# Yoneda lemma for enriched categories

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.

• Have a look at this. – Pece Oct 3 '14 at 5:09
• @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? – Daniel Robert-Nicoud Oct 3 '14 at 9:12

## 1 Answer

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.

• 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". – Vladimir Sotirov Aug 27 '16 at 0:54
• 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 – Vladimir Sotirov Oct 21 '16 at 19:00