

CLEMENS BERGER AND DOMINIQUE BOURN

When D is pointed, such a Mal’tsev operation on X induces (and is induced by)
an abelian group structure on X. For a pointed Mal’tsev category D, the category
Aff(D) of affine objects is thus the category Ab(D) of abelian group objects of D.
Remark 2.2. When D is a regular Mal’tsev category, any nilindex f : X → Y factors
as a regular epimorphism f˜ : X ։ f (X) followed by a monomorphism f (X) ֌ Y
with codomain in C; therefore, if C is closed under taking subobjects in D, this defines
a strongly epimorphic nilindex f˜ for X with same central kernel relation as f . In
other words, for regular Mal’tsev categories D and subcategories C which are closed
under taking subobjects in D, the C-affine objects of D are precisely the objects which
are obtained as central extensions of objects of C.
Proposition 2.3. For any full subcategory C of a Mal’tsev category D, the subcategory
Aff C (D) is closed under taking subobjects in D. If C is closed under taking binary
products in D then Aff C (D) as well, so that Aff C (D) is finitely complete.
Proof. Let m : X ֌ X ′ be a monomorphism with C-affine codomain X ′ . If f ′ :
X ′ → Y ′ is a nilindex for X ′ , then f m : X → Y ′ is a nilindex for X, since central
equivalence relations are stable under pointed base-change along monomorphisms and
we have R[f m] = m−1 (R[f ]).
If X and Y are C-affine with nilindices f and g then f × g is a nilindex for X × Y
since maps with central kernel relations are stable under products.

A Birkhoff subcategory [44] of a regular category D is a subcategory C which is
closed under taking subobjects, products and quotients in D. A Birkhoff subcategory of an exact (resp. Mal’tsev) category is exact (resp. Mal’tsev), and regular
epimorphisms in C are those morphisms in C which are regular epimorphisms in D.
If D is a variety (in the single-sorted monadic sense) then Birkhoff subcategories
of D are precisely subvarieties of D, cf. the proof of Lemma 5.11 below.
Proposition 2.4. Let C be a full subcategory of an exact Mal’tsev category D.
If C is closed under taking subobjects and quotients in D then Aff C (D) as well. In
particular, if C is a Birkhoff subcategory of D, then Aff C (D) as well.
Proof. Let X be a C-affine object of D with nilindex f : X → Y . We can suppose f
is a regular epimorphism, cf. Remark 2.2. Thanks to Proposition 2.3 it remains to
establish closure under quotients. Let g : X ։ X ′ be a regular epimorphism in D.
Since D is exact, the pushout of f along g exists in D
X

f

g


X′

//Y
h

f′


// Y′

and f ′ is a central extension since f is, cf. Corollary 1.8. By hypothesis C is stable
under quotients. Therefore the quotient Y ′ belongs to C, and f ′ is a nilindex for X ′
so that X ′ is C-affine as required.


CENTRAL REFLECTIONS AND NILPOTENCY IN EXACT MAL’TSEV CATEGORIES

19

2.2. The C-lower central sequence. –
Definition 2.1 is clearly the beginning of an iterative process. We write C = Nil0C (D)
and define inductively NilnC (D) to be the category Aff Niln−1 (D) (D). The objects of this
C
category NilnC (D) are called the C-nilpotent objects of order n of D, and we get the
following diagram
♦7 = DO a❇h◗◗◗
♦♦⑤♦⑤⑤ ❇❇❇◗◗◗◗◗
♦
♦
❇❇ ◗◗◗
♦♦ ⑤⑤
◗◗◗
❇❇
♦♦♦ ⑤⑤⑤
♦
◗◗◗
❇❇
♦
♦
⑤
◗◗◗
♦
❇
❇a
♦♦
=⑤
O
⑤
h
♦
♦
7/♦ /
n
2
1
NilC (D) / / Niln+1
C
NilC (D) / / NilC (D)
(D)
C
which we call the C-lower central sequence of D.
If C = {1D }, we obtain the (absolute) lower central sequence of D:
7 D h◗
♥♥⑤♥⑤> O a❇❇◗❇◗◗◗◗◗
♥
♥
❇❇ ◗◗
♥♥ ⑤⑤
◗◗◗
❇❇
♥♥♥ ⑤⑤⑤
◗◗◗
♥
❇❇
♥
♥
⑤
◗◗◗
♥
❇
⑤
♥
❇
♥
◗◗h
⑤
❇a
>⑤
O
7♥♥♥
2
n
1
/ Nil (D) / / Nil (D)
Nil (D) / / Niln+1 (D)
{1D } /
Remark 2.5. It follows from Remark 2.2 and an iterative application of Proposition
2.4 that for an exact Mal’tsev category D, the nilpotent objects of order n are precisely
those which can be obtained as an n-fold central extension of the terminal object 1D
and that moreover Niln (D) is a Birkhoff subcategory of D.
If D is the category of groups (resp. Lie algebras) then Niln (D) is precisely the full
subcategory spanned by nilpotent groups (resp. Lie algebras) of class ≤ n. Indeed,
it is well-known that a group (resp. Lie algebra) is nilpotent of class ≤ n precisely
when it can be obtained as an n-fold “central extension” of the trivial group (resp.
Lie algebra), and we have seen in Section 1.5 that the group (resp. Lie) theorist’s
definition of central extension agrees with ours. We will see in Proposition 2.14
below that the equivalence between the central extension definition and the iterated
commutator definition of nilpotency carries over to our general context of finitely
cocomplete exact Mal’tsev categories. Huq [43] had foreseen a long time ago that a
categorical approach to nilpotency was possible. Everaert-Van der Linden [25] recast
Huq’s approach in modern language in the context of semi-abelian categories.
Definition 2.6. A Mal’tsev category D with full subcategory C is called C-nilpotent
of order n (resp. of class n) if D = NilnC (D) (resp. if n is the least such integer).
When C = {1D } the prefix C will be dropped, and instead of “nilpotent of order
n” we also just say “n-nilpotent”.
Proposition 2.7. A Mal’tsev category is n-nilpotent if and only if each morphism is
n-fold centrally decomposable.
A regular Mal’tsev category is n-nilpotent if and only if each morphism factors as
an n-fold central extension followed by a monomorphism.

20

CLEMENS BERGER AND DOMINIQUE BOURN

Proof. The second statement follows from the first by Lemma 1.1. If each morphism
is n-fold centrally decomposable, then this holds for terminal maps ωX : X → 1D , so
that all objects are n-nilpotent. Conversely, assume that all objects are n-nilpotent,
i.e. that for all objects X, the terminal map ωX is n-fold centrally decomposable.
Then, for each morphism f : X → Y , the identity ωX = ωY f together with Lemma
1.3 imply that f is n-fold centrally decomposable as well.

2.3. Epireflections, Birkhoff reflections and central reflections. –
We shall see that if C is a reflective subcategory of D, then the categories NilnC (D)
are again reflective subcategories of D, provided D and the reflection fulfill suitable
conditions. In order to give precise statements we need to fix some terminology.
A full replete subcategory C of D is called reflective if the inclusion C ֒→ D admits
a left adjoint functor I : D → C, called reflection. The unit of the adjunction at
an object X of D will be denoted by ηX : X → I(X). Reflective subcategories C
are stable under formation of limits in D. In particular, reflective subcategories of
Mal’tsev categories are Mal’tsev categories.
A reflective subcategory C of D is called strongly epireflective and the reflection
I is called a strong epireflection if the unit ηX : X → I(X) is pointwise a strong
epimorphism. Strongly epireflective subcategories are characterised by the property
that C is closed under taking subobjects in D. In particular, strongly epireflective
subcategories of regular categories are regular categories.
A Birkhoff reflection (cf. [14]) is a strong epireflection I : D → C such that for
each regular epimorphism f : X → Y in D, the following naturality square
X
f


Y

ηX

/ / I(X)
I(f )


/ / I(Y )
ηY

is a regular pushout (see Section 1.3 and Proposition 1.6).
A subcategory of D defined by a Birkhoff reflection is a Birkhoff subcategory of
D, and is thus exact whenever D is. It follows from Corollary 1.8 that a reflective
subcategory of an exact Mal’tsev category is a Birkhoff subcategory if and only if the
reflection is a Birkhoff reflection.
A central reflection is a strong epireflection I : D → C with the property that the
unit ηX : X ։ I(X) is pointwise a central extension.
The following exactness result will be used at several places. In the stated generality, it is due to Diana Rodelo and the second author [14], but the interested reader
can as well consult [44, 31, 32] for closely related statements.
Proposition 2.8. In a regular Mal’tsev category, strong epireflections preserve pullback squares of split epimorphisms, and Birkhoff reflections preserve pullbacks of split
epimorphisms along regular epimorphisms.
Proof. See Proposition 3.4 and Theorem 3.16 in [14].



CENTRAL REFLECTIONS AND NILPOTENCY IN EXACT MAL’TSEV CATEGORIES

21

Lemma 2.9. Let C be a reflective subcategory of D with reflection I, and assume that
ηX : X → I(X) factors through an epimorphism f : X ։ Y as in:
ηX

/ I(X)
♦7
♦
♦
♦
f 
 ♦♦♦♦♦ η
Y
X

Then I(f ) is an isomorphism and we have η = I(f )−1 ηY .
Proof. Consider the following diagram
ηX

/ I(X)
♣8
♣
♣
f 
I(f )
♣♣

 ♣♣♣♣ η
/
Y
I(Y )
ηY
X

where the lower triangle commutes because f is an epimorphism. If we apply the
reflection I to the whole diagram we get two horizontal isomorphisms I(ηX ) and I(ηY ).
It follows that I(η) is an isomorphism as well, hence so is I(f ), and η = I(f )−1 ηY . 
Lemma 2.10. For any reflective subcategory C of a Mal’tsev category D the C-affine
objects of D are those X for which the unit ηX has a central kernel relation.
Proof. If ηX : X → I(X) has a central kernel relation then X is C-affine. Conversely,
let X be C-affine with nilindex f : X → Y . Then Y is an object of the reflective
subcategory C so that f factors through ηX : X → I(X). Accordingly, we get
R[ηX ] ⊂ R[f ], and hence R[ηX ] is central because R[f ] is.

Corollary 2.11. A reflection I : D → C of a regular Mal’tsev category D is central
if and only if D is C-nilpotent of order 1 (i.e. all objects of D are C-affine).
Theorem 2.12 (cf. [8], Sect. 4.3 in [43], Prp. 7.8 in [25] and Thm. 3.6 in [49]). –
For a reflective subcategory C of a finitely cocomplete regular Mal’tsev category D,
the category Aff C (D) is a strongly epireflective subcategory of D.
The associated strong epireflection IC1 : D → Aff C (D) is obtained by factoring the
unit ηX : X → I(X) universally through a map with central kernel relation.
If C is a reflective Birkhoff subcategory of a finitely cocomplete exact Mal’tsev
category D, then the reflection IC1 : D → Aff C (D) is a Birkhoff reflection.
Proof. Proposition 1.4 yields the following factorisation of the unit:
X
1
ηX



IC1 (X)

ηX

/ I(X)
♥7
♥
♥
♥♥
♥♥♥ η̄X

Since η̄X has a central kernel relation and I(X) is an object of C, the object IC1 (X)
1
belongs to Aff C (D). We claim that the maps ηX
: X ։ IC1 (X) have the universal
1
property of the unit of an epireflection IC : D → Aff C (D).

22

CLEMENS BERGER AND DOMINIQUE BOURN

Let f : X → T be a map with C-affine codomain T which means that ηT has a
central kernel relation. Then consider the following diagram:
f

X■
■■ η1
■■ X
■■
■$ $
ηX
IC1 (X)
✈
✈✈
✈✈
 {✈✈ η̄X
I(X)
I(f )

/T ❈
❈❈❈❈
❈❈❈❈
❈❈❈❈
❈❈❈
f¯
/T
⑤
⑤
⑤
⑤⑤
 ~⑤⑤ ηT
/ I(T )

According to Proposition 1.4, there is a unique factorisation f¯ making the diagram
commute. If D is exact and C a Birkhoff subcategory, the subcategory Aff C (D) is
closed under taking subobjects and quotients by Proposition 2.4. The reflection IC1 is
thus a Birkhoff reflection in this case.

Remark 2.13. A reflective Birkhoff subcategory C of a semi-abelian category D satisfies all hypotheses of the preceding theorem. In this special case, the Birkhoff reflection
IC1 : D → Aff C (D) is given by the formula
IC1 (X) = X/[X, K[ηX ]]
where [X, K[ηX ]] is the Huq commutator of X and K[ηX ], cf. Section 1.5.
Indeed, the pointed protomodularity of D implies (cf. [33, Proposition 2.2]) that
the kernel of the quotient map X → X/[∇X , R[ηX ]] is canonically isomorphic to the
Huq commutator [X, K[ηX ]] so that the formula follows from Proposition 1.4.
2.4. The Birkhoff nilpotency tower. –
According to Theorem 2.12, any reflective Birkhoff subcategory C of a finitely
cocomplete exact Mal’tsev category D produces iteratively the following commutative
diagram of Birkhoff reflections:
D ◗◗
♦♦⑤♦⑤ ❇❇❇◗❇◗◗◗◗
♦
♦
❇❇ ◗◗◗◗
♦♦ ⑤⑤
❇❇ n ◗◗◗
♦1♦♦ ⑤⑤⑤
♦
♦
◗◗◗ I n+1
❇❇ IC
I
♦ C ⑤⑤
IC2
◗◗C◗
❇❇
I ♦♦♦♦
⑤

}
(
⑤
♦
!
♦
w♦
1
2
n
NilC (D) o
NilC (D)
NilC (D) o
Niln+1
(D)
C o
C
A Birkhoff subcategory of an exact Mal’tsev category is an exact Mal’tsev category
so that the subcategories NilnC (D) are all exact Mal’tsev categories, and the horizontal
n
reflections Niln+1
C (D) → NilC (D) are central reflections by Corollary 2.11.
In the special case C = {1D } we get the following commutative diagram of Birkhoff
subcategories and Birkhoff reflections:

CENTRAL REFLECTIONS AND NILPOTENCY IN EXACT MAL’TSEV CATEGORIES

23

D◗
♥♥⑤♥⑤ ❇❇◗❇◗◗◗◗◗
♥
♥
❇❇ ◗◗
♥ ⑤
◗◗
❇❇
♥♥♥ ⑤⑤⑤
♥
♥
❇❇ n ◗◗◗◗◗ n+1
♥ I 1 ⑤⑤
♥
2
♥
❇
◗◗I◗
I
⑤
I
I ♥♥
❇❇
◗(
~⑤⑤

!
w♥♥♥
Nil1 (D) o
Nil2 (D)
Niln (D) o
Niln+1 (D)
{1D } o
1
If D is pointed, then the first Birkhoff reflection I 1 = I{⋆
: D → Nil1 (D) can be
D}
identified with the classical abelianisation functor D → Ab(D). In particular, the
abelian group objects of D are precisely the nilpotent objects of order 1.
When C is a reflective Birkhoff subcategory of a finitely cocomplete exact Mal’tsev
category D, then D is C-nilpotent of class n if and only if n is the least integer such
that either the unit of the n-th Birkhoff reflection ICn is invertible, or equivalently, the
(n − 1)st Birkhoff reflection ICn−1 is a central reflection, see Corollary 2.11.

Proposition 2.14. For an exact Mal’tsev category D with binary sums, the unit of
n
the n-th Birkhoff reflection ηX
: X → I n (X) is given by quotienting out the iterated
Smith commutator [∇X , [∇X , [∇X , · · · , ∇X ]]] of length n + 1.
If D is semi-abelian, this unit is also given as the quotient of X by the iterated
Huq commutator [X, [X, [X, . . . , X]]] of length n + 1.
Proof. The second statement follows from the first by Remark 2.13. The first staten
ment follows from the inductive construction of ηX
in the proof of Theorem 2.12
together with Proposition 1.4.

A finite limit and finite colimit preserving functor is called exact. A functor between
exact Mal’tsev categories with binary sums is exact if and only if it preserves finite
limits, regular epimorphisms and binary sums, cf. Section 1.4.
Lemma 2.15. Any exact functor F : D → E between exact Mal’tsev categories with
binary sums commutes with the n-th Birkhoff reflections, i.e. IEn ◦ F ∼
= F|Niln (D) ◦ IDn .
Proof. According to Theorem 2.12 and Proposition 1.4 it suffices to show that F
takes the canonical factorisation of f : X ։ Y through the central extension ζf :
X/[∇X , R[f ]] ։ Y to the factorisation of F (f ) : F (X) ։ F (Y ) through the central
extension ζF (f ) : F (X)/[∇F (X) , R[F (f )]] ։ F (Y ). Since F is left exact, we have
F (∇X ) = ∇F (X) and F (R[f ]) = R[F (f )], and since F preserves regular epimorphisms, we have F (X/[∇X , R[f ]]) = F (X)/F ([∇X , R[f ]]). It remains to be shown
that F preserves Smith commutators. This follows from exactness of F and the fact
that in a finitely cocomplete exact Mal’tsev category the Smith commutator is given
by an explicit formula involving only finite limits and finite colimits.

3. Affine morphisms and central reflections
We have seen that the nilpotency tower of a σ-pointed exact Mal’tsev category is
a tower of central reflections. In this section we establish a useful general property
of central reflections in exact Mal’tsev categories, namely that the unit of a central

24

CLEMENS BERGER AND DOMINIQUE BOURN

reflection is pointwise affine. Since this property might be useful in other contexts as
well, we first discuss possible weakenings of the notion of exactness.
3.1. Quasi-exact and efficiently regular Mal’tsev categories. –
An exact category is a regular category in which equivalence relations are effective,
i.e. arise as kernel relations of some morphism. In general, effective equivalence
relations R on X have the property that the inclusion R ֌ X × X is a strong
monomorphism. Equivalence relations with this property are called strong. A regular
category in which strong equivalence relations are effective is called quasi-exact.
Any quasi-topos (cf. Penon [58]) is quasi-exact so that there are plenty examples
of quasi-exact categories which are not exact. There are also quasi-exact Mal’tsev
categories which are not exact, as for instance the category of topological groups
and continuous group homomorphisms. Further weakenings of exactness occur quite
naturally as shown in the following chain of implications:
exact

+3 quasi-exact

+3 efficiently regular

+3 fibrational kernel relations

A category is efficiently regular [14] if every equivalence relation (X, S), which
is a regular refinement of an effective equivalence relation (X, R), is itself effective.
By regular refinement we mean any map of equivalence relations (X, S) → (X, R)
inducing the identity on X and a regular monomorphism S → R.
We call a kernel relation (X, R) fibrational if for each fibrant map of equivalence
relations (Y, S) → (X, R) the domain is an effective equivalence relation as well.
According to Janelidze-Sobral-Tholen [48] a kernel relation (X, R) is fibrational precisely when its quotient map X ։ X/R has effective descent, i.e. base-change along
X ։ X/R is a monadic functor. A regular category has thus fibrational kernel relations precisely when all regular epimorphisms have effective descent. The careful
reader will observe that in all proofs of this section where we invoke efficient regularity
we actually just need that the considered kernel relations are fibrational.
The second implication above follows from the fact that any regular monomorphism
is a strong monomorphism, while the third implication follows from the facts that for
any fibrant map of equivalence relations f : (Y, S) → (X, R) the induced map on
relations S → f ∗ (R) is a regular (even split) monomorphism, and that in any regular
category, effective equivalence relations are closed under inverse image.
3.2. Fibration of points and essentially affine categories. –
Recall [6] that for any category D, we denote by Pt(D) the category whose objects
are split epimorphisms with chosen section (“genereralised points”) of D and whose
morphisms are natural transformations between such (compatible with the chosen
sections), and that ¶D : Pt(D) → D denotes the functor associating to a split epimorphism its codomain.
The functor ¶D : Pt(D) → D is a fibration (the so-called fibration of points) whenever D has pullbacks of split epimorphisms. The ¶D -cartesian maps are precisely
pullbacks of split epimorphisms. Given any morphism f : X → Y in D, base-change
along f with respect to the fibration ¶D is denoted by f ∗ : PtY (D) → PtX (D), and will

CENTRAL REFLECTIONS AND NILPOTENCY IN EXACT MAL’TSEV CATEGORIES

25

be called pointed base-change in order to distinguish it from the classical base-change
D/Y → D/X on slices.
Pointed base-change f ∗ has a left adjoint pointed cobase-change f! if and only if
pushouts along f of split monomorphisms with domain X exist in D. In this case
pointed cobase-change along f is given by precisely this pushout, cf. [5]. Accordingly,
the unit (resp. counit) of the (f! , f ∗ )-adjunction is an isomorphism precisely when for
each natural transformation of split epimorphisms
′
X
O
s

f′

r


X

/ Y′
O
s′

f


/Y

r′

the downward square is a pullback as soon as the upward square is a pushout (resp.
upward square is a pushout as soon as the downward square is a pullback).
It is important that in a regular Mal’tsev category pointed base-change along a
regular epimorphism is fully faithful : in a pullback square like above with regular
epimorphism f : X ։ Y , the upward-oriented square is automatically a pushout. This
follows from fact that the induced morphism on kernel relations R(s, s′ ) : R[f ] → R[f ′ ]
together with the diagonal X ′ → R[f ′ ] forms a strongly epimorphic cospan because
the kernel relation R[f ′ ] is the product of R[f ] and X ′ in the fibre PtX (D), and all
fibres are unital in virtue of the Mal’tsev condition, cf. [4, 6].
Recall [6] that a category is called essentially affine if pushouts of split monomorphisms and pullbacks of split epimorphisms exist, and moreover for any morphism
f : X → Y the pointed base-change adjunction (f! , f ∗ ) is an adjoint equivalence.
Additive categories with pullbacks of split epimorphisms are essentially affine. This
follows from the fact that in this kind of category every split epimorphism is a projection, and every split monomorphism is a coprojection. Conversely, a pointed essentially affine category is an additive category with pullbacks of split epimorphisms.
Any slice or coslice category of an additive category with pullbacks of split epimorphisms is an example of an essentially affine category that is not pointed, and hence
not additive. Therefore, the property of a morphism f : X → Y in D to induce an
adjoint equivalence f ∗ : PtY (D) → PtX (D) expresses somehow a “relative additivity”
of f . This motivates the following definition:
Definition 3.1. In a category D with pullbacks of split epimorphisms, a morphism
f : X → Y will be called ¶D -affine when the induced pointed base-change functor
f ∗ : PtD (Y ) → PtD (X) is an equivalence of categories.
An affine extension in D is any regular epimorphism which is ¶D -affine.
Clearly any isomorphism is ¶D -affine. It follows from the analogous property of
equivalences of categories that for any composable morphisms f, g in D, if two among
f , g and gf are ¶D -affine then so is the third, i.e. ¶D -affine morphisms fulfill the
so-called two-out-of-three property.
It might be confusing that we use the term “affine” in two different contexts, namely
as well for objects as well for morphisms. Although their respective definitions seem

26

CLEMENS BERGER AND DOMINIQUE BOURN

unrelated at first sight, this isn’t the case. We will see in Proposition 3.2 that every
affine extension is a central extension, and it will follow from Theorem 3.3 that for a
reflective subcategory C of an efficiently regular Mal’tsev category D, every object of
D is C-affine if and only if every object of D is an affine extension of an object of C.
We hope that this justifies our double use of the term “affine”.
Proposition 3.2. In any Mal’tsev category D, the kernel relation of a ¶D -affine
morphism is central. In particular, each affine extension is a central extension.
Proof. Let f : X → Y be an ¶D -affine morphism with kernel relation
(p0 , p1 ) : R[f ] ⇒ X
and let
X
(pX
0 , p1 ) : X × X ⇒ X

be the indiscrete equivalence relation on X with section sX
0 : X → X × X. Since
pointed base-change f ∗ is an equivalence of categories, there is a split epimorphism
X
(r, s) : Y ′ ⇄ Y such that f ∗ (r, s) = (pX
0 , s0 ) and we get the right hand pullback of
diagram
R[fˇ] o
O
R(pX
0 ,r)

p̌1
p̌0

R(sX
0 ,s)


R[f ] o

p1
p0

/

/ X ×O X
pX
0

fˇ

pX
1

/ 
/X

/ Y′
O
s

r

f


/Y

in which the left hand side are the respective kernel relations. Therefore the left hand
side consists of two pullbacks, and the map pX
1 p̌0 produces the required connector
between R[f ] and the indiscrete equivalence relation ∇X on X.

In particular, if an epireflection I : D → C of a Mal’tsev category D has a pointwise
¶D -affine unit ηX : X → I(X), then it is a central reflection. The following converse
will be essential in understanding nilpotency.
Theorem 3.3. A central reflection I of an efficiently regular Mal’tsev category D has
a unit which is pointwise an affine extension. In particular, morphisms f : X → Y
with invertible image I(f ) : I(X) → I(Y ) are necessarily ¶D -affine.
Proof. The second assertion follows from the first since ¶D -affine morphisms fulfill the
two-out-of-three property and isomorphisms are ¶D -affine.
For the first assertion note that in a regular Mal’tsev category pointed base-change
along a regular epimorphism is fully faithful, and hence ηY∗ : PtI(Y ) (D) → PtY (D) is
a fully faithful functor. Corollary 3.6 below shows that ηY∗ is essentially surjective,
hence ηY∗ is an equivalence of categories for all objects Y in D.


CENTRAL REFLECTIONS AND NILPOTENCY IN EXACT MAL’TSEV CATEGORIES

27

3.3. Centralising double relations. Given a pair (R, S) of equivalence relations
on Y , we denote by RS the inverse image of the reflexive relation S × S under
R
(pR
0 , p1 ) : R ֌ Y × Y . This defines a double relation
q1S

o
RS
O

/

/ SO

q0S

q1R

q0R

 
Ro

pS
1

pS
0
pR
1

/ 
/Y

pR
0

actually the largest double relation relating R and S. In set-theoretical terms, this
double relation RS corresponds to the subset of elements (u, v, u′ , v ′ ) of Y 4 such
that the relations uRu′ , vRv ′ , uSv, u′ Sv ′ hold.
Lemma 3.4. Any split epimorphism (r, s) : X ⇄ Y of a regular Mal’tsev category
with epireflection I and unit η induces the following diagram
R[ηR[r] ] o
O
R(pr0 ,Ipr0 )

R(pr1 ,Ipr1 )

 
R[ηX ] o
O
R(r,Ir)

R(s,Is)


R[ηY ] o

/

/ R[r]
O

ηR[r]

pr1

pr0

/ 
/ XO
r

/ 
/Y

/ / I(R[r])
O
Ipr0

ηX
s

 
/ / IX
O
Ir

ηY

Ipr1

Is


/ / IY

in which the rows and the two right vertical columns represent kernel relations. The
left most column represents then the kernel relation of the induced map R(r, Ir) :
R[ηX ] → R[ηY ], and we have R[ηR[r] ] = R[ηX ]R[r] = R[R(r, Ir)].
Proof. This follows from Proposition 2.8 which shows that I(R[r]) may be identified
with the kernel relation R[I(r)] of the split epimorphism Ir : IX → IY .

For sake of simplicity a split epimorphism (r, s) : X ⇄ Y is called a C-affine point
over Y whenever its domain X is C-affine.
Proposition 3.5. Let C be a reflective subcategory of an efficiently regular Mal’tsev
category D with reflection I and unit η.
Any C-affine point (r, s) over Y is the image under ηY∗ of a C-affine point (r̄, s̄)
over IY such that both points have isomorphic reflections in C.

28

CLEMENS BERGER AND DOMINIQUE BOURN

Proof. Since by Lemma 2.10 ηX has a central kernel relation, the kernel relations
R[ηX ] and R[r] centralise each other. This induces a centralising double relation
/

R[ηX ] ×X R[r] o
O

/ R[r]
O
pr0

 
R[ηX ] o

pr1

/ 
/X

which we consider as a fibrant split epimorphism of equivalence relations (disregarding
the dotted arrows). Pulling back along the monomorphism
/

R[ηX ] o
O

/ XO
s

R(s,Is)

/
/Y

R[ηY ] o

yields on the left hand side of the following diagram
/

RI [(r, s)] o
O

/ XO
s

r


R[ηY ] o

q

/

/Y

/ / X̄
O
r̄

ηY

s̄


/ / IY

another fibrant split epimorphism of equivalence relations. Since D is efficiently regular, the upper equivalence relation is effective with quotient q : X ։ X̄. We claim
that the induced point (r̄, s̄) : X̄ ⇄ IY has the required properties.
Indeed, the right square is a pullback by a well-known result of Barr and Kock,
cf. [4, Lemma A.5.8], so that ηY∗ (r̄, s̄) = (r, s). The centralising double relation
R[ηX ] ×X R[r] is coherently embedded in the double relation R[ηX ]R[r] of Lemma
3.4, cf. [4, Proposition 2.6.13]. This induces an inclusion RI [(r, s)] ֌ R[ηX ] and
hence a morphism φ : X̄ ։ IX such that φq = ηX .
According to Lemma 2.9, we get an isomorphism Iq : IX ∼
= I X̄ compatible with
the units. The kernel relation R[ηX̄ ] is thus the direct image of the central kernel
relation R[ηX ] under the regular epimorphism q : X → X̄ and as such central as well.
In particular, (r̄, s̄) is a C-affine point with same reflection in C as (r, s).

Corollary 3.6. For an efficiently regular Mal’tsev category D with central reflection
I and unit η, pointed base-change ηY∗ : PtI(Y ) (D) → PtY (D) is essentially surjective.
Proof. Since the reflection is central, Corollary 2.11 shows that Proposition 3.5 applies
to the whole fibre PtY (D) whence essential surjectivity of ηY∗ .


CENTRAL REFLECTIONS AND NILPOTENCY IN EXACT MAL’TSEV CATEGORIES

29

3.4. Affine extensions in efficiently regular Mal’tsev categories. –
A functor G : E → E′ is called saturated on quotients if for each object A in E and
each strong epimorphism g ′ : G(A) → B ′ in E′ , there exists a strong epimorphism
g : A → B in E such that G(g) and g ′ are isomorphic under G(A). Note that a
right adjoint functor G : E → E′ is essentially surjective whenever it is saturated on
quotients and each object B ′ of E′ is the quotient of an object B ′′ for which the unit
ηB ′′ : B ′′ → GF (B ′′ ) is invertible.
Lemma 3.7. In an efficiently regular Mal’tsev category, pointed base-change along a
regular epimorphism is saturated on quotients.
Proof. Let f : X ։ Y be a regular epimorphism, let (r, s) be a point over Y , and
l : f ∗ ((r, s)) ։ (r′ , s′ ) be a quotient map over X. Consider the following diagram
pf1

′

′

/

f′

′
/ X O ❇❇
❇❇ l
❇❇
❇❇!
/ ! ′′
/X
⑤⑤
⑤
∗
∗
⑤
f (r) f (s)
⑤⑤
 }⑤} ⑤⑤ r′
/
/X

R[f ] o
O ❈❈
❈❈ pf0 ′
❈❈
❈❈!
!
So
④
④
④
④④
 }④} ④④
pf1
o
R[f ]

// Y′
O
l̄
f ′′
r

s

 ~ ~
//Y

f

pf0

/ / Y ′′
ρ

in which the right hand side square is a pullback, and the left hand side is defined by
factoring the induced map on kernel relations R[f ′ ] → R[f ] through the direct image
S = l(R[f ′ ]) under l. Since the right square is a pullback, the left square represents
a fibrant split epimorphism of equivalence relations. The factorisation of this fibrant
morphism induced by l yields two fibrant maps of equivalence relations, cf. Lemma
1.17. Note that the second (X ′′ , S) → (X, R[f ]) is a fibrant split epimorphism. Efficient regularity implies then that the equivalence relation S is effective with quotient
f ′′ : U ′′ ։ V ′′ , defining a point (ρ, σ) over Y .
This induces (by a well-known result of Barr and Kock, cf. [4, Lemma A.5.8]) a
decomposition of the right pullback into two pullbacks. The induced regular epimorphism l̄ : (r, s) ։ (ρ, σ) has the required properties, namely f ∗ (l̄) = l.

Proposition 3.8. In an efficiently regular Mal’tsev category with binary sums, a
regular epimorphism f : X ։ Y is an affine extension if and only if for each object
Z either of the following two diagrams
X +O Z
Z
πX

f +Z

ιZ

 X
X

//Y +Z
O
Z
πY

f



ιZ
Y

//Y

is a downward-oriented pullback square.

X +Z

f +Z

/ / Y +Z

θX,Z


X ×Z

θY,Z



f ×Z

/ / Y ×Z

30

CLEMENS BERGER AND DOMINIQUE BOURN

Proof. If f is an affine extension then the downward-oriented left square is a pullback
because the upward-oriented left square is a pushout, cf. Section 3.2. Moreover, the
outer rectangle of the following diagram
f +Z

X +Z

/ Y +Z
θY,Z

θX,Z


X ×Z


/ Y ×Z

f ×Z

pY

pX


X


/Y

f

is a pullback if and only if the upper square is a pullback, because the lower square
is always a pullback.
Assume conversely that the downward oriented left square is a pullback. In a
regular Mal’tsev category, pointed base-change along a regular epimorphism is fully
faithful so that f is affine whenever f ∗ is essentially surjective. Lemma 3.7 shows that
in an efficiently regular Mal’tsev category f ∗ is saturated on quotients. It suffices thus
to show that in the fibre over X each point is the quotient of a point for which the
unit of the pointed base-change adjunction is invertible.
Since for each object Z, the undotted downward-oriented square
X +O Z
Z
πX

f +Z

Z
πY

h1X ,ri

 
X

/ / Y +Z
O
h1Y ,f ri

 
//Y

f

is a pullback, the dotted downward-oriented square (which is induced by an arbitrary
morphism r : Z → X) is a pullback as well. This holds in any regular Mal’tsev
category, since the whole diagram represents a natural transformation of reflexive
graphs, cf. [6]. It follows that the point (h1X , ri, ιZ
X ) : X + Z ⇄ X has an invertible
unit with respect to the pointed base-change adjunction (f! , f ∗ ).
Now, an arbitrary point (r, s) : Z ⇄ X can be realised as a quotient
Z ❅o o
_❅❅❅❅
❅❅❅❅r
❅❅❅❅
s ❅❅❅

hs,1Z i

X; + Z
①①①①
①
①①①①
①①①①
①{①①①① h1X ,ri
ιZ
X

X

of the latter point (h1X , ri, ιZ
X ) : X + Y ⇄ X with invertible unit.



We end this section with several properties of affine extensions in semi-abelian
categories. They will only be used in Section 6.

CENTRAL REFLECTIONS AND NILPOTENCY IN EXACT MAL’TSEV CATEGORIES

31

Proposition 3.9. In a semi-abelian category, a regular epimorphism f : X ։ Y is
an affine extension if and only if either of the following conditions is satisfied:
(a) for each object Z, the induced map f ⋄ Z : X ⋄ Z → Y ⋄ Z is invertible, where
X ⋄ Z stands for the kernel of θX,Z : X + Z ։ X × Z;
(b) every pushout of f along a split monomorphism is a pullback.
Proof. That condition (a) characterises affine extensions follows from Proposition 3.8
and protomodularity. The necessity of condition (b) follows from Section 3.2. The
sufficiency of condition (b) follows from the “pullback cancellation property” in semi
abelian categories, cf. [4, Proposition 4.1.4].
Remark 3.10. This product X ⋄ Z is often called the co-smash product of X and Z,
since it is the dual of the smash product as investigated by Carboni-Janelidze [16] in
the context of lextensive categories. The co-smash product X ⋄ Z coincides in semiabelian categories with the second cross-effect cr2 (X, Z) of the identity functor, cf.
Definition 5.1 and [54, 38, 39]. Since the co-smash product is in general not associative
(cf. [16]), parentheses should be used with care.
Proposition 3.11. Let Y −→ W ←− Z and Ȳ −→ W̄ ←− Z̄ be cospans in the fibre
PtX (D) of a semi-abelian category D. Let f : Y ։ Ȳ , g : Z ։ Z̄, h : W ։ W̄ be
affine extensions in D inducing a map of cospans in PtX (D). Assume furthermore
that the first cospan realises W as the binary sum of Y and Z in PtX (D).
Then the second cospan realises W̄ as the binary sum of Ȳ and Z̄ in PtX (D) if
and only if the kernel cospan K[f ] −→ K[h] ←− K[g] is strongly epimorphic in D.
Proof. Let us consider the following commutative diagram
X /

i

/Y


f

/ / Ȳ


j


Z /
g


Z̄ /



/ W h1 / / W1
❈❈
❈❈h
❈❈
h2
❈! ! 

/ W2
/ / W̄

in which i (resp. j) denotes the section of the point Y (resp. Z) over X, and all
little squares except the lower right one are pushouts. It follows that the outer square
is a pushout (i.e. W̄ = Ȳ +X Z̄) if and only if the lower right square is a pushout.
According to Carboni-Kelly-Pedicchio [17, Theorem 5.2] this happens if and only if
the kernel relation R[h] is the join of the kernel relations R[h1 ] and R[h2 ]. In a semiabelian category this is the case if and only if the kernel K[h] is generated as normal
subobject of W by the kernels K[h1 ] and K[h2 ], resp. (since h1 and h2 are affine
extensions) by the kernels K[f ] and K[g], cf. Proposition 3.9b.
Now, h is also an affine extension so that by Proposition 3.2, the kernel K[h] is a
central subobject of W . In particular, any subobject of K[h] is central and normal
in W (cf. the characterisation of normal subobjects in semi-abelian categories by
Mantovani-Metere [54, Theorem 6.3]). Therefore, generating K[h] as normal subobject of W amounts to the same as generating K[h] as subobject of W .


32

CLEMENS BERGER AND DOMINIQUE BOURN

4. Aspects of nilpotency
Recall that a morphism is called n-fold centrally decomposable if it is the composite
of n morphisms with central kernel relation.
For consistency, a monomorphism is called 0-fold centrally decomposable, and an
isomorphism a 0-fold central extension.
Proposition 4.1. For all objects X, Y of a σ-pointed n-nilpotent Mal’tsev category,
the comparison map θX,Y : X + Y → X × Y is (n − 1)-fold centrally decomposable.
Proof. In a pointed n-nilpotent Mal’tsev category, each object maps to an abelian
group object through an (n − 1)-fold centrally decomposable morphism, cf. Proposition 2.7. Since the codomain of such a morphism φX,Y : X + Y → A is an abelian
group object, the restrictions to the two summands commute and φX,Y factors
θX,Y

X + Y❍
❍❍
❍❍
φX,Y ❍$

A

/ / X ×Y
✈✈
✈
✈ψ
✈
z✈ X,Y

so that θX,Y is (n − 1)-fold centrally decomposable by Lemma 1.3.



Proposition 4.2. For any finitely cocomplete regular pointed Mal’tsev category, the
following pushout square
X +X
h1X ,1X i


X

θX,X

/ / X ×X

/ / A(X)

defines the abelianisation A(X) of X. In particular, the lower row can be identified
1
with the unit ηX
: X → I 1 (X) of the strong epireflection of Theorem 2.12.
Proof. The first assertion follows by combining [4, Proposition 1.7.5, Theorems 1.9.5
and 1.9.11] with the fact that pointed Mal’tsev categories are strongly unital in the
sense of the second author, cf. [4, Corollary 2.2.10]. The second assertion expresses
the fact that X ։ A(X) and X ։ I 1 (X) share the same universal property.

Theorem 4.3. A σ-pointed exact Mal’tsev category is n-nilpotent if and only if for
all objects X, Y the comparison map θX,Y : X + Y → X × Y is an (n − 1)-fold central
extension.
Proof. By Proposition 4.1 n-nilpotency implies that θX,Y is an (n − 1)-fold central
extension. For the converse, consider the pushout square of Proposition 4.2, which is
1
regular by Corollary 1.8. The unit ηX
: X → I 1 (X) is thus an (n − 1)-fold central
extension by Proposition 1.13 so that all objects are n-nilpotent.

Corollary 4.4. For a σ-pointed exact Mal’tsev category, the following three properties
are equivalent:
(a) the category is 1-nilpotent;
(b) the category is linear (cf. Definition 5.1);

CENTRAL REFLECTIONS AND NILPOTENCY IN EXACT MAL’TSEV CATEGORIES

33

(c) the category is abelian.
Proof. The equivalence of (a) and (b) follows from Theorem 4.3. The equivalence of
(b) and (c) follows from the fact that a σ-pointed Mal’tsev category is additive if and
only if it is linear (cf. [4, Theorem 1.10.14]) together with the well-known fact (due
to Miles Tierney) that abelian categories are precisely the additive categories among
exact categories.

Theorem 4.5. For a σ-pointed exact Mal’tsev category, the following five properties
are equivalent:
(a)
(b)
(b′ )
(c)
(c′ )

all objects are 2-nilpotent;
1
for all X, abelianisation ηX
: X → I 1 (X) is a central extension;
1
for all X, abelianisation ηX : X → I 1 (X) is an affine extension;
for all X, Y , the map θX,Y : X + Y → X × Y is a central extension;
for all X, Y , the map θX,Y : X + Y → X × Y is an affine extension.

Proof. Properties (a) and (b) are equivalent by definition of 2-nilpotency. Theorem
4.3 shows that (b) and (c) are equivalent. Theorem 3.3 and Proposition 3.2 imply
that (b) and (b′ ) are equivalent.
Finally, since the first Birkhoff reflection I 1 preserves binary sums and binary
products (cf. Proposition 2.8), we get I 1 (θX,Y ) = θI 1 (X),I 1 (Y ) which is invertible
in the subcategory of 1-nilpotent objects by Proposition 4.1. It follows that under
assumption (a), the map θX,Y is an affine extension by Theorem 3.3, which is property
(c′ ). Conversely, (c′ ) implies (c) by Proposition 3.2.

4.1. Niltensor products. In order to extend Proposition 4.5 to higher n we introduce here a new family of binary tensor products, called niltensor products.
For any finitely cocomplete pointed regular Mal’tsev category (D, ⋆D ) the n-th niltensor product X ⊗n Y is defined by factorizing the comparison map θX,Y universally
n
into a regular epimorphism θX,Y
: X + Y → X ⊗n Y followed by an (n − 1)-fold
central extension
ω n−1

ω2

X,Y
X,Y
/ / X ⊗n−1 Y
X ⊗3 Y❞❞❞❞❞/2 /2 X ⊗2 Y◆
X; ⊗n Y
❞
4
❞
4
◆◆◆ ω1
✇;
n
θX,Y
❞❞❞❞❞❞❞
❥❥❥❥
✇✇
◆◆◆X,Y
❞❞❞❞❞❞❞
n−1
❞
❞
✇✇ ❥❥❥❥❥❥❥θX,Y
❞
❞
◆◆◆
❞
✇
❞
❞
❞
2
❞
✇
❥
❞
❞
❥
❞
θX,Y
◆& &
❥
❞❞❞
✇✇❥
❞❥❞❞❞❞❞❞
/ /X ×Y
X +Y

θX,Y

as provided by Proposition 1.5. This n-th niltensor product is symmetric and has ⋆D
as unit, but it does not seem to be associative in general.

34

CLEMENS BERGER AND DOMINIQUE BOURN

Proposition 4.6. In a σ-pointed exact Mal’tsev category, the following diagram is
an iterated pushout diagram

X: ⊗n X
n
θX,X
✉✉:
✉
✉
✉✉
✉✉
X +X

X●
●●
●●
●●
n
●●# 
ηX
# 
I n (X)

n−1
ωX,X

/ / X ⊗n−1 X

X ⊗3 X

θX,X

ηX


/ / I n−1 (X)


I (X)
3

2
ωX,X

/ / X ⊗2 X
◆◆◆ ω1
◆◆◆X,X
◆◆◆
◆' '
/ / X ×X

/ / I 1 (X)
q8 8
qqq
q
q

qqq
/ / I 2 (X)

where the left vertical map is the folding map h1X , 1X i : X + X → X.
Proof. This follows from Corollary 1.8 and Propositions 1.13 and 4.2.



Theorem 4.7. For a σ-pointed exact Mal’tsev category, the following five properties
are equivalent:
(a)
(b)
(b′ )
(c)
(c′ )

all objects are n-nilpotent;
n−1
for all X, the (n − 1)th unit ηX
: X ։ I n−1 (X) is a central extension;
n−1
for all X, the (n − 1)th unit ηX : X ։ I n−1 (X) is an affine extension;
n−1
for all X, Y , the map θX,Y
: X + Y → X ⊗n−1 Y is a central extension;
n−1
for all X, Y , the map θX,Y : X + Y → X ⊗n−1 Y is an affine extension.

Proof. Properties (a) and (b) are equivalent by definition of n-nilpotency. Theorem
4.3 shows that (a) implies (c) while Proposition 4.6 shows that (c) implies (b). Therefore, (a), (b) and (c) are equivalent. Proposition 3.2 and Theorem 3.3 imply that (b)
and (b′ ) are equivalent.
Finally, Proposition 1.13 implies that the Birkhoff reflection I n−1 takes the comn−1
parison map θX,Y
to the corresponding map θIn−1
n−1 (X),I n−1 (Y ) for the (n − 1)-nilpotent
n−1
n−1
objects I
(X) and I
(Y ). Since the (n − 1)-nilpotent objects form an (n − 1)nilpotent Birkhoff subcategory, Theorem 4.3 shows that the latter map must be invertible; therefore, (a) implies (c′ ) by Theorem 3.3. Conversely, (c′ ) implies (c) by
Proposition 3.2.

Definition 4.8. A σ-pointed Mal’tsev category is said to be pseudo-additive (resp.
n
pseudo-n-additive) if for all X, Y, the map θX,Y : X + Y ։ X × Y (resp. θX,Y
:
X + Y ։ X ⊗n Y ) is an affine extension.

CENTRAL REFLECTIONS AND NILPOTENCY IN EXACT MAL’TSEV CATEGORIES

35

Proposition 4.9. A σ-pointed exact Mal’tsev category is pseudo-additive (i.e. 2nilpotent) if and only if the following diagram
(X + Y ) + Z

θX,Y +Z

/ / (X × Y ) + Z
θX×Y,Z

θX+Y,Z


(X + Y ) × Z



/ / (X × Y ) × Z

θX,Y ×Z

is a pullback for all objects X, Y, Z.
Proof. This follows from Theorem 4.5 and Proposition 3.8.



Proposition 4.10. A σ-pointed exact Mal’tsev category is pseudo-(n − 1)-additive
(i.e. n-nilpotent) if and only if the following diagram
(X + Y ) + Z
n−1
θX,Y
+Z

θX+Y,Z



(X ⊗n−1 Y ) + Z

/ / (X + Y ) × Z


θX⊗n−1 Y,Z

n−1
θX,Y
×Z

/ / (X ⊗n−1 Y ) × Z

is a pullback for all objects X, Y, Z.
Proof. This follows from Theorem 4.7 and Proposition 3.8.



We end this section with a general remark about the behaviour of n-nilpotency
under slicing and passage to the fibres. Note that any left exact functor between
Mal’tsev categories preserves central equivalence relations, morphisms with central
kernel relation, and consequently n-nilpotent objects.
Proposition 4.11. If D is an n-nilpotent Mal’tsev category, then so are any of its
slice categories D/Y and of its fibres PtY (D).
Proof. The slices D/Y of a Mal’tsev category D are again Mal’tsev categories. Moreover, base-change ωY∗ : D → D/Y is a left exact functor so that the objects of D/Y
of the form ωY∗ (X) = pY : Y × X → Y are n-nilpotent provided D is an n-nilpotent
Mal’tsev category. We can conclude with Proposition 2.3 by observing that any object
f : X → Y of D/Y may be considered as a subobject
(f,1X )
/ Y ×X
X ❆/
❆❆
✇✇
❆❆
✇✇
✇
❆
❆ {✇✇✇ pY
f
Y

36

CLEMENS BERGER AND DOMINIQUE BOURN

of ωY∗ (X) in D/Y . The proof for the fibres is the same as for the slices, since any
object (r, s) of PtY (D) may be considered as a subobject
(r,1X )
/ Y ×X
X `❆/
❆❆❆❆ s
①①①①<
①
p
❆❆❆❆
Y ①① ①
①
❆❆❆❆
①①①①①
①
❆
❆
①
①
(1
❆
❆
Y ,s)
r
❆❆❆ |①①①①①
Y

of the projection pY : Y × X → Y splitted by (1Y , s) : Y → Y × X.



5. Quadratic identity functors
We have seen that 1-nilpotency has much to do with linear identity functors (cf.
Corollary 4.4). We now investigate the relationship between 2-nilpotency and quadratic identity functors, and below in Section 6, the relationship between n-nilpotency
and identity functors of degree n. While a linear functor takes binary sums to binary
products, a quadratic functor takes certain cubes constructed out of triple sums to
limit cubes. This is the beginning of a whole hierarchy assigning degree ≤ n to a
functor whenever the functor takes certain (n + 1)-dimensional cubes constructed out
of iterated sums to limit cubes.
This definition of degree of a functor is much inspired by Goodwillie [29] who described polynomial approximations of a homotopy functor in terms of their behaviour
on certain cubical diagrams. Eilenberg-Mac Lane [23] defined the degree of a functor
with values in an abelian category by a vanishing condition of so-called cross-effects.
Our definition of degree does not need cross-effects. Yet, a functor with values in
a semi-abelian (or homological [4]) category is of degree ≤ n precisely when all its
cross-effects of order n + 1 vanish, cf. Corollary 6.17. Our cubical cross-effects agree
up to isomorphism with those of Hartl-Loiseau [38] and Hartl-Van der Linden [39],
which are defined as kernel intersections.
There are several other places in literature where degree n functors, especially quadratic functors, are studied in a non-additive context, most notably Baues-Pirashvili
[1], Johnson-McCarthy [50] and Hartl-Vespa [40]. In all these places, the definition of
a degree n functor is based on a vanishing condition of cross-effects, closely following
the original approach of Eilenberg-Mac Lane. It turned out that for us Goodwillie’s
cubical approach to functor calculus was more convenient. Our Definition 6.3 of an
n-folded object and the resulting characterisation of degree n identity functors in
terms of n-folded objects (cf. Proposition 6.5) rely in an essential way on cubical
combinatorics.
5.1. Degree and cross-effects of a functor. –
An n-cube in a category E is given by a functor Ξ : [0, 1]n → E with domain the
n-fold cartesian product [0, 1]n of the arrow category [0, 1].
The category [0, 1] has two objects 0,1 and exactly one non-identity arrow 0 → 1.
Thus, an n-cube in E is given by objects Ξ(ǫ1 , . . . , ǫn ) in E with ǫi ∈ {0, 1}, and

CENTRAL REFLECTIONS AND NILPOTENCY IN EXACT MAL’TSEV CATEGORIES

37

ǫ′ ,...,ǫ′

arrows ξǫ11,...,ǫnn : Ξ(ǫ1 , . . . , ǫn ) → Ξ(ǫ′1 , . . . , ǫ′n ) in E, one for each arrow in [0, 1]n ,
which compose in an obvious way.
To each n-cube Ξ we associate a punctured n-cube Ξ̌ obtained by restriction of Ξ
to the full subcategory of [0, 1]n spanned by the objects (ǫ1 , . . . , ǫn ) 6= (0, . . . , 0).
Definition 5.1. Let (E, ⋆E ) be a σ-pointed category. For each n-tuple of objects
(X1 , . . . , Xn ) of E we denote by ΞX1 ,...,Xn the following n-cube:
• ΞX1 ,...,Xn (ǫ1 , . . . , ǫn ) = X1 (ǫ1 ) + · · · + Xn (ǫn )
with X(0) = X and X(1) = ⋆E ;
′n
ǫ′
ǫ′ ,...,ǫ′
• ξǫ11,...,ǫnn = jǫ11 + · · · + jǫǫn
′
where jǫǫ is the identity if ǫ = ǫ′ , resp. the null morphism if ǫ 6= ǫ′ .
A functor of σ-pointed categories F : (E, ⋆E ) → (E′ , ⋆E′ ) is called of degree ≤ n if
• F (⋆E ) ∼
= ⋆ E′ ;
• for each (n + 1)-cube ΞX1 ,...,Xn+1 in E, the image-cube F ◦ ΞX1 ,...,Xn+1 is a
limit-cube in E′ , i.e. F (X1 + · · · + Xn+1 ) may be identified with the limit of
the punctured image-cube F ◦ Ξ̌X1 ,...,Xn+1 .
A functor of degree ≤ 1 (resp. ≤ 2) is called linear (resp. quadratic).
A σ-pointed category is called linear (resp. quadratic) if its identity functor is.
If E′ has pullbacks, the limit over the punctured image-cube is denoted
F
=
PX
1 ,...,Xn+1

lim
←−

F ◦ Ξ̌X1 ,...,Xn+1

[0,1]n+1 −(0,...,0)

and the associated comparison map is denoted
F
F
.
: F (X1 + · · · + Xn+1 ) → PX
θX
1 ,...,Xn+1
1 ,...,Xn+1

The (n + 1)-st cross-effects of the functor F : E → E′ are the total kernels of the
F
image-cubes, i.e. the kernels of the comparison maps θX
:
1 ,...,Xn+1
F
F
].
crn+1
(X1 , . . . , Xn+1 ) = K[θX
1 ,...,Xn+1

If F is the identity functor, the symbol F will be dropped from the notation. A
functor F : E → E′ has degree ≤ n if and only if for all (n + 1)-tuples (X1 , . . . , Xn+1 )
F
of objects of E, the comparison maps θX
are invertible.
1 ,...,Xn+1
F
Our total-kernel definition of the cross-effect crn+1
(X1 , . . . , Xn+1 ) is directly inspired by Goodwillie [29, pg. 676] but agrees up to isomorphism with the kernel
intersection definition of Hartl-Loiseau [38] and Hartl-Van der Linden [39]. Their
kernel intersection is dual to the (n + 1)-fold smash product of Carboni-Janelidze
[16], cf. Remark 3.10 and also Remark 6.2, where the duality between cross-effects
and smash-products is discussed in more detail.

Indeed, each of the n + 1 “contraction morphisms”
F
ci + · · · + Xn+1 ), 1 ≤ i ≤ n + 1,
πX̂
: F (X1 + · · · + Xn+1 ) → F (X1 + · · · + X
i

38

CLEMENS BERGER AND DOMINIQUE BOURN

F
factors through θX
so that we get a composite morphism
1 ,...,Xn+1

rF : F (X1 + · · · + Xn+1 )

F
θX
1 ,...,X

−→

n+1

F
֌
PX
1 ,...,Xn+1

n+1
Y
i=1

ci + · · · + Xn+1 )
F (X1 + · · · + X

F
embedding the limit construction PX
into an (n + 1)-fold cartesian product.
1 ,...,Xn+1
F
Therefore, the kernel of θX1 ,...,Xn+1 coincides with the kernel of rF

K[rF ] =

n+1
\

F
K[πX̂
],
i

i=1

which is precisely the kernel intersection of [38, 39] serving as their definition for the
F
(n + 1)st cross-effect crn+1
(X1 , . . . , Xn+1 ) of the functor F .
For n = 1 and F = idE we get the following 2-cube ΞX,Y
X +Y
πŶ


X

πX̂

/Y

/ ⋆E

so that the limit PX,Y of the punctured 2-cube is X × Y and the comparison map
θX,Y : X + Y → X × Y
is the one already used before. In particular, the just introduced notion of linear
category is the usual one.
For n = 2 and F = idE we get the following 3-cube ΞX,Y,Z
π

Ŷ
/: Z
Y
7 +O Z
♥
πX̂ ✈✈ O
♥
♥
✈
♥
✈
✈✈
♥♥♥
πŶ
/
X + YO + Z
X +O Z
π

πX̂

Ẑ

πẐ


X +Y


Y
7
♦
πX̂ ♦♦♦
♦♦
♦
♦
♦
πŶ


/ ⋆E
:
✈
πẐ
✈✈✈
 ✈✈✈
/X

which induces a split natural transformation of 2-cubes:
ΞX,Y + Z ⇄ ΞX,Y
For sake of simplicity, we denote by +Z the functor E → PtZ (E) which takes an
object X to X + Z → Z with obvious section, and similarly, we denote by ×Z the
functor E → PtZ (E), which takes an object X to X × Z → Z with obvious section.

CENTRAL REFLECTIONS AND NILPOTENCY IN EXACT MAL’TSEV CATEGORIES

39

The previous split natural transformation of 2-cubes induces a natural transformation of split epimorphisms
X +YO +Z
πẐ


X +Y

+Z
θX,Y

'&%$
!"#
2
θX,Y

/ (X + Z) ×Z (Y + Z)
O

/ X ×Y

the comparison map of which may be identified with the comparison map
θX,Y,Z : X + Y + Z → PX,Y,Z
of ΞX,Y,Z . In particular, the category E is quadratic if and only if square (2) is a
pullback square. Notice that in a regular Mal’tsev category, the downward-oriented
square (2) is necessarily a regular pushout by Corollary 1.7.
Proposition 5.2. In a σ-pointed regular Mal’tsev category, the comparison map
θX,Y,Z is a regular epimorphism with kernel relation R[πX̂ ] ∩ R[πŶ ] ∩ R[πẐ ].
Proof. The first assertion expresses the regularity of pushout (2), the second follows
+Z
+Z
from identities R[θX,Y
] ∩ R[πẐ ] which hold
] = R[πX̂ ] ∩ R[πŶ ] and R[θX,Y,Z ] = R[θX,Y
+Z
because both, θX,Y and θX,Y,Z , are comparison maps.

Lemma 5.3. A σ-pointed category with pullbacks is quadratic if and only if square
(2 ′ ) of the following diagram
X +Y +Z
+Z
θX,Y


(X + Z) ×Z (Y + Z)

θX+Y,Z

/.-,
()*+
2′

/ (X + Y ) × Z

/ X +Y

×Z
θX,Y

θX,Z ×Z θY,Z


/ (X × Z) ×Z (Y × Z)

7654
0123
2′′

θX,Y


/ X ×Y

is a pullback square.
Proof. Composing squares (2′ ) and (2′′ ) yields square (2) above. Square (2′′ ) is a
pullback since (X × Z) ×Z (Y × Z) is canonically isomorphic to (X × Y ) × Z.

5.2. The main diagram. We shall now give several criteria for quadraticity. For
this we consider the following diagram
(X + Y ) ⋄ Z /
θX,Y ⋄Z


(X × Y ) ⋄ Z /
ϕZ
X,Y


(X ⋄ Z) × (Y ⋄ Z) /

θX+Y,Z

/ (X + Y ) + Z
θX,Y +Z


/ (X × Y ) + Z
φZ

/ / (X + Y ) × Z

'&%$
!"#
a

θX,Y ×Z


/ / (X × Y ) × Z

θX×Y,Z

 X,Y
/ (X + Z) ×Z (Y + Z)

'&%$
!"#
b

µZ

 X,Y
/ / (X × Z) ×Z (Y × Z)

θX,Z ×Z θY,Z

+Z
×Z
⋄Z
in which the vertical composite morphisms from left to right are θX,Y
, θX,Y
, θX,Y
, the
horizontal morphisms on the left are the kernel-inclusions of the horizontal regular
epimorphisms on their right, and µZ
X,Y is the canonical isomorphism.

40

CLEMENS BERGER AND DOMINIQUE BOURN

Observe that square (b) is a pullback if and only if the canonical map
φZ
X,Y : (X × Y ) + Z → (X + Z) ×Z (Y + Z)
is invertible. This is the case if and only if for each Z pointed cobase-change
(αZ )! : D → PtZ (D)
along the initial map αZ : ⋆D → Z preserves binary products, cf. Section 3.2.
Proposition 5.4. A σ-pointed regular Mal’tsev category is quadratic if and only if
squares (a) and (b) of the main diagram are pullback squares.
Proof. Since composing squares (a) and (b) yields square (2′ ) of Lemma 5.3, the
condition is sufficient. Lemma 1.17 and Corollary 1.16 imply that the condition is
necessary as well.

Theorem 5.5. A σ-pointed exact Mal’tsev category is quadratic if and only if it is
2-nilpotent and pointed cobase-change along initial maps preserves binary products.
Proof. By Proposition 5.4, the category is quadratic if and only if the squares (a)
and (b) are pullback squares, i.e. the category is 2-nilpotent by Proposition 4.9, and
pointed cobase-change along initial maps preserves binary products.

Corollary 5.6. A semi-abelian category is quadratic1 if and only if either of the
following three equivalent conditions is satisfied:
(a) all objects are 2-nilpotent, and the comparison maps
ϕZ
X,Y : (X × Y ) ⋄ Z → (X ⋄ Z) × (Y ⋄ Z)
are invertible for all objects X, Y, Z;
(b) the third cross-effects of the identity functor cr3 (X, Y, Z) = K[θX,Y,Z ] vanish
for all objects X, Y, Z;
(c) the co-smash product is linear, i.e. the canonical comparison maps
⋄Z
θX,Y
: (X + Y ) ⋄ Z → (X ⋄ Z) × (Y ⋄ Z)

are invertible for all objects X, Y, Z.
Proof. Theorem 5.5 shows that condition (a) amounts to quadraticity.
For condition (b) note that by protomodularity the cross-effect K[θX,Y,Z ] vanishes
if and only if the regular epimorphism θX,Y,Z is invertible.
The equivalence of conditions (b) and (c) follows from the isomorphism of kernels
⋄Z
K[θX,Y,Z ] ∼
]. The latter is a consequence of the 3 × 3-lemma which, applied
= K[θX,Y
to main diagram 5.2 and square (2), yields the chain of isomorphisms
K[θ⋄Z ] ∼
= K[K[θ+Z ] ։ K[θ×Z ]] ∼
= K[θX,Y,Z ].
X,Y

X,Y

X,Y


1The authors of [20, 38, 39] call a semi-abelian category two-nilpotent if each object has a vanishing
ternary Higgins commutator, cf. Remark 6.4. By Proposition 6.5 this means that the identity functor
is quadratic. Corollaries 5.6 and 5.16 describe how to enhance a 2-nilpotent semi-abelian category
in our sense so as to get a two-nilpotent semi-abelian category in their sense.

CENTRAL REFLECTIONS AND NILPOTENCY IN EXACT MAL’TSEV CATEGORIES

41

5.3. Algebraic distributivity and algebraic extensivity. –
We shall see that in a pseudo-additive regular Mal’tsev category (D, ⋆D ), pointed
cobase-change along initial maps αZ : ⋆D → Z preserves binary products if and only
if pointed base-change along terminal maps ωZ : Z → ⋆D preserves binary sums. The
latter condition means that for all objects X, Y, Z, the following square
Z

/ Y ×Z


X ×Z


/ (X + Y ) × Z

is a pushout, inducing thus for all objects X, Y, Z an isomorphism
(X × Z) +Z (Y × Z) ∼
= (X + Y ) × Z
which can be considered as an algebraic distributivity law. This suggests the following definitions, where the added adjective “algebraic” means here that the familiar
definition has to be modified by replacing the slices of the category with the fibres of
the fibration of points, cf. Section 3.2 and Carboni-Lack-Walters [18].
Definition 5.7. A category with pullbacks of split epimorphisms is algebraically distributive if pointed base-change along terminal maps preserves binary sums.
A category with pullbacks of split epimorphisms and pushouts of split monomorphisms is algebraically extensive if any pointed base-change preserves binary sums.
We get the following implications between several in literature studied “algebraic”
notions, where we assume that pullbacks of split epimorphisms and (whenever needed)
pushouts of split monomorphisms exist:
(5.11)

local alg. cartesian closure

+3 alg. extensivity

(5.8)

+3 alg. coherence

(5.9)
(5.9)
qy


+3 alg. distributivity
protomodularity


alg. cartesian
T\ closure
(5.11)

The existence of centralisers implies algebraic cartesian closure [13] and hence
algebraic distributivity, cf. Section 1.5. The categories of groups and of Lie algebras
are not only algebraically cartesian closed, but also locally algebraically cartesian
closed [35, 36], which means that any pointed base-change admits a right adjoint.
Algebraic coherence, cf. Cigoli-Gray-Van der Linden [20], requires any pointed basechange to be coherent, i.e. to preserve strongly epimorphic cospans.
Lemma 5.8. An algebraically extensive regular category is algebraically coherent.
Proof. In a regular category, pointed base-change preserves regular epimorphisms.
Henceforth, if the fibres have binary sums and pointed base-change preserves them,
pointed base-change also preserves strongly epimorphic cospans.


42

CLEMENS BERGER AND DOMINIQUE BOURN

Lemma 5.9 (cf. [10], Theorem 3.10 and [20], Theorem 6.1). An algebraically coherent
pointed Mal’tsev category is protomodular and algebraically distributive.
Proof. To any split epimorphism (r, s) : Y ⇄ X we associate the split epimorphism
(r̄ = r × 1X , s̄ = s × 1X ) : Y × X ⇄ X × X
r̄

Y × Xc● o
●●●●
●●●●(s,1X )
●●●●
●●
p2 ●●●●●
●●
#

/

X; × X
①①①①
①
①
①①①①①
①①①①①p2
①
①{①①①①

s̄
(1X ,1X )

X
in the fibre over X. The kernel of (r̄, s̄) may be identified with the given point
(r, s) over X where the kernel-inclusion is defined by (1Y , r) : Y ֌ Y × X. Kernelinclusion and section strongly generate the point Y × X over X, cf. [10, Proposition
3.7]. Pointed base-change along αX : ⋆ → X takes (r̄, s̄) back to (r, s), so that by
algebraic coherence, section and kernel-inclusion of (r, s) strongly generate Y . In a
pointed category this amounts to protomodularity.
For the second assertion observe that if F and G are composable coherent functors
such that G is conservative and GF preserves binary sums, then F preserves binary
sums as well; indeed, the isomorphism GF (X) + GF (Y ) → GF (X + Y ) decomposes
into two isomorphisms GF (X) + GF (Y ) → G(F (X) + F (Y )) → GF (X + Y ). This
∗
applies to F = ωZ
and G = α∗Z (where αZ : ⋆ → Z and ωZ : Z → ⋆) because
∗
∗
preserves binary sums for all Z. 
ωZ αZ = idZ and αZ is conservative, so that ωZ
Lemma 5.10. A σ-pointed exact Mal’tsev category is algebraically extensive if and
only if it is a semi-abelian category with exact pointed base-change functors.
Proof. This follows from Lemmas 5.8 and 5.9 and the fact that a left exact and regular
epimorphism preserving functor between semi-abelian categories is right exact if and

only if it preserves binary sums, cf. Section 1.4.
For the following lemma a variety means a category equipped with a forgetful
functor to sets which is monadic and preserves filtered colimits. Every variety is
bicomplete, cowellpowered, and has finite limits commuting with filtered colimits.
Lemma 5.11 (cf. [35], Theorem 2.9). A semi-abelian variety is (locally) algebraically
cartesian closed if and only if it is algebraically distributive (extensive).
Proof. Since the fibres of a semi-abelian category are semi-abelian, the pointed basechange functors preserve binary sums if and only if they preserve finite colimits, cf.
Section 1.4. Since any colimit is a filtered colimit of finite colimits, and pointed basechange functors of a variety preserve filtered colimits, they preserve binary sums if and
only if they preserve all colimits. It follows then from Freyd’s special adjoint functor
theorem that a pointed base-change functor of a semi-abelian variety preserves binary
sums if and only if it has a right adjoint.

A pointed category D with pullbacks is called fibrewise algebraically cartesian
closed (resp. distributive) if for all objects Z of D the fibres PtZ (D) are algebraically

CENTRAL REFLECTIONS AND NILPOTENCY IN EXACT MAL’TSEV CATEGORIES

43

cartesian closed (resp. distributive). This is the case if and only if pointed base-change
along every split epimorphism has a right adjoint (resp. preserves binary sums). Any
algebraically coherent pointed Mal’tsev category is fibrewise algebraically distributive,
cf. the proof of Lemma 5.9.
Proposition 5.12. For a pointed regular Mal’tsev category, fibrewise algebraic cartesian closure is preserved under strong epireflections.
Proof. Let (r, s) : X ⇄ Y be a point in a strongly epireflective subcategory C of a
fibrewise algebraically cartesian closed regular Mal’tsev category D. Let f : Y → Z
be a split epimorphism in C. We shall denote f∗ : PtY (D) → PtZ (D) the right adjoint
of pointed base-change f ∗ : PtZ (D) → PtY (D). Consider the following diagram
φ̄

f¯
¯
/ X̄
X̄
O ❈❈
O ❈o ❈
❈❈ ηX̄¯
❈❈ ηX̄
❈❈
❈❈
r̄
s̄
r̄¯ s̄¯
❈! !
❈! !
¯)
I(ε)
I(
f
¯
o
/
X
I(X̄ )
I(X̄)
③
③
⑧⑧
③
③
⑧
③③ I(r̄¯)
③③
⑧⑧
 }③③③ I(r̄)
 }③③③
 ⑧⑧ r
f
/Z
Y
Y o

XO ❂❂o
❂❂❂❂
❂
r
s ❂❂❂❂
❂❂❂

ε

where (r̄, s̄) = f∗ (r, s), the downward-oriented right square is a pullback, and ε :
(r̄¯, s̄¯) = f ∗ f∗ (r, s) → (r, s) is the counit at (r, s). Since D is a regular Mal’tsev
category, the strong epireflection I preserves this pullback of split epimorphisms (cf.
¯ ) is isomorphic to f ∗ (I(X̄)). Since by adjunction, maps
Proposition 2.8) so that I(X̄
¯ ) = f ∗ (I(X̄)) → X there is a
I(X̄) → f∗ (X) = X̄ correspond bijectively to maps I(X̄
∗
unique dotted map φ̄ : I(X̄) → X̄ such that ε ◦ f (φ̄) = I(ε).
Accordingly we get φ̄ηX̄ = 1X̄ so that ηX̄ is invertible and hence X̄ belongs to C.
This shows that the right adjoint f∗ : PtY (D) → PtZ (D) restricts to a right adjoint
f∗ : PtY (C) → PtZ (C) so that C is fibrewise algebraically cartesian closed.

For regular Mal’tsev categories, algebraic cartesian closure amounts to the existence
of centralisers for all (split) subobjects, see Section 1.5. Part of Proposition 5.12 could
thus be reformulated by saying that in this context the existence of centralisers is
preserved under strong epireflections, which can also be proved directly. In a varietal
context, Proposition 5.12 also follows from Lemmas 5.11 and 5.17.
Lemma 5.13. If an algebraically extensive semi-abelian (or homological [4]) category
D has an identity functor of degree ≤ n, then all its fibres PtZ (D) as well.
Proof. The kernel functors (αZ )∗ : PtZ (D) → D are conservative and preserve binary
Pt (D)
sums. Therefore, the kernel functors preserve the limits PX1Z,...,Xn+1 and the comparPt (D)

ison maps θX1Z,...,Xn+1 . Accordingly, if the identity functor of D is of degree ≤ n, then
Pt (D)

α∗Z (θX1 ,...,Xn+1 ) is invertible, hence so is θX1Z,...,Xn+1 , for all objects X1 , . . . , Xn+1 of
the fibre PtZ (D).


44

CLEMENS BERGER AND DOMINIQUE BOURN

It should be noted that in general neither algebraic extensivity nor local algebraic
cartesian closure is preserved under Birkhoff reflection. This is in neat contrast to
(fibrewise) algebraic distributivity and algebraic coherence, which are preserved under
strong epireflections, cf. Proposition 5.12 and [20, Proposition 3.7].
5.4. Duality for pseudo-additive regular Mal’tsev categories.
Lemma 5.14. For any pointed category D with binary sums and binary products
consider the following commutative diagram
(X × Y ) + Z
O
Y
jX
+Z

ρX,Y,Z

pY
X +Z


X +Z

/ X × (Y + Z)
O
X×ιY
Z

Y
X×πZ


/ X ×Z

θX,Z

Y
in which ρX,Y,Z is induced by the pair X × ιZ
Y : X × Y → X × (Y + Z) and αX × ιZ :
Z → X × (Y + Z).
(1) pointed base-change (ωX )∗ : D → PtX (D) preserves binary sums if and only
if the upward-oriented square is a pushout for all objects Y, Z;
(2) pointed cobase-change (αZ )! : PtZ (D) → D preserves binary products if and
only if the downward-oriented square is a pullback for all objects X, Y .

Proof. The left upward-oriented square of the following diagram
X×ιZ
Y

Y
jX

/

/ (X × Y ) + Z
O
ιZ
X×Y

X O× Y

Y
jX
+Z

pY
X


X

ιZ
X

ρX,Y,Z

pY
X +Z

/ X × (Y + Z)
O
X×ιY
Z


/X +Z

θX,Z

/

Z
jX

Y
X×πZ


/ X ×Z

is a pushout so that the whole upward-oriented rectangle is a pushout (i.e. (ωX )∗
preserves binary sums) if and only if the right upward-oriented square is a pushout.
The right downward-oriented square of the following diagram
pX
Y +Z

(X × Y ) + Z
O
Y
jX
+Z

/
+ Z) X
ρX,Y,Z X × (Y
O
pY +Z

pY
X +Z


X +Z

X×ιY
Z
θX,Z

Y
X×πZ


/ X ×Z
X
πZ

/
/ Y +Z
O
ιY
Z

pX
Z

Y
πZ


/Z
/

is a pullback so that the whole downward-oriented rectangle is a pullback (i.e. (αZ )!
preserves binary products) if and only if the left downward-oriented square is a pullback.


CENTRAL REFLECTIONS AND NILPOTENCY IN EXACT MAL’TSEV CATEGORIES

45

Proposition 5.15. In a σ-pointed regular Mal’tsev category D, pointed base-change
(ωZ )∗ : D → PtZ (D) preserves binary sums for all objects Z as soon as pointed cobasechange (αZ )! : D → PtZ (D) preserves binary products for all objects Z. The converse
implication holds if D is pseudo-additive (cf. Definition 4.8).
Proof. According to the previous lemma pointed cobase-change (αX )! preserves binary products if and only if the downward-oriented square is a pullback which implies
that the upward-oriented square is a pushout, and hence pointed base-change (ωZ )∗
preserves binary sums. If D is pseudo-additive, the comparison map θX,Z is an affine
extension. Therefore, the downward-oriented square is a pullback if and only if the
upward-oriented square is a pushout, whence the converse.

Corollary 5.16. A σ-pointed exact Mal’tsev category is quadratic if and only if it is
2-nilpotent and algebraically distributive.
Lemma 5.17. In a σ-pointed regular Mal’tsev category, (fibrewise) algebraic distributivity is preserved under strong epireflections.
Proof. This follows from Proposition 2.8 which shows that strong epireflections preserve besides pushouts and binary sums also binary products (in the fibres).

Theorem 5.18. For any algebraically distributive, σ-pointed exact Mal’tsev category,
the Birkhoff subcategory of 2-nilpotent objects is quadratic.
Proof. The Birkhoff subcategory is pointed, exact, 2-nilpotent, and algebraically distributive by Lemma 5.17, and hence quadratic by Corollary 5.16.

Corollary 5.19 (cf. Cigoli-Gray-Van der Linden [20], Corollary 7.2). For each object
X of an algebraically distributive semi-abelian category, the iterated Huq commutator
[X, [X, X]] coincides with the ternary Higgins commutator [X, X, X].
Proof. Recall (cf. [38, 39]) that [X, X, X] is the direct image of K[θX,X,X ] under the
ternary folding map X + X + X → X. In general, the iterated Huq commutator
[X, [X, X]] is contained in [X, X, X], cf. Corollary 6.12. In a semi-abelian category,
2
the unit of second Birkhoff reflection I 2 takes the form ηX
: X → X/[X, [X, X]],
cf. Remark 2.13. Since in the algebraically distributive case, the subcategory of 2nilpotent objects is quadratic by Theorem 5.18, the image of [X, X, X] in X/[X, [X, X]]
is trivial by Corollaries 5.6b and 6.20, whence [X, [X, X]] = [X, X, X].

Remark 5.20. The category of groups (resp. Lie algebras) has centralisers for subobjects and is thus algebraically distributive. Therefore, the category of 2-nilpotent
groups (resp. Lie algebras) is a quadratic semi-abelian variety.
The reader should observe that although on the level of 2-nilpotent objects there is a
perfect symmetry between the property that pointed base-change along terminal maps
preserves binary sums and the property that pointed cobase-change along initial maps
preserves binary products (cf. Proposition 5.15), only the algebraic distributivity
carries over to the category of all groups (resp. Lie algebras) while the algebraic
“codistributivity” fails in either of these categories. “Codistributivity” is a quite
restrictive property, which is rarely satisfied without assuming 2-nilpotency.

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CLEMENS BERGER AND DOMINIQUE BOURN

6. Identity functors with bounded degree
In the previous section we have seen that quadraticity is a slightly stronger property
than 2-nilpotency, insofar as it also requires a certain compatibility between binary
sum and binary product (cf. Theorem 5.5 and Proposition 5.15). In this last section,
we relate n-nilpotency to identity functors of degree ≤ n.
6.1. Degree n functors and n-folded objects. –
Any (n + 1)-cube ΞX1 ,...,Xn+1 (cf. Definition 5.1) defines a split natural transformation of n-cubes inducing a natural transformation of split epimorphisms
X1 + · · · + Xn+1
O
πX̂

n+1


X1 + · · · + Xn

+Xn+1
1 ,...,Xn

θX

/ P +Xn+1
X1 ,...,Xn
O

+ωX
n+1
1 ,...,Xn

'&%$
!"#
n

PX
θX1 ,...,Xn

+αX
n+1
1 ,...,Xn

PX


/ PX1 ,...,Xn

the comparison map of which may be identified with the comparison map
θX1 ,...,Xn+1 : X1 + · · · + Xn+1 → PX1 ,...,Xn+1
of the given (n + 1)-cube. In particular, our category has an identity functor of degree
≤ n if and only if square (n) is a pullback square for all objects X1 , . . . , Xn+1 .
Proposition 6.1. In a σ-pointed regular Mal’tsev category, the comparison map
θX1 ,...,Xn+1 is a regular epimorphism with kernel relation R[πX̂1 ] ∩ · · · ∩ R[πX̂n+1 ].
Proof. This follows by induction on n like in the proof of Proposition 5.2.



Remark 6.2. The previous proposition shows that in a σ-pointed regular Mal’tsev
category, the intersection of the kernel relations of the contraction maps may be considered as the “total kernel relation” of the cube. This parallels the more elementary
fact that the total-kernel definition of the cross-effects crn+1 (X, . . . , Xn+1 ) coincides
with the kernel-intersection definition of Hartl-Loiseau [38] and Hartl-Van der Linden
[39]. In particular, in any σ-pointed regular Mal’tsev category, the image of the morQn+1
ci + · · · + Xn+1 coincides with the
phism rid : X1 + · · · + Xn+1 → i=1 X1 + · · · + X
limit PX1 ,...,Xn+1 of the punctured (n + 1)-cube.
We already mentioned that these kernel intersections are dual to the (n + 1)fold smash products of Carboni-Janelidze [16]. An alternative way to describe the
duality between cross-effects and smash-products is to consider the limit construction
PX1 ,...,Xn as the dual of the so-called fat wedge T X1 ,...,Xn , cf. Hovey [42]. Settheoretically, the fat wedge is the subobject of the product X1 × · · · × Xn formed by
the n-tuples having at least one coordinate at a base-point. If base-point inclusions
behave “well” with respect to cartesian product, the fat wedge is given by a colimit
construction, strictly dual to the limit construction defining PX1 ,...,Xn . The n-fold
smash-product X1 ∧ · · · ∧ Xn is then the cokernel of the monomorphism T X1 ,...,Xn ֌
X1 × · · · × Xn while the n-th cross-effect crn (X1 , . . . , Xn ) is the kernel of the regular
epimorphism X1 + · · · + Xn ։ PX1 ,...,Xn .

CENTRAL REFLECTIONS AND NILPOTENCY IN EXACT MAL’TSEV CATEGORIES

47

The cubical cross-effects are just the algebraic version of Goodwillie’s homotopical cross-effects [29, pg. 676]. Nevertheless, for functors taking values in abelian
categories, the cubical cross-effects agree with the original cross-effects of EilenbergMac Lane [23, pg. 77]. Indeed, by [23, Theorems 9.1 and 9.6], for a based functor
F : D → E from a σ-pointed category (D, +, ⋆D ) to an abelian category (E, ⊕, 0E ), the
latter are completely determined by the following decomposition formula
M
M
F (X1 + · · · + Xn ) ∼
crkF (Xi1 , . . . , Xik )
=
1≤k≤n 1≤i1 <···<ik ≤n

for any objects X1 , . . . , Xn in D. It suffices thus to show that the cubical cross-effects
satisfy the decomposition formula if values are taken in an abelian category.
F
F
For n = 2 we get PX
= F (X1 ) ⊕ F (X2 ) from which it follows that θX
:
1 ,X2
1 ,X2
F (X1 +X2 ) ։ F (X1 )⊕F (X2 ) is a split epimorphism. Henceforth, we get the asserted
isomorphism F (X1 + X2 ) ∼
= F (X1 ) ⊕ F (X2 ) ⊕ cr2F (X1 , X2 ).
The 3-cube F (ΞX1 ,X2 ,X3 ) induces a natural transformation of split epimorphisms
F (X1 + X2 + X3 )
O

/ P3
O


F (X1 + X2 )


/ / F (X1 ) ⊕ F (X2 )

F
θX
1 ,X2

in which P3 is isomorphic to F (X1 ) ⊕ F (X2 ) ⊕ F (X3 ) ⊕ cr2F (X1 , X3 ) ⊕ cr2F (X2 , X3 ).
F
From this, we get for PX
= F (X1 + X2 ) ×F (X1 )⊕F (X2 ) P3 the formula
1 ,X2 ,X3
F
∼
PX
= F (X1 ) ⊕ F (X2 ) ⊕ F (X3 ) ⊕ cr2F (X1 , X2 ) ⊕ cr2F (X1 , X3 ) ⊕ cr2F (X2 , X3 )
1 ,X2 ,X3
F
so that θX
is again a split epimorphism inducing the asserted decomposition
1 ,X2 ,X3
of F (X1 + X2 + X3 ). The same scheme keeps on for all positive integers n.


Definition 6.3. An object X of a σ-pointed regular Mal’tsev category will be called
X
n-folded if the folding map δn+1
factors through the comparison map θX,...,X
θX,...,X

X + · · · ❉+ X
❉❉
❉❉
❉❉
X
δn+1 ❉❉!
!

/ / PX,...,X
mX

X

X
i.e. if the folding map δn+1
annihilates the kernel relation R[θX,...,X ].

An object X is 1-folded if and only if the identity of X commutes with itself. In
a σ-pointed regular Mal’tsev category this is the case if and only if X is an abelian
group object, cf. the proof of Proposition 4.2.
Remark 6.4. In a semi-abelian (or homological [4]) category, an object X is n-folded
X
if and only if the image of the kernel K[θX,...,X ] under the folding map δn+1
:X+
· · · + X ։ X is trivial. Recall [38, 39, 41, 54] that this image is by definition the
so-called Higgins commutator [X, . . . , X] of length n + 1. Therefore, an object of a

48

CLEMENS BERGER AND DOMINIQUE BOURN

semi-abelian category is n-folded precisely when its Higgins commutator of length
n + 1 vanishes. Under this form n-folded objects have already been studied by Hartl
emphasising their role in his theory of polynomial approximation.
In a varietal context, n-foldedness can be expressed in more combinatorial terms.
For instance, a group X is n-folded if and only if (n + 1)-reducible elements of X are
trivial. An element w ∈ X is called (n + 1)-reducible if there is an element v in the
free group F (X ⊔ · · · ⊔ X) on n + 1 copies of X (viewed as a set) such that
(a) w is the image of v under the composite map
n+1

n+1

n+1

X
}|
{
z
z
z
}|
{
}|
{ δn+1
F (X ⊔ · · · ⊔ X) ∼
= F (X) + · · · + F (X) ։ X + · · · + X ։ X

F (X)

: F (X)+(n+1) ։ F (X)+n , cf. Section
(b) for each of the n + 1 contractions πi
F (X)
(v) maps to the neutral element of X under
6.2, the image πi
n

n

z
}|
{
}|
{ δnX
z
F (X) + · · · + F (X) ։ X + · · · + X ։ X.
Indeed, since the evaluation map F (X) ։ X is a regular epimorphism, and the evaluation maps in (a) and (b) are compatible with the contraction maps πi : X +(n+1) ։
X +n , Proposition 6.1, Section 6.2 and Corollary 6.20 imply that we get in this way
X
the image of the kernel K[θX,...,X ] under the folding map δn+1
.
Qk
Qk
−1 −1
Any product of commutators i=1 [xi , yi ] = i=1 xi yi xi yi in X is 2-reducible
by letting the xi (resp. yi ) belong to the first (resp. second) copy of X. Conversely,
a direct computation shows that any 2-reducible element of X can be rewritten as a
product of commutators of X. This recovers in a combinatorial way the aforementioned fact that X is abelian (i.e. 1-nilpotent) if and only if X is 1-folded.
The relationship between n-nilpotency and n-foldedness is more subtle, closely
related to the cross-effects of the identity functor (cf. Theorem 6.8). For groups and
Lie algebras the two concepts coincide (cf. Theorem 6.23c) but, though any n-folded
object is n-nilpotent, the converse is wrong in general (cf. Section 6.5).
Proposition 6.5. Let F : D → E be a based functor between σ-pointed categories and
assume that E is a regular Mal’tsev category.
(a) If F is of degree ≤ n then F takes values in n-folded objects of E;
(b) If F preserves binary sums and takes values in n-folded objects of E then F
is of degree ≤ n;
(c) The identity functor of E is of degree ≤ n if and only if all objects of E are
n-folded.
F (X)

Proof. Clearly, (c) follows from (a) and (b). For (a) note that δn+1 factors through
X
F (δn+1
), and that by definition of a functor of degree ≤ n, the comparison map
F
X
θX,...,X is invertible so that F (δn+1
) gets identified with mF (X) .
For (b) observe first that preservation of binary sums yields the isomorphisms
F
F
∼
∼
PX
= θF (X1 ),...,F (Xn+1 ) . We shall show
= PF (X1 ),...,F (Xn+1 ) and θX
1 ,...,Xn+1
1 ,...,Xn+1
F
that if moreover F takes values in n-folded objects of E then θX
is invertible
1 ,...,Xn+1
for all (n + 1)-tuples (X1 , . . . , Xn+1 ) of objects of D.

CENTRAL REFLECTIONS AND NILPOTENCY IN EXACT MAL’TSEV CATEGORIES

49

T
Consider any family (fi : Xi → T )1≤i≤n+1 of morphisms in E, and let φ = δn+1
◦
(f1 + · · · + fn+1 ) : X1 + · · · + Xn+1 → T be the induced map. We have the following
F
factorisation of F (φ) through θX
:
1 ,...,Xn+1
θF (X1 ),...,F (Xn+1 )

F (X1 ) + · · · + F (Xn+1 )

/ / PF (X ),...,F (X )
1
n+1
PF (f1 ),...,F (fn+1 )

F (f1 )+···+F (fn+1 )



θF (T ),...,F (T )
/ / PF (T ),...,F (T )
F (T ) + · · · + F (T )
◆◆◆
t
◆◆◆
tt
◆◆◆
tt
t
T
◆◆◆
tt mF (T )
δn+1
ztt
&
F (T )
In particular, if T = X1 + · · · + Xn+1 and fi is the inclusion of the ith summand,
we get a retraction of θF (X1 ),...,F (Xn+1 ) which accordingly is a monomorphism. Since
θF (X1 ),...,F (Xn+1 ) is also a regular epimorphism, it is invertible.

Proposition 6.6. The full subcategory Fldn (E) of n-folded objects of a σ-pointed
regular Mal’tsev category E is closed under products, subobjects and quotients.
Proof. For any two n-folded objects X and Y the following diagram
(X × Y ) + · · · + (X × Y )

θX×Y,...,X×Y

/ / PX×Y,...,X×Y



θX,...,X ×θY,...,Y
/ / PX,...,X × PY,...,Y
(X + · · · + X) × (Y + · · · + Y )
❘❘❘
r
❘❘❘
rr
❘❘❘
rr
r
❘
r
❘
X
Y
❘❘❘
r m ×m
δn+1
×δn+1
(
yrr X Y
X ×Y
X×Y
induces the required factorisation of δn+1
through θX×Y,...,X×Y .
For a subobject n : U ֌ X of an n-folded object X consider the diagram

U + · · · + U ❩❩❩❩❩
❩❩❩❩❩θ❩U,···
❩❩❩,U❩❩❩❩❩
θ
❩❩❩❩❩❩❩- '
ν
/ PU,...,U
W /
U
δn+1

 
U /

Pn,...,n
n+···+n

n


(
θX,...,X
/ / PX,...,X
X + ···+ X
X
✐✐✐✐
δn+1
 t✐✐✐✐✐✐✐mX
/X

in which the dotted quadrangle is a pullback. The commutatitvity of the diagram
induces a morphism θ such that νθ = θU,...,U . Since θU,...,U is a regular epimorphism,
the monomorphism ν is invertible, whence the desired factorisation.

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CLEMENS BERGER AND DOMINIQUE BOURN

Finally, for a regular epimorphism f : X ։ Y with n-folded domain X consider
the following diagram
f +···+f

X + · · · +❖ X
❖❖❖θX,...,X
❖❖❖
❖❖❖
''
X
δn+1
PX,...,X
♦
♦
♦
♦♦♦
 w♦♦♦♦♦ mX
X
f

/ / Y + ··· + Y
❖❖❖
❖❖θ❖Y,...,Y
❖❖❖
Y
δn+1
❖' '
Pf,...,f
/ / PY,...,Y

//Y w

mY

in which the existence of the dotted arrow has to be shown. According to Lemma 6.7
the induced morphism on kernel relations
R(f + · · · + f, Pf,...,f ) : R[θX,...,X ] → R[θY,...,Y ]
Y
is a regular epimorphism. A diagram chase shows then that δn+1
annihilates R[θY,...,Y ]
Y
whence the required factorisation of δn+1 through θY,...,Y .


Lemma 6.7. In a σ-pointed regular Mal’tsev category, any finite family of regular
epimorphisms fi : Xi ։ Yi (i = 1, . . . , n) induces a regular epimorphism on kernel
relations R(f1 + · · · + fn , Pf1 ,...,fn ) : R[θX1 ,...,Xn ] ։ R[θY1 ,...,Yn ].
Proof. Since regular epimorphisms compose (in any regular category) it suffices to
establish the assertion under the assumption fi = 1Xi for i = 2, . . . , n. Moreover we
can argue by induction on n since for n = 1 the comparison map is the terminal map
θX : X ։ ⋆ and a binary product of regular epimorphisms is a regular epimorphism.
Assume now that the statement is proved for n−1 morphisms. Using the isomorphism
of kernel relations
∼ R[R[θ+Xn
R[θX ,...,X ,X ] =
] ։ R[θX ,...,X ]]
1

n−1

n

X1 ,...,Xn−1

1

n−1

and Propositon 1.6 it suffices then to show that the following by f1 : X1 ։ Y1 induced
commutative square
+Xn
R[θX
]
1 ,X2 ,...,Xn−1
O

/ / R[θ+Xn
Y1 ,X2 ,...,Xn−1 ]
O


R[θX1 ,X2 ,...,Xn−1 ]


/ / R[θY1 ,X2 ,...,Xn−1 ]

is a downward-oriented regular pushout. This in turn follows from Corollary 1.7 since
the vertical arrows above are split epimorphisms by construction and the horizontal
arrows are regular epimorphisms by induction hypothesis.

Special instances of the following theorem have been considered in literature: if E
is the category of groups and n = 2, the result can be deduced from Baues-Pirashvili
[1] and Hartl-Vespa [40]; if E is a semi-abelian category, under the identification of
n-folded objects given in Remark 6.4 and with the kernel intersection definition of
degree, the result has been announced by Manfred Hartl in several of his talks.

CENTRAL REFLECTIONS AND NILPOTENCY IN EXACT MAL’TSEV CATEGORIES

51

Theorem 6.8. The full subcategory Fldn (E) of n-folded objects of a σ-pointed exact Mal’tsev category E is a reflective Birkhoff subcategory. The associated Birkhoff
reflection J n : E → Fldn (E) is the universal endofunctor of E of degree ≤ n.
Proof. Observe that the second assertion is a consequence of the first and of Proposition 6.5a-b. In virtue of Proposition 6.6 it suffices thus to construct the reflection
J n : E → Fldn (E). The latter is obtained by the following pushout
θX,··· ,X

X + ···+ X
X
δn+1

/ / PX,...,X
µn X


X

ǫn X


/ / J n (X)

which is regular by Corollary 1.8 so that J n (X) = X/Hn+1 [X] where Hn+1 [X] is the
X
direct image of R[θX,...,X ] under the folding map δn+1
. We will show that J n (X) is
n-folded and that any morphism X → T with n-folded codomain T factors uniquely
through J n (X). For this, consider the following diagram
θH

[X],...,H

[X]

n+1
n+1
/ / PH [X],...,H [X]
Hn+1 [X] + · · · + H❚n+1
n+1
n+1
❚❚❚[X]
❚❚❚❚ ❚❚p❚1❚+···+p
Pp1 ,··· ,p1
Pp0 ,··· ,p0
❚❚❚❚ ❚❚) 1
 
θX,··· ,X
p0 +···+p0 ❚❚)
/ / PX,...,X
X + ···+ X
Hn+1 [X]
❯❯❯P❯ǫn X,...,ǫn X
δn+1
❯❯❯* *
X
P n

δn+1



p1

Hn+1 [X]

p0

µn X

/ 
/X

ǫn X

J (X),...,J n (X)


/ / J n (X) s

in which the existence of the dotted arrow has to be shown. By Lemma 6.9 PJ n (X),...,J n (X)
is the coequaliser of the reflexive pair (Pp0 ,...,p0 , Pp1 ,...,p1 ). It suffices thus to check
that µn X coequalises the same pair. This follows by precomposition with the regular
epimorphism θHn+1 [X],...,Hn+1 [X] using the commutativity of the previous diagram.
For the universal property of ǫn X : X ։ J n (X) let us consider a morphism
f : X → T with n-folded codomain T . By construction of J n (X), the following
commutative diagram
θX,...,X

/ / PX,...,X
X + · · · + ❙X
❙❙f ❙+···+f
◆◆P◆f,...,f
❙❙❙
◆◆&
)
θ
T ,...,T
X
/ / PT,...,T
δn+1
T + · · · +❘T δT
❘❘❘n+1
♦
♦
❘
♦
❘

❘❘( w♦♦♦ mT
/T
X
f

induces the desired factorisation.



Lemma 6.9. In a σ-pointed exact Mal’tsev category, the functor PX1 ,...,Xn+1 preserves reflexive coequalisers in each of its n + 1 variables.

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CLEMENS BERGER AND DOMINIQUE BOURN

Proof. By exactness, it suffices to show that P preserves regular epimorphisms in each
variable, and that for a regular epimorphism fi : Xi ։ Xi′ the induced map on kernel
relations PX1 ,...,R[fi ],...,Xn+1 → R[PX1 ,...,fi ,...,Xn+1 ] is a regular epimorphism as well.
By symmetry, it is even sufficient to do so for the first variable.
We shall argue by induction on n (since for n = 1 there is nothing to prove) and
consider the following downward-oriented pullback diagram
PX1 ,...,Xn ,Xn+1
O

/ / P +Xn+1
X1 ,...,Xn
O


X1 + · · · + Xn


/ / PX1 ,...,Xn

θX1 ,...,Xn

which derives from square (n) of the beginning of this section. By induction hypothesis, the two lower corners and the upper right corner are functors preserving regular
epimorphisms in the first variable. It follows then from the cogluing lemma (cf. the
proof of Theorem 6.23a) and Corollary 1.7 that the upper left corner also preserves
regular epimorphisms in the first variable.
It remains to be shown that for f : X1 ։ X1′ we get an induced regular epimorphism on kernel relations. For this we denote by F, G, H the functors induced on the
lower left, lower right and upper right corners, and consider the following commutative
diagram
/ H(R[f ])
P (R[f ])
◆◆ρ◆H
tR[f ]
ρP
@
✁✁@
◆◆◆'
✁
✁
'
✁
'
'
✁✁✁
✁
/ R[H(f )]
✁✁✁
R[P (f )]
✁
@
✁@
✁✁✁✁
✁✁✁✁✁
✁✁✁✁✁
✁
R(t
)
X
✁✁
/ G(R[f ])
F (R[f ])
✁✁✁✁✁
θR[f ]
◆
◆◆◆
✁
◆
◆
✁✁
ρG ◆◆◆' '
ρF ◆◆◆' '
✁✁✁✁✁
/ R[G(f )]
R[F (f )]
R(θX )

in which the back vertical square is a downward-oriented pullback by definition of P .
By commutation of limits the front vertical square is a downward oriented pullback
as well. Again, according to the cogluing lemma and Corollary 1.7, the induced
arrow ρP is then a regular epimorphisms, since ρF , ρG and ρH are so by induction
hypothesis.

6.2. Higgins commutator relations and their normalisation. –
We shall now concentrate on the case X = X1 = X2 = · · · = Xn+1 . Accordingly,
we abbreviate the n + 1 “contractions” as follows:
n+1

πX̂i

n

z
}|
{
}|
{
z
= πi : X + · · · + X → X + · · · + X,

i = 1, . . . , n + 1.

Proposition 6.1 reads then R[θX,...,X ] = R[π1 ] ∩ · · · ∩ R[πn+1 ].

CENTRAL REFLECTIONS AND NILPOTENCY IN EXACT MAL’TSEV CATEGORIES

53

We denote the direct image of the kernel relation R[θX,...,X ] under the folding map
X
δn+1
: X + · · · + X → X by a single bracket [∇X , . . . , ∇X ] of length n + 1 and call it
the (n + 1)-ary Higgins commutator relation on X.
The proof of Theorem 6.8 shows that in σ-pointed exact Mal’tsev categories the
universal n-folded quotient J n (X) of an object X is obtained by quotienting out the
(n + 1)-ary Higgins commutator relation. The binary Higgins commutator relation
coincides with the Smith commutator [∇X , ∇X ] (cf. Section 1.1, Corollary 1.8 and
Proposition 4.2) which ensures consistency of our notation.
Recall that in a pointed category the normalisation of an effective equivalence
relation R on X is the kernel of its quotient map X ։ X/R. In σ-pointed exact
Mal’tsev categories normalisation commutes with direct image, cf. Corollary 1.8. In
particular, the normalisation of the Higgins commutator relation yields precisely the
Higgins commutator of same length, cf. Remark 6.4.
Proposition 6.10. In a σ-pointed exact Mal’tsev category, the image of R[θX,X,X ]
under δ2X + 1X is the kernel relation of the pushout of θX,X,X along δ2X + 1X
X +X +X

θX,X,X

δ2X +1X


X +X

X
ζX,X

/ / PX,X,X

/ / JX
X,X

X
which may be computed as an intersection: R[ζX,X
] = [R[π1 ], R[π1 ]] ∩ R[π2 ].

In particular, we get the inclusion [∇X , [∇X , ∇X ]] ⊂ [∇X , ∇X , ∇X ].
Proof. By Corollary 1.8, the pushout is regular so that the first assertion follows from
Proposition 1.6. Consider the following diagram
δ2X +1X

X + X + X❱
❱❱❱❱ θ+X
⑤>
❱❱❱X,X
⑤⑤⑤⑤⑤
❱❱* *
⑤
π3 ⑤⑤⑤⑤
(X
+ X) ×X (X + X)
⑤
⑤
⑤⑤⑤⑤⑤
r8
⑤
rrrrrrr δX
~⑤⑤⑤⑤⑤
r
r
2
r rr
X +X
rrrrrrr
r
r
PPP
r
rr
P
xrrrrrrr
θX,X PPP( (
X ×X

/ / X +X
✠✠D ▼▼▼η▼1 (π1 )
✠
▼& &
✠
✠✠✠✠
✠
/
/ I 1 (π1 )
✠
✠✠
π2 ✠✠✠
✠
✄✄✄A
✠✠✠✠
′
✄✄✄✄✄
f
✄
//X
✄✄
❍❍ η1 (X) ✄✄✄✄✄✄
❍❍
✄
✄
✄✄✄
❍# #
/ / I 1 (X)

in which top and bottom are regular pushouts by Corollary 1.8. The bottom square
constructs the associated abelian object I 1 (X) of X, while the top square constructs
the associated abelian object I 1 (π1 ) of π1 : X + X ⇄ X in the fibre over X. The
upward oriented back and front faces are pushouts of split monomorphisms. The left
face is a specialisation of square (2) just before Proposition 5.2. We can therefore

54

CLEMENS BERGER AND DOMINIQUE BOURN

apply Corollary 1.19 and we get diagram
δ X +1

X
2
/ / X +X
X + X Z✺+ ❘X
✲✲V✲✲ ❑❑❑ X
✺✺✺ ❘❘❘❘
✺
❘❘θ❘X,X,X
✺✺✺
✲✲✲✲ ❑❑❑ζ❑X,X
❘
✺
❘❘❘
❑❑❑
✺✺✺
✲✲✲✲
❘❘❘
❑% %
✺✺✺✺
❘) )
✲✲✲✲
✺
✺✺✺
✲✲
/ / JX
✲
P
✺
X,X,X
X,X
✲✲ ✲
π3 ✺✺✺
✺
✺✺✺
π2 ✲ ✲✲
⑤⑤=
✞✞C
⑤
✲
✺
✞
✺✺✺
✲✲✲✲
⑤⑤⑤⑤
✞✞✞✞
✺✺✺✺
⑤⑤⑤⑤⑤
✲ ✲ ✞✞✞✞✞
⑤

}⑤ ⑤
/ / X ✞✞
X + X⑤
X

δ2

X
in which the kernel relation of the regular epimorphism ζX,X
is given by

R[η 1 (π1 )] ∩ R[π2 ] = [R[π1 ], R[π1 ]] ∩ R[π2 ].
For the second assertion, observe first that the ternary folding map δ3 may be
identified with the composition δ2X ◦ (δ2X + 1X ). Therefore, the ternary Higgins commutator relation [∇X , ∇X , ∇X ] is the direct image under δ2X : X + X → X of the
X
kernel relation of ζX,X
. Now we have the following chain of inclusions, where for
shortness we write R1 = R[π1 ], R2 = R[π2 ], R12 = R1 ∩ R2 :
[R12 , [R12 , R12 ]] ⊂ R12 ∩ [R12 , R12 ] ⊂ R2 ∩ [R1 , R1 ].
By exactness, the direct image of the leftmost relation is [∇X , [∇X , ∇X ]], while
the direct image of the right most relation is [∇X , ∇X , ∇X ].

Proposition 6.11. In a σ-pointed exact Mal’tsev category, the image of R[θX,...,X ]
X
X
+ 1X
+ 1X is the kernel relation of the pushout of θX,...,X along δn−1
under δn−1
X + ···+ X

θX,...,X

X
δn−1
+1X


X +X

X
ζX,...,X

/ / PX,...,X

/ / JX
X,...,X
n−1

which may be computed as an intersection

X
R[ζX,...,X
]
n−1

z
}|
{
= [R[π1 ], . . . , R[π1 ]] ∩ R[π2 ].
n

z
}|
{
}|
{
z
In particular, we get the inclusion [∇X , [∇X , . . . , ∇X ]] ⊂ [∇X , . . . , ∇X ].

CENTRAL REFLECTIONS AND NILPOTENCY IN EXACT MAL’TSEV CATEGORIES

55

Proof. The first assertion follows from Proposition 1.6 and the following diagram
X
δn−1
+1X

X + · · · +PX +X
PPθPX,...,X
①①①;
PP( (
①①①①
πn ①①①①
+X
①
①
①①
PX,...,X
①①①①①
①
=
①①
④④④X
{①①①①①
④④④④④δn−1
④
X + ···+ X
④④④④
❚❚❚❚
④④④④④
④
❚
}④④④
θX,...,X ❚❚❚) )
PX,...,X

// X +X
❚❚❚❚ q
❚❚❚❚)
⑦⑦⑦⑦>
⑦
)
⑦⑦⑦⑦⑦
X+X
⑦
//
⑦
⑦
[∇π1 ,...,∇π1 ]
⑦⑦⑦
π2 ⑦⑦⑦
✉
✉✉ ✉:
~⑦⑦⑦⑦
✉✉✉✉✉✉
//X ⑦
✉
✉✉ ✉
◗◗◗
✉✉✉✉✉✉
◗◗◗
✉
z
✉
◗( (
✉✉
/ / X/[∇
X , . . . , ∇X ]

in which top and bottom are regular pushouts by Corollary 1.8. The bottom square
constructs the quotient of X by the (n−1)-ary Higgins commutator relation [∇X , . . . , ∇X ].
The top square constructs the quotient of π1 : X + X ⇄ X by the (n − 1)-ary Higgins
commutator relation [∇π1 , . . . , ∇π1 ] in the fibre over X. The upward oriented back
and front faces are pushouts of split monomorphisms. The left face is a specialisation
of square (n) of the beginning of this section. We can therefore apply Corollary 1.19
and we get the following diagram
X
δn−1
+1X

X + · · ·\✿+ X
✿✿✿✿ ❯❯❯❯❯❯
❯❯❯θ❯X,...,X
✿✿✿✿
❯❯❯❯
✿✿✿✿
❯❯❯❯
✿✿✿✿
❯❯* *
✿✿✿✿
PX,...,X
✿
πn ✿✿✿
✿✿✿✿
✈
✿✿✿✿
✈✈ ✈:
✈✈✈✈✈✈
✿✿✿✿
✈
✈✈ ✈
✿
z✈✈✈✈✈✈
X + ···+ X
X

δn−1

/ / X +X
✲✲V✲✲ ▲▲▲ X
✲ ▲▲▲ζX,...,X
π2 ✲✲ ✲
▲▲▲
✲✲✲✲
▲▲& &
✲
✲✲ ✲
✲
/ / JX
✲✲ ✲
X,...,X
✲✲✲✲
✄A
✲✲✲✲
✄
✄✄✄✄
✄
✲✲✲✲
✄✄✄✄✄

/ / X ✄✄

X
in which the kernel relation of the regular epimorphism ζX,...,X
is given by

R[q] ∩ R[π2 ] = [∇π1 , . . . , ∇π1 ] ∩ R[π2 ] = [R[π1 ], . . . , R[π1 ]] ∩ R[π2 ].
X
Since δnX = δ2X ◦ (δn−1
+ 1X ), the proof of the second assertion is completely
analogous to the proof of the corresponding part of Proposition 6.10.


The semi-abelian part of the following corollary can also be derived from a direct
analysis of “iterated” Higgins commutators, cf. [39, Proposition 2.21(iv)].
Corollary 6.12. For any object X of a σ-pointed exact Mal’tsev category, the iterated
Smith commutator [∇X , [∇X , [∇X , · · · , [∇X , ∇X ] · · · ]]] is a subobject of the Higgins
commutator relation [∇X , . . . , ∇X ] of same length.
In a semi-abelian category, the iterated Huq commutator [X, [X, · · · , [X, X] · · · ]] is
a subobject of the Higgins commutator [X, . . . , X] of same length.
Proof. The first statement follows inductively from Propositions 6.10 and 6.11.
The second statement follows from the first and the fact that in a semi-abelian
category, the iterated Huq commutator is the normalisation of the iterated Smith

commutator by Remark 2.13 and [33, Proposition 2.2].

56

CLEMENS BERGER AND DOMINIQUE BOURN

Proposition 6.13. In a σ-pointed exact Mal’tsev category E, each n-folded object is
an n-nilpotent object, i.e. Fldn (E) ⊂ Niln (E). In particular, endofunctors of degree
≤ n take values in n-nilpotent objects.
Proof. The second assertion follows from the first and from Proposition 6.5. For any
n-folded object X, the (n + 1)-ary Higgins commutator relation of X is discrete and
hence, by Corollary 6.12, the iterated Smith commutator of same length is discrete as
well. By an iterated application of Theorem 2.12 and Proposition 1.4, this iterated
n
Smith commutator is the kernel relation of ηX
: X ։ I n (X), and hence X ∼
= I n (X),
i.e. X is n-nilpotent.

The following theorem generalises Theorem 4.5 to all positive integers.
Theorem 6.14. For a σ-pointed exact Mal’tsev category D such that the identity
functor of Niln−1 (D) is of degree ≤ n − 1, the following properties are equivalent:
(a) all objects are n-nilpotent;
(b) for all objects X1 , . . . , Xn , the map θX1 ,...,Xn is an affine extension;
(c) for all objects X1 , . . . , Xn , the map θX1 ,...,Xn is a central extension.
Proof. For an n-nilpotent category D, the Birkhoff reflection I n−1 : D → Niln−1 (D)
is a central reflection. Since all limits involved in the construction of PX1 ,...,Xn
are preserved under I n−1 by an iterative application of Proposition 2.8, we get
I n−1 (θX1 ,...,Xn ) = θI n−1 (X1 ),...,I n−1 (Xn ) . Since by assumption the identity functor
of Niln−1 (D) is of degree ≤ n − 1, the latter comparison map is invertible so that by
Theorem 3.3, the comparison map θX1 ,...,Xn is an affine extension, i.e. (a) implies (b).
By Proposition 3.2, (b) implies (c).
Specializing (c) to the case X = X1 = X2 = · · · = Xn we get the following
commutative diagram
R[θX,...,X ]

[∇X , . . . , ∇X ]

/

/ X + ··· + X
X
δn


/ 
/X

θX,...,X

/ / PX,...,X

/ / X/[∇X , . . . , ∇X ]

in which is the right square is a regular pushout by Corollary 1.8 so that the lower
row represents the kernel relation of a central extension. We have already seen that
the iterated Smith commutator [∇X , [∇X , [∇X , · · · , [∇X , ∇X ] · · · ]]] of length n is the
kernel relation of the unit η n−1 (X) : X ։ I n−1 (X) of the (n−1)st Birkhoff reflection.
Corollary 6.12 implies thus that this unit is a central extension as well so that D is
n-nilpotent, i.e. (c) implies (a).

Definition 6.15. A σ-pointed category (D, ⋆D ) with pullbacks is said to satisfy condition Pn if for all X1 , . . . , Xn , Z, pointed cobase-change (αZ )! : D → PtZ (D) takes
+Z
.
the object PX1 ,...,Xn to the object PX
1 ,...,Xn
In particular, since PX = ⋆ condition P1 is void and just expresses that (αZ )!
preserves the null-object. Since PX,Y = X × Y condition P2 expresses that (αZ )!

CENTRAL REFLECTIONS AND NILPOTENCY IN EXACT MAL’TSEV CATEGORIES

57

preserves binary products. Therefore, the following result extends Corollary 4.4 (n =
1) and Theorem 5.5 (n = 2) to all positive integers.
Proposition 6.16. The identity functor of a σ-pointed exact Mal’tsev category D is
of degree ≤ n if and only if all objects are n-nilpotent and the Birkhoff subcategories
Nilk (D) satisfy condition Pk for 1 ≤ k ≤ n.
Proof. Since the statement is true for n = 1 by Corollary 4.4 we can argue by induction
on n and assume that the statement is true up to level n − 1. In particular, we can
assume that Niln−1 (D) has an identity functor of degree ≤ n − 1. Let us then consider
the following substitute of the main diagram 5.2:

(X1 + · · · + Xn ) ⋄ Z /
θX1 ,...,Xn ⋄Z


PX1 ,...,Xn ⋄ Z /

ϕZ
X1 ,...,Xn


⋄Z
/
PX
,...,X
1
n

θ

/ (X1 + · · · + Xn ) + ZX1 +···+Xn/ / ,Z
(X1 + · · · + Xn ) × Z
!"#
θX1 ,...,Xn +Z '&%$
θ
×Z
a

 X1 ,...Xn
θPX ,...,X ,Z
n
1
/ / PX1 ,...,Xn × Z
/ PX1 ,...,Xn + Z
Z
Z
'&%$
!"#
b
 φX1 ,...,Xn
 µX1 ,...,Xn
/ / P ×Z
/ P +Z
X1 ,...,Xn

X1 ,...,Xn

in which the composite vertical morphisms from left to right are respectively
+Z
×Z
⋄Z
and θX
θX
and θX
,
1 ,...,Xn
1 ,...,Xn
1 ,...,Xn

and the morphism µZ
X1 ,...,Xn is the canonical isomorphism. Exactly as in the proof of
Proposition 5.4 it follows that the identity functor of D is of degree ≤ n if and only
if squares (a) and (b) are pullback squares.
Square (b) is a pullback if and only if φZ
X1 ,...,Xn is invertible which is the case
precisely when condition Pn holds. By Proposition 3.8 and Theorem 6.14, square (a)
is a pullback if and only if θX1 ,...,Xn is an affine (resp. central) extension, which is
the case for all objects X1 , . . . , Xn precisely when D is n-nilpotent.

Corollary 6.17. A semi-abelian category has an identity functor of degree ≤ n if
and only if either of the following three equivalent conditions is satisfied:
(a) all objects are n-nilpotent, and the comparison maps
⋄Z
ϕZ
X1 ,...,Xn : PX1 ,...,Xn ⋄ Z → PX1 ,...,Xn

are invertible for all objects X1 , . . . , Xn , Z;
(b) the (n + 1)st cross-effects of the identity functor
crn+1 (X1 , . . . , Xn , Z) = K[θX1 ,...,Xn ,Z ]
vanish for all objects X1 , . . . , Xn , Z;
(c) the co-smash product is of degree ≤ n − 1, i.e. the comparison maps
⋄Z
⋄Z
: (X1 + · · · + Xn ) ⋄ Z → PX
θX
1 ,...,Xn
1 ,...,Xn

are invertible for all objects X1 , . . . , Xn , Z.

58

CLEMENS BERGER AND DOMINIQUE BOURN

Proof. Condition (a) expresses that squares (a) and (b) of the main diagram are
pullbacks. By Proposition 6.16 this amounts to an identity functor of degree ≤ n.
For (b) note that by protomodularity the cross-effect crn+1 (X1 , . . . , Xn , Z) is trivial
if and only if the regular epimorphism θX1 ,...,Xn ,Z is invertible.
The equivalence of conditions (b) and (c) follows from the isomorphism of kernels
⋄Z
K[θX1 ,...,Xn ,Z ] ∼
]. The latter is a consequence of the 3 × 3-lemma which,
= K[θX
1 ,...,Xn
applied to main diagram 6.16 and to square (n), yields a chain of isomorphisms:
×Z
+Z
⋄Z
]] ∼
] ։ K[θX
]∼
K[θX
= K[θX1 ,...,Xn ,Z ].
= K[K[θX
1 ,...,Xn
1 ,...,Xn
1 ,...,Xn


6.3. Higher duality and multilinear cross-effects. –
In Section 5 we obtained a precise criterion for when 2-nilpotency implies quadraticity, namely algebraic distributivity, cf. Corollary 5.16. We now look for a similar
criterion for when n-nilpotency implies an identity functor of degree ≤ n. Proposition
6.16 gives us an explicit exactness condition in terms of certain limit-preservation
properties (called Pn ) of pointed cobase-change along initial maps. In order to exploit
the latter we first need to dualise condition Pn into a colimit-preservation property,
extending Proposition 5.15. Surprisingly, this dualisation process yields the simple
condition that in each variable, the functor PX1 ,...,−,...,Xn takes binary sums to binary
sums in the fibre over PX1 ,...,⋆,...,Xn . For n-nilpotent semi-abelian categories, this
in turn amounts to the condition that the n-th cross-effect of the identity functor
crn (X1 , . . . , Xn ) = K[θX1 ,...,Xn ] is multilinear.
Such a characterisation of degree n functors in terms of the multilinearity of their nth cross-effect is already present in the original treatment of Eilenberg-Mac Lane [23]
for functors between abelian categories. It plays also an important role in Goodwillie’s
[29] homotopical context (where however linearity has a slightly different meaning).
The following lemma is known in contexts close to our’s.
Lemma 6.18. Let D be a σ-pointed category and let E be an abelian category. Any
∆n

F

D
multilinear functor F : Dn → E has a diagonal G : D −→
Dn −→ E of degree ≤ n.

Proof. This is a consequence of the decomposition formula of Eilenberg-Mac Lane for
functors taking values in abelian categories, cf. Remark 6.2. Indeed, an induction on
k shows that the k-th cross-effect of the diagonal crkG (X1 , . . . , Xk ) is the direct sum of
all terms F (Xj1 , . . . , Xjn ) such that the sequence (j1 , . . . , jn ) contains only integers
1, 2, . . . , k, but each of them at least once. In particular,
M
crnG (X1 , . . . , Xn ) ∼
F (Xσ(1) , . . . , Xσ(n) )
=
σ∈Σn

and the cross-effects of G of order > n vanish, whence G is of degree ≤ n.



Lemma 6.19 (cf. Proposition 2.9 in [39]). For each n ≥ 1, the n-th cross-effect of
the identity functor of a semi-abelian category preserves regular epimorphisms in each
variable.

CENTRAL REFLECTIONS AND NILPOTENCY IN EXACT MAL’TSEV CATEGORIES

59

Proof. The first cross-effect is the identity functor and the second cross-effect is the
co-smash product. Proposition 1.15 and Lemma 1.11 imply that the co-smash product
preserves regular epimorphisms in both variables.
The general case n + 1 follows from the already treated case n = 1. By symmetry
it suffices to establish the preservation property for the last variable which we shall
denote Z. We have the following formula:
crn+1 (X1 , . . . , Xn , Z) = crn⋄Z (X1 , . . . , Xn )

(n ≥ 1)

⋄Z
where crn⋄Z (X1 , . . . , Xn ) = K[θX
] denotes the n-th cross-effect of the functor
1 ,...,Xn
− ⋄ Z. Indeed, this kernel has already been identified with K[θX1 ,...,Xn ,Z ] in the
proofs of Corollaries 5.6 and 6.17. It is now straightforward to deduce preservation
of regular epimorphisms in Z using that (−) ⋄ (−) preserves regular epimorphisms in
both variables.


Corollary 6.20 (cf. Proposition 2.21 in [39]). In a semi-abelian category, the image
of a Higgins commutator [X, . . . , X] of X under a regular epimorphism f : X ։ Y is
the corresponding Higgins commutator [Y, . . . , Y ] of Y .
Proof. The Higgins commutator of length n is the image of the diagonal n-th crosseffect K[θX,...,X ] under the folding map δnX : X + · · · + X → X, cf. Section 6.2. By
Lemma 6.19, any regular epimorphism f : X ։ Y induces a regular epimorphism
K[θX,...,X ] ։ K[θY,...,Y ] on diagonal cross-effects, whence the result.

Note that the commutative square (n) of the beginning of this section induces the
following pullback square
χX1 ,...,Xn+1

PX1 ,...,Xn+1
O

PX1 ,...,Xn ,ωX

n+1

PX1 ,...,Xn ,⋆

/ P +Xn+1
X1 ,...,Xn
O

+ωX
n+1
1 ,...,Xn

PX1 ,...,Xn ,αX

PX

n+1


X1 + · · · + Xn



+αX
n+1
1 ,...,Xn

PX

/ PX1 ,...,Xn

θX1 ,...,Xn

in which the identification PX1 ,...,Xn−1 ,⋆ = X1 + · · · + Xn−1 is exploited to give the
left vertical morphisms names. Recall that αX : ⋆ → X and ωX : X → ⋆ denote the
initial and terminal maps.
Proposition 6.21. For any objects X1 , . . . , Xn−1 , Y, Z of a σ-pointed category (D, ⋆D )
with pullbacks consider the following diagram
ρX1 ,...,Xn−1 ,Y,Z
/ PX1 ,...,Xn−1 ,Y +Z

PX1 ,...,Xn−1 ,Y + Z
O

O

PX

PX1 ,...,Xn−1 ,ωY +Z


X1 + · · · + Xn−1 + Z

Y
1 ,...,Xn−1 ,πZ

PX



Y
1 ,...,Xn−1 ,ιZ

/ PX1 ,...,Xn−1 ,Z

θX1 ,...,Xn−1 ,Z

in which the horizontal map ρX1 ,...,Xn−1 ,Y,Z is induced by the pair PX1 ,...,Xn−1 ,ιZY :
PX1 ,...,Xn−1 ,Y → PX1 ,...,Xn−1 ,Y +Z and PαX1 ,...,αXn−1 ,ιZY : Z → PX1 ,...,Xn−1 ,Y +Z ;

60

CLEMENS BERGER AND DOMINIQUE BOURN

(1) The functor PX1 ,...,Xn−1 ,− : D → PtX1 +···+Xn−1 (D) preserves binary sums if
and only if the upward-oriented square is a pushout for all objects Y, Z;
(2) the category D satisfies condition Pn (cf. Definition 6.15) if and only if the
downward-oriented square is a pullback for all objects X1 , . . . , Xn−1 , Y, Z.
In particular, (1) and (2) hold simultaneously whenever θX1 ,...,Xn−1 ,Z is an affine
extension for all objects X1 , . . . , Xn−1 , Z.
Proof. The second assertion follows from the discussion in Section 3.2. For (1), observe
that the left upward-oriented square of the following diagram
PX

Z
1 ,...,Xn−1 ,ιY

/
/ PX1 ,...,Xn−1 ,Y +Z
/ PX1 ,...,Xn−1 ,Y + Z
ρ
X1 ,...,Xn−1 ,Y,Z
O
O

PX1 ,...,Xn−1 ,Y
O

PX

PX1 ,...,Xn−1 ,αY


X1 + · · · + Xn−1

PX

Y
1 ,...,Xn−1 ,πZ


θ
/ X1 + · · · + Xn−1 + ZX1 ,...,Z

/

PX1 ,...,Xn−1 ,αZ

Y
1 ,...,Xn−1 ,ιZ



/ PX1 ,...,Xn−1 ,Z

is a pushout so that the whole upward-oriented rectangle is a pushout if and only if
the right upward-oriented square is a pushout, establishing (1).
For (2) observe that the right downward-oriented square of the following diagram
ρX1 ,...,Xn−1 ,Y,Z
χX1 ,...,Xn−1 ,Y +Z
/ PX1 ,...,Xn−1 ,Y +Z
/ P +Y +Z
X1 ,...,Xn−1

PX1 ,...Xn−1 ,Y + Z
O

O

Y
+πZ
1 ,...,Xn−1

PX

O



+ιY
Z
1 ,...,Xn−1

PX



θX1 ,...,Xn−1 ,Z
χX ,...,Xn−1 ,Z
/ PX1 ,...,Xn−1 ,Z 1
/ P +Z
X1 + · · · + Xn−1 + Z
/ X1 ,...,Xn−1
+Z
θX
,··· ,X
1

n−1

is a pullback (see below) so that the whole downward-oriented rectangle is a pullback
if and only if the left downward-oriented square is a pullback. The whole downwardoriented rectangle is a pullback if and only if the comparaison map
+Z
PX1 ,...,Xn−1 ,Y + Z → PX
1 ,...,Xn−1 ,Y

is invertible (i.e. if and only if condition Pn holds) since the following square is by
definition a pullback in the fibre PtZ (D):

P +Z

+Z
PX
1 ,...,Xn−1 ,Y
O

χ+Z
X ,...,X
1

n−1 ,Y

Y
+πZ
1 ,...,Xn−1

PX

X1 ,...,Xn−1 ,π Y
Z


X1 + · · · + Xn−1 + Z

/ P +Y +Z
X1 ,...,Xn−1
O

+Z
θX
,...,X
1

n−1


/ P +Z
X1 ,...,Xn−1

CENTRAL REFLECTIONS AND NILPOTENCY IN EXACT MAL’TSEV CATEGORIES

61

Thus (2) is established. The pullback property of the right square above follows finally
from considering the following diagram
χX1 ,...,Xn−1 ,Y +Z
/ P +Y +Z
X1 ,...,Xn−1

PX1 ,...,Xn−1 ,Y +Z
O

Y
+πZ
PX ,...,X
1
n−1



/ P +Z

O



+ιY
Z
1 ,...,Xn−1

PX

PX1 ,...,Xn−1 ,Z
χX1 ,...,Xn−1 ,Z X1 ,...,Xn−1
O
O

X1 + · · · + Xn−1


/ PX1 ,...,Xn−1

θX1 ,...,Xn−1

in which whole rectangle and lower square are downward-oriented pullbacks.



Theorem 6.22. Let D be an n-nilpotent σ-pointed exact Mal’tsev category such that
the identity functor of Niln−1 (D) is of degree ≤ n − 1.
The following properties are equivalent:
(a) the identity functor of D is of degree ≤ n;
(b) the category D satisfies condition Pn (cf. Definition 6.15);
(c) the functor PX1 ,...,Xn−1 ,− : D → PtX1 +···+Xn−1 (D) preserves binary sums for
all objects X1 , . . . , Xn−1 .
If D is semi-abelian then the former properties are also equivalent to:
(d) the n-th cross-effect of the identity is coherent in each variable;
(e) the n-th cross-effect of the identity is linear in each variable;
(f) the diagonal n-th cross-effect of the identity is a functor of degree ≤ n.
Proof. It follows from Proposition 6.16 that properties (a) and (b) are equivalent,
while properties (b) and (c) are equivalent by Theorem 6.14 and Proposition 6.21.
For the equivalence between (c) and (d), note first that the n-th cross-effect preserves
regular epimorphisms in each variable by Lemma 6.19 so that coherence (in the last
variable) amounts to the property that the canonical map
crn (X1 , . . . , Xn−1 , Y ) + crn (X1 , . . . , Xn−1 , Z) → crn (X1 , . . . , Xn−1 , Y + Z)
is a regular epimorphism. Since by Theorem 6.14 for W = Y, Z, Y + Z the regular
epimorphism X1 +· · ·+Xn−1 +W ։ PX1 ,...,Xn−1 ,W is an affine extension, Proposition
3.11 establishes the equivalence between (c) and (d). Finally, consider the following
commutative diagram in Nil1 (D) = Ab(D)
I 1 (crn (X1...n−1 , Y ) + crn (X1...n−1 , Z))

crn (X1...n−1 , Y + Z)

∼
=

/ crn (X1...n−1 , Y ) × crn (X1...n−1 , Z)
O

crn (X1 ,...,Xn−1 ,θY,Z )

/ crn (X1...n−1 , Y × Z)

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CLEMENS BERGER AND DOMINIQUE BOURN

in which the upper horizontal map is invertible because the n-th cross-effect takes
values in abelian group objects. It follows that the left vertical map is a section so
that property (d) is equivalent to the invertibility of this left vertical map. Therefore,
(d) is equivalent to the invertibility of the diagonal map
crn (X1 , . . . , Xn−1 , Y + Z) → crn (X1 , . . . , Xn−1 , Y ) × crn (X1 , . . . , Xn−1 , Z)
which expresses linearity in the last variable, i.e. property (e).
Property (e) implies property (f) by Lemma 6.18. It suffices now to prove that (f)
implies (a). The Higgins commutator [X, . . . , X] of length n is the image of diagonal
n-th cross-effect crn (X, . . . , X) under the n-th folding map δnX : X + · · · + X → X.
The Higgins commutator of length n is thus a quotient-functor of the diagonal n-th
cross-effect and as such a functor of degree ≤ n by Theorem 6.23a. Corollary 6.12
n−1
and Remark 2.13 imply that the kernel K[ηX
: X ։ I n−1 (X)] (considered as a
functor in X) is a subfunctor of the Higgins commutator of length n and hence, again
by Theorem 6.23a, a functor of degree ≤ n. It follows then from the short exact
sequence of endofunctors
⋆ −→ K[η n−1 ] −→ idD −→ I n−1 −→ ⋆
(by a third application of Theorem 6.23a) that the identity functor of D is also of
degree ≤ n, whence (f) implies (a).

6.4. Homogeneous nilpotency towers. –
One of the starting points of this article was the existence of a functorial nilpotency
tower for any σ-pointed exact Mal’tsev category E, cf. Section 2.4. It is not surprising
that for a semi-abelian category E the successive kernels of the nilpotency tower
capture the essence of the whole tower.
To make this more precise, we denote by
M
M
LE (X) =
LnE (X) =
K[I n (X) ։ I n−1 (X)] ∈ Ab(E)
n≥1

n≥1

the graded abelian group object defined by the successive kernels. This construction
is a functor in X. The nilpotency tower of E is said to be homogeneous if for each n,
the n-th kernel functor LnE : E → Ab(E) is a functor of degree ≤ n.
The degree of a functor does not change under composition with conservative left
exact functors. We can therefore consider LnE as an endofunctor of E. Observe also
that the binary sum in Niln (E) is obtained as the reflection of the binary sum in E.
This implies that the degree of LnE is the same as the degree of LnNiln (E) . We get the
following short exact sequence of endofunctors of Niln (E)
⋆ −→ LnNiln (E) −→ idNiln (E) −→ IEn,n−1 −→ ⋆
where the last term is the relative Birkhoff reflection IEn,n−1 : Niln (E) → Niln−1 (E).
A more familiar way to express the successive kernels LnE (X) of the nilpotency
tower of X is to realise them as subquotients of the lower central series of X. Indeed,

CENTRAL REFLECTIONS AND NILPOTENCY IN EXACT MAL’TSEV CATEGORIES

63

the 3 × 3-lemma implies that there is a short exact sequence
⋆ −→ LnE (X) = γn (X)/γn+1 (X) −→ X/γn+1 (X) −→ X/γn (X) −→ ⋆
where γn+1 (X) denotes the iterated Huq commutator of X of length n + 1, i.e. the
n
kernel of the n-th Birkhoff reflection ηX
: X ։ I n (X), cf. Remark 2.13.
The conclusion of the following theorem is folklore among those who are familiar
with Goodwillie calculus in homotopy theory (cf. [3, 29]). Ideally, we would have liked
to establish Theorem 6.23c by checking inductively one of the conditions of Theorem
6.22 without using any computation involving elements.
Theorem 6.23. Let E be a semi-abelian category.
(a) For any short exact sequence ⋆ −→ F1 −→ F −→ F2 −→ ⋆ of endofunctors
of E, F is of degree ≤ n if and only if F1 and F2 are both of degree ≤ n;
(b) The nilpotency tower of E is homogeneous if and only if the identity functors
of Niln (E) are of degree ≤ n for all n;
(c) The category of groups and the category of Lie algebras have homogeneous
nilpotency towers.
Proof. For (a) we need the following cogluing lemma for regular epimorphisms in
regular categories: for any quotient-map of cospans
X
f


X′

//Zo
h

Y
g


Y′



/ / Z′ o

in which the left naturality square is a regular pushout (cf. Section 1.3), the induced
map on pullbacks f ×h g : X ×Z Y → X ′ ×Z ′ Y ′ is again a regular epimorphism.
Indeed, a diagram chase shows that the following square
Xo

X ×Z ′ Z o
′

X ×Z Y
′

(X ×

Z′


Y ′ ) ×Y ′ Y

is a pullback. The left vertical map is a regular epimorphism by assumption so that the
right vertical map is a regular epimorphism as well. Since g is a regular epimorphism,
the projection (X ′ ×Z ′ Y ′ ) ×Y ′ Y → X ′ ×Z ′ Y ′ is again a regular epimorphism so that
f ×h g is the composite of two regular epimorphisms.
F
The limit construction PX
is an iterated pullback along split epimorphisms.
1 ,...,Xn+1
Therefore, Corollary 1.7 and the cogluing lemma show inductively that the morphism

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CLEMENS BERGER AND DOMINIQUE BOURN

F2
F
PX
→ PX
induced by the quotient-map F ։ F2 is a regular epimor1 ,...,Xn+1
1 ,...,Xn+1
phism. The 3 × 3-lemma yields then the exact 3 × 3-square

⋆

⋆

⋆

⋆


/ crF1 (X1 , . . . , Xn+1 )
n+1


F
/ crn+1
(X1 , . . . , Xn+1 )


/ crF2 (X1 , . . . , Xn+1 )
n+1

/⋆

⋆


/ F1 (X1 + · · · + Xn+1 )


/ F (X1 + · · · + Xn+1 )


/ F2 (X1 + · · · + Xn+1 )

/⋆

⋆


/ P F1
X1 ,...,Xn+1


/ PF
X1 ,...,Xn+1


/ P F2
X1 ,...,Xn+1

/⋆


⋆


⋆


⋆

from which (a) immediately follows, cf. Corollary 6.17b.
For (b) we can assume inductively that Niln−1 (E) has an identity functor of degree
≤ n − 1 so that the Birkhoff reflection IEn,n−1 : Niln (E) → Niln−1 (E) is of degree
≤ n−1, and finally IEn,n−1 is also of degree ≤ n−1 when considered as an endofunctor
of Niln (E). Statement (b) follows then from (a) by induction on n.
For (c) we treat the group case, the Lie algebra case being very similar. In the
category of groups, the graded object LGrp (X) is a graded Lie ring with Lie bracket
m+n
n
[−, −] : Lm
Grp ⊗ LGrp (X) → LGrp (X) induced by the commutator map (x, y) 7→
−1 −1
xyx y
in X. This graded Lie ring is generated by its elements of degree 1, cf.
Lazard [51, Section I.2]. In particular, there is a regular epimorphism of abelian
groups L1Grp (X)⊗n → LnGrp (X) which is natural in X. The functor which assigns
to X the tensor power L1Grp (X)⊗n is the diagonal of a multilinear abelian-groupvalued functor in n variables, and hence a functor of degree ≤ n by Lemma 6.18. It
follows from (a) that its quotient-functor LnGrp is of degree ≤ n as well, whence the
homogeneity of the nilpotency tower in the category of groups.

Theorem 6.24. Let E be a semi-abelian category.
The following conditions are equivalent:
(a)
(b)
(c)
(d)

The nilpotency tower of E is homogeneous;
For each n, the n-th Birkhoff reflection I n : E → Niln (E) is of degree ≤ n;
For each n, an object of E is n-nilpotent if and only if it is n-folded;
For each object X of E, iterated Huq commutator [X, [X, · · · , X] · · · ]] and
Higgins commutator [X, X, . . . , X] of same length coincide.

If E satisfies one and hence all of these conditions, then so does any reflective
Birkhoff subcategory of E. If E is algebraically extensive and satisfies one and hence
all of these conditions then so does the fibre PtX (E) over any object X of E.
Proof. We have already seen that (b) is equivalent to condition (b) of Theorem 6.23,
which implies the equivalence between (a) and (b). Propositions 6.5 and 6.13 show

CENTRAL REFLECTIONS AND NILPOTENCY IN EXACT MAL’TSEV CATEGORIES

65

that (b) implies (c), while Theorem 6.8 shows that (c) implies (b). The equivalence
between (c) and (d) is proved in exactly the same way as Corollary 5.19.
Let D be a reflective Birkhoff subcategory of E. We shall show that D inherits (c)
from E. By Proposition 6.13, it suffices to show that in D each n-nilpotent object X
is n-folded. Since the inclusion D ֒→ E is left exact, it preserves n-nilpotent objects
so that X is n-nilpotent in E, and hence by assumption n-folded in E. The Birkhoff
reflection E → D preserves sums and the limit construction PX1 ,...,Xn+1 by an iterated
application of Proposition 2.8. Therefore, X is indeed n-folded in D.
By Lemma 5.10 algebraic extensivity implies that all pointed base-change functors
are exact. By Lemma 2.15 this implies that the following square of functors
PtX (E)

n
IPt

X (E)

/ Niln (PtX (E))
∗
ωX

∗
ωX


E



IEn

/ Niln (E)

commutes up to isomorphism. The vertical functors are exact and conservative.
n
Therefore, if IEn is of degree ≤ n then IPt
is of degree ≤ n as well.

X (E)
6.5. On Moufang loops and triality groups. –
We end this article by giving an example of a semi-abelian category in which
2-foldedness is not equivalent to 2-nilpotency, namely the semi-abelian variety of
Moufang loops. In particular, the semi-abelian subvariety of 2-nilpotent Moufang
loops is neither quadratic (cf. Proposition 6.5) nor algebraically distributive (cf.
Corollaries 5.16 and 5.19). The nilpotency tower of the semi-abelian category of
Moufang loops is thus inhomogeneous (cf. Theorem 6.24). Nevertheless, the category
of Moufang loops fully embeds into the category of triality groups [22, 28, 37] which,
as we will see, is a semi-abelian category with homogeneous nilpotency tower.
Recall [15] that a loop is a unital magma (L, ·, 1) such that left and right translation
by any element z ∈ L are bijective. A Moufang loop [56] is a loop L such that
(z(xy))z = (zx)(yz) = z((xy)z) for all x, y, z ∈ L. Moufang loops form a semiabelian variety which contains the variety of groups as a reflective Birkhoff subvariety.
Moufang loops share many properties of groups, but the lack of a full associative law
complicates the situation. The main example of a non-associative Moufang loop is the
set of invertible elements of a non-associative alternative algebra (i.e. in characteristic
6= 2 a unital algebra in which the difference (xy)z − x(yz) alternates in sign whenever
two variables are permuted). In particular, the set O∗ of non-zero octonions forms
a Moufang loop. Taking the standard real basis of the octonions together with their
additive inverses yields a Moufang subloop
O16 = {±1, ±e1, . . . , ±e7 }
with sixteen elements. We will see that O16 is 2-nilpotent, but not 2-folded.

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CLEMENS BERGER AND DOMINIQUE BOURN

By Moufang’s theorem [56], any Moufang loop, which can be generated by two
elements, is associative and hence a group. In particular, for any element of a Moufang loop, left and right inverse coincide. The kernel of the reflection of a Moofang
loop L into the category of groups is the so-called associator subloop [L, L, L]ass of
L. For a Moufang loop L, the associator subloop is generated by the elements of the
form [x, y, z] = ((xy)z)(x(yz))−1 . Such an “associator” satisfies [1, y, z] = [x, 1, z] =
[x, y, 1] = 1 and is thus 3-reducible, cf. Remark 6.4. This implies that for a Moufang
loop L, the associator subloop [L, L, L]ass is contained in the ternary Higgins commutator [L, L, L], cf. Proposition 6.1 and Section 6.2. In conclusion, any 2-folded
Moufang loop has a trivial associator subloop and is therefore a 2-folded group. In
particular, O16 cannot be 2-folded since O16 is not a group. One can actually show
that [O16 , O16 , O16 ] = {±1}. On the other hand, the centre of O16 is also {±1}, and
the quotient by the centre O16 /{±1} is isomorphic to (Z/2Z)3 . This implies that O16
is 2-nilpotent, i.e. [O16 , [O16 , O16 ]] = {1}.
The variety of Moufang loops is interesting with respect to the existence of centralisers. Since algebraic distributivity fails, such centralisers do not exist for general
subloops, cf. [13]. Nevertheless, each Moufang loop L has a centre Z(L) in the sense
of Section 1.5, i.e. a centraliser Z(1L ) for its identity 1L : L → L. This centre
Z(L) is a normal subloop of L, and is the intersection Z(L) = M (L) ∩ N (L) of
the Moufang centre M (L) = {z ∈ L | zx = xz ∀x ∈ L} with the so-called nucleus
N (L) = {z ∈ L | [z, x, y] = [x, z, y] = [x, y, z] = 1 ∀x, y ∈ L}, cf. Bruck [15].
Groups with triality have been introduced in the context of Moufang loops by
Glauberman [28] and Doro [22]. A triality on a group G0 is an action (by automorphisms) of the symmetric group S3 on three letters such that for all g ∈ G0 and
σ ∈ S3 (resp. ρ ∈ S3 ) of order 2 (resp. 3), the identity
[σ, g](ρ.[σ, g])(ρ2 .[σ, g]) = 1
holds where [σ, g] = (σ.g)g −1 . We denote the split epimorphism associated to the
group action by p : G0 ⋊ S3 ⇆ S3 : i and call it the associated triality group. The
defining relations for a group with triality are equivalent to the following condition
on the associated triality group p : G ⇆ S3 : i (cf. Liebeck [52] and Hall [37]):
for any two special elements g, h ∈ G such that p(g) 6= p(h) one has (gh)3 = 1
where g ∈ G is called special if g is conjugate in G to some element of order 2 in
i(S3 ). For the obvious notion of morphism, the category TriGrp⋆ of triality groups
is a full subcategory of the fibre PtS3 (Grp) over the symmetric group S3 .
The category TriGrp⋆ is closed under taking subobjects, products and quotients
in PtS3 (Grp). Moreover, quotienting out the normal subgroup generated by the
products (gh)3 for all pairs of special elements (g, h) such that p(g) 6= p(h) defines
a reflection PtS3 (Grp) → TriGrp⋆ . Therefore, TriGrp⋆ is a reflective Birkhoff subcategory of PtS3 (Grp). Since the category of groups is an algebraically extensive
semi-abelian category (cf. Section 5.3) with homogeneous nilpotency tower (cf. Theorem 6.23), so is its fibre PtS3 (Grp) by Lemma 5.13 and Theorem 6.24. The reflective
Birkhoff subcategory TriGrp⋆ formed by the triality groups is thus also a semi-abelian
category with homogeneous nilpotency tower, again by Theorem 6.24.

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67

This result is remarkable because the category of triality groups contains the category of Moufang loops as a full coreflective subcategory, and the latter has an inhomogeneous nilpotency tower. The embedding of Moufang loops and its right adjoint
have been described by Doro [22] for groups with triality, and by Hall [37] for the
associated triality groups, see also Grishkov-Zavarnitsine [34]. Moufang loops can
thus up to equivalence of categories be identified with triality groups for which the
counit of the adjunction is invertible. Considering them inside the category of triality
groups permits the construction of a homogeneous nilpotency tower.
Acknowledgements
We are grateful to Georg Biedermann, Rosona Eldred, Marino Gran, James Gray,
Jonathan Hall, George Janelidze, Daniel Tanré and Tim Van der Linden for helpful
discussions. We are also grateful to the referee for his careful reading of our manuscript. Special thanks are due to Manfred Hartl whose seminar talk in September
2013 in Nice was the starting point for this work. The first author acknowledges
financial support of the French ANR grant HOGT.
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Université de Nice, Lab. J. A. Dieudonné, Parc Valrose, 06108 Nice Cedex, France
E-mail address: cberger@math.unice.fr
Lab. J. Liouville, CNRS Fed. Rech. 2956, Calais, France
E-mail address: bourn@univ-littoral.fr

