Let $\mathcal{M}$ be a monoidal category. Is there a generalization of the Yoneda lemma to categories enriched over $\mathcal{M}$?

In the specific case I need, $\mathcal{M}$ would be the category $k\operatorname{-Vect}$ of $k$-vector spaces.

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    $\begingroup$ Have a look at this. $\endgroup$ – Pece Oct 3 '14 at 5:09
  • $\begingroup$ @Pece That was exactly what I was looking for. Thanks! Would you like to summarize the statement and give the link in an answer, so that I can accept it? $\endgroup$ – Daniel Robert-Nicoud Oct 3 '14 at 9:12

Yes, see here. A standard reference is Kelly's Basic concepts of enriched category theory, TAC reprint. The case of $k$-linear categories is very simple (basically because here the monoidal unit is a generator). Here, $k$ can be any commutative ring. For $k$-linear functors $F : \mathcal{C} \to \mathsf{Mod}_k$ and objects $A \in \mathrm{Ob}(\mathcal{C})$ there is a canonical $k$-linear map $$\mathrm{Hom}\bigl(\mathrm{Hom}(A,-),F\bigr) \to F(A)$$ which is an isomorphism since by the usual Yoneda Lemma the underlying map of sets is an isomorphism.

  • $\begingroup$ You should indicate that this generalization is to enrichments in (monoidal) closed categories (unless I'm mistaken and there is a way to generalize the Yoneda lemma for enrichments in non-closed monoidal categories, but I don't think I am and would love a reference); in particular the correct answer to the question asked would be "No, unless the monoidal category is closed, in which case yes". $\endgroup$ – Vladimir Sotirov Aug 27 '16 at 0:54
  • $\begingroup$ What do you know, a reference was published 4 days after I left the comment... tac.mta.ca/tac/volumes/31/29/31-29abs.html $\endgroup$ – Vladimir Sotirov Oct 21 '16 at 19:00

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