# Cocomplete $R$-linear categories are tensored : adjoint functor theorem?

Let $$B$$ be an abelian category which is actually $$Mod_R$$-enriched for some ring $$R$$ (say unital commutative ring).

For $$b\in B$$, we have a functor $$\hom(b,-) : B\to Mod_R$$ which preserves limits, so one might one want to try to apply the adjoint functor theorem to get a left adjoint $$-\otimes b$$. If $$B$$ is complete and if $$\hom(b,-)$$ satisfies the solution set condition, we can just apply the adjoint functor theorem to get the existence of $$-\otimes b: Mod_R\to B$$ which is a left adjoint.

But if $$B$$ is cocomplete, one can actually define this left adjoint "by hand", without any kind of solution set condition :

if $$M$$ is a module, it has a free presentation $$R^{(I)}\to R^{(J)}\to M$$, then it's easy to define an associated map $$b^{(I)}\to b^{(J)}$$, and define $$M\otimes b$$ as its cokernel : in other words we just define $$R\otimes b$$ and let the rest be guided by the fact that $$-\otimes b$$ preserves colimits. One then checks that this is independent of the presentation and indeed defines a left adjoint.

But this seems to be an extremely ad hoc construction, I was wondering if I was missing something of the "adjoint functor theorem"-type, which allows to get a tensoring over a base monoidal category if you're cocomplete.

is there a general principle (perhaps with some hypotheses, less restrictive than "the base category is $$Mod_R$$" if possible) that implies that a complete enriched category is automatically tensored over the base category ?

• @ArnaudD. : uhm yes, I guess you're right - for some reason I was thinking noncommutative but anyways since I'm after a more general thing, it doesn't really matter, so I'll say commutative to be sure I'm not saying anything stupid. Thanks ! Nov 14, 2019 at 14:30

You can find a result of that type as Proposition 3.46 in Kelly's "Basic concepts of enriched category theory" (available here). It's given for cotensors, but the dual result would be :

Proposition : If $$\mathcal{V}$$ is a monoidal category such that the functor $$\mathcal{V}(I,\_)$$ is conservative and each object of $$\mathcal{V}$$ has only a set of extremal epimorphic quotients, then a $$\mathcal{V}$$-enriched category $$\mathcal{B}$$ is tensored if its underlying category $$\mathcal{B}_0$$ is cocomplete.

In particular, this holds whenever $$\mathcal{V}$$ is monadic over $$\mathbf{Set}$$ and the unit object $$I$$ is the free object on one element, as in the case of $$\mathcal{V}=\mathbf{Mod}_R$$. Kelly also gives the category of Banach spaces with contractions as an example where $$\mathcal{V}(I,\_)$$ is conservative (see page 8).

• Thank you for the reference ! Looking at the proof, one may notice that it uses that $\mathcal V$ is the closure under colimits of $\{I\}$, which is the same kind of reasoning that goes into the specific proof; and still in that specific proof, conservativity is needed for obvious reasons. Nov 14, 2019 at 19:18