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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 ?

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  • $\begingroup$ @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 ! $\endgroup$ Nov 14, 2019 at 14:30

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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).

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  • $\begingroup$ 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. $\endgroup$ Nov 14, 2019 at 19:18

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