Looking for substitutes for co-free modules in a topological setting

Question

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.

Answer 1

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, 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 by Girard, who applies them to the semantics of his 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, 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).

Answer 2

This is a long comment (so an answer) which is really just a thank you to Todd Trimble-- this idea of Chu duality (or perhaps, Barr-Chu duality) seems really interesting. For Banach spaces, it basically captures the idea of a "dual pair", which is certainly an idea out there in the literature. But the key difference seems (as ever with category theory!) to worry about the morphisms.

So an object is $(E_1,E_2,\langle\cdot,\cdot,\rangle)$ which is a pair of Banach spaces together a bilinear pairing to $\mathbb C$. A morphism $(E_1,E_2,\langle\cdot,\cdot,\rangle)\rightarrow(F_1,F_2,\langle\cdot,\cdot,\rangle)$ is a pair of maps $f:E_1\rightarrow F_1$ and $g:F_2\rightarrow E_2$ which commutes across the pairing: $$ \langle f(x), \mu \rangle = \langle x, g(\mu) \rangle \qquad (x\in E_1, \mu\in F_2). $$

Then you can carry out tensor product, and internal hom-space, constructions. What's remarkable, to me, is that the obvious notion of a module in this setting automatically gives me this concept of "normality"-- something which has always seemed convenient, but slightly ad-hoc. So for that alone, this is a nice discovery!