I should say that I'm not a category theorist or an abstract algebraist, so maybe this will be very pedestrian. I have the following, somewhat vague question:



> 
> I have categories C and D, a forgetful functor $U:C\rightarrow D$. This has a left adjoint, but does not have a right adjoint. Are there other situations where this occurs, and what "workarounds" are there.
> 
> 
> 


Let's be more precise with an example which is close to my actual situation. Let $A$ be a $k$-algebra ($k=\mathbb C$ if you like) and let $U:A{\sf -mod} \rightarrow k{\sf -vect}$ the forgetful functor from left $A$-modules to $k$-vector spaces. This has a left adjoint $A \otimes \underline{\ \ }$; you have a natural bijection, for a vector space $V$ and a module $M$,


`\[ \text{Hom}_{k{\sf -vect}}(V,U(M)) \cong \text{Hom}_{A{\sf-mod}}(A\otimes V,M), \]`


which sends $T:V\rightarrow U(M)$ to $a\otimes x\mapsto a\cdot T(x)$. Then a module of the form $A\otimes V$ is "free", and this leads to consideration of projective modules etc. In my situation (dealing with topological algebras etc.) I can parallel all of this.


Similarly, we have a right adjoint $\text{Hom}\_{k{\sf -vect}}(A,\underline{\ \ })$. This gives a natural bijection


$$ \text{Hom}\_{A{\sf -mod}}(M,\text{Hom}\_{k{\sf -vect}}(A,V)) \cong
\text{Hom}\_{k{\sf -vect}}(U(M),V),
$$
which identifies $T:U(M)\rightarrow V$ with $x \mapsto (a\mapsto T(a\cdot x))$. Then one might call a module of the form $\text{Hom}\_{k{\sf -vect}}(A,V)$ "cofree", and this leads to injective modules, etc.


In my situation, my category of modules is not a "closed category"; if cofree modules existed, the naturality of the above bijection would force them to $\text{Hom}(A,V)$ (for a suitable meaning of $\text{Hom}$), but this object is not in my category. I'm hence missing a good notion of "cofree"; but really I'm interested in injectives. Are there situations (say, in algebra) where something similar occurs?


**Edit:** People demand a more explicit example. Suppose $A$ is a von Neumann algebra, and my category is those modules which are dual Banach spaces $E$, with a bounded left action of $A$, and such that for each $x\in E$, the orbit map $A\rightarrow E; x\mapsto a\cdot x$ is weak$^\*$-continuous. The forgetful functor is to the category of dual Banach spaces with weak$^\*$-continuous bounded linear maps as morphisms. The right adjoint "should be" $B^\sigma(A,\underline{\ \ })$, the weak$^\*$-cts bounded linear maps. But this is rarely itself even a dual Banach space.



Looking for substitutes for co-free modules in a topological setting
I did say I would respond, so here are some thoughts. I'll repeat the caveat that I am not a functional analyst by any means. 


There's something in category theory called the [Chu construction](http://ncatlab.org/nlab/show/Chu+construction), which has very nice categorical properties and which seems to be a natural place in which to embed your category of dual Banach spaces and weak-\* continuous maps, and also an environment in which cofree module constructions could be carried out. 


Let $\mathbf{B}$ be a symmetric monoidal closed complete and cocomplete category, (in our situation, $\mathbf{B}$ will be the category of Banach spaces) and let $D$ be a putative "dualizing object" (which here will be the scalar field, let's say $\mathbb{C}$ for complex Banach spaces). The Chu construction $Chu(\mathbf{B}, D)$ is the category whose objects are triples $(B, C, \phi: B \otimes C \to D)$ and whose morphisms $(B, C, \phi) \to (B', C', \phi')$ consist of pairs of maps $(f: B \to B', g: C' \to C)$ such that 


$$\phi'\circ (f \otimes 1\_{C'}) = \phi \circ (1\_B \otimes g)$$ 


There is a certain canonical symmetric monoidal closed structure on $Chu(\mathbf{B}, D)$ such that every object of $Chu$ is reflexive with respect to the triple $(I, D, \lambda: I \otimes D \to D)$, where $I$ is the monoidal unit of $\mathbf{B}$ and $\lambda$ is the canonical action of the unit. (The dual of an object $(B, C, \phi: B \otimes C \to D)$, by homming into $(I, D, \lambda)$, is just $(C, B, \phi \circ \sigma)$ where $\sigma: C \otimes B \to B \otimes C$ is the canonical symmetry.) The Chu category is also complete and cocomplete, if $\mathbf{B}$ is. 


There is also an embedding $i: \mathbf{B} \to Chu(\mathbf{B}, D)$ taking an object $B$ to the triple $(B, \hom(B, D), eval: B \otimes \hom(B, D) \to D)$. This embedding preserves the tensor product up to canonical isomorphism. 


Now let $C'$, $C$ be dual Banach spaces: $C = \hom(B, D)$ as above. Theo quotes in one of his comments the fact that a map $g: C' \to C$ between dual Banach spaces is an adjoint, $g = f^\ast$ for some $f: B \to B'$, if and only if $g$ is weak-$\ast$ continuous. (I hope I've got that right.) So if $DualBan$ is the category of dual Banach spaces and weak-$\ast$ continuous maps, we have another embedding 


$$j: DualBan \to Chu(Ban, \mathbb{C})$$ 


which takes a dual Banach space $C$ to the triple $(C, B, eval: C \otimes B \to \mathbb{C})$ where $B$ is the predual of $C$; it takes a weak-$\ast$ continuous map $f$ to the pair $(f, g)$, where $g$ is the adjoint to $f$. The embedding $j$ is full and faithful. 


The point I'm driving at is that if your category $DualBan$ lacks certain good properties (for purposes of co-free modules, etc.), you can nevertheless fully embed it in the very nice category $Chu(Ban, \mathbb{C})$ where the co-free modules are available. 


For example, if $A$ is a Von Neumann algebra with predual $A^\flat$, there is a monoid object $j(A) = (A, A^\flat, eval: A \otimes A^\flat \to \mathbb{C})$ in $Chu = Chu(Ban, \mathbb{C})$. One could study the category of Chu-modules over $j(A)$, i.e., the category of monoid actions of $j(A)$ in the monoidal category $Chu$. 


Since $Chu$ is symmetric monoidal closed, one can mimic the usual constructions from commutative algebra, and in particular form the internal hom $\hom\_{Chu}(A, M)$ for any Chu-module $M$ over $A$. So the forgetful functor 


$$ChuMod\_A \to Chu$$ 


would have both left and right adjoints. 


In summary, this is following a philosophy of Grothendieck: instead of working with a category of 'nice' objects but with bad categorical properties, try expanding to a wider context where the objects might be 'wilder' (e.g., Chu triples) but which has good categorical properties. It is certainly my categorical reflex in this case. 


**Edit:** Here are some references. The Chu category I described is called the category of [coherent Banach spaces](http://www.google.com/url?sa=t&source=web&cd=4&ved=0CDAQFjAD&url=http%3A%2F%2Fiml.univ-mrs.fr%2F~girard%2Fbanach.ps.gz&rct=j&q=%22Barr%22%20%22Chu%22%20%22Banach%22&ei=U8xuTfq2F8H7lwfroIRY&usg=AFQjCNEa6sLRqqRfUkNcIxIPahkHn-k1wA&sig2=pCHbsTsJ0T6JepRz_nMHpQ&cad=rja) by Girard, who applies them to the semantics of his [linear logic](http://en.wikipedia.org/wiki/Linear_logic). The Chu construction is very well-known in category theory, and has been studied extensively by Michael Barr and Vaughan Pratt (just to name two authors; Barr was Chu's adviser for his master's thesis, where this construction first appeared). In answer to Yemon, Barr does mention the connection with Mackey spaces in numerous places, for example [here](http://www.tac.mta.ca/tac/volumes/17/1/17-01.pdf), and the references therein. 


The basic idea is that in functional analysis, there is a multiplicity of possible topologies (weak, weak-$\ast$, Mackey, etc.), and the Chu construction provides an environment for discussing all of them within a single category, much as Theo surmised in one of his comments below. As I understand it, similar considerations were paramount in Barr's mind when he was doing seminal work on $\ast$-autonomous categories (see his book, Springer Lecture Notes in Mathematics 752). 


