# Yoneda lemma for monoidal categories

I am looking at the Yoneda lemma trying to see where the assumption of "locally small" really comes in. Obviously in order to define a functor to the category sets using $$Hom$$-spaces we need our $$Hom$$-spaces to be sets. However if we consider a enriched-category, enriched over some non-locally small monoidal category M, then for any element of the category, our $$Hom$$-sets give us a functor $$Hom(A,-)$$ to M.

In particlar, in the statement $$Hom(Hom(-,A),F) \simeq F(A),$$ where $$F$$ is a set-valued functor, where does the assumption of "smallness" play a role.

In the answer to this question, it is stated that the category of sets can be replaced by any Grothendieck universe $$U$$. However, the definition of a Grothendieck universe assumes that $$U$$ is a set. Moreover, the enriched Yoneda lemma again assumes "smallness". In these answers, is smallness a necessary assumption?

• Yes, there is an enriched Yoneda lemma: see this question, or this nLab page. Commented Nov 26, 2020 at 15:01
• But in the nLab version they require that the monoidal category is locally small. This seems like a more special version rather than a generalisation. Commented Nov 26, 2020 at 15:14
• I would need to check more carefully, but I think you are right: I think the only size assumption are in order for the enriched ends defining $V^C(F,G)$ to exists. But assuming that C has a set of objects, or rather that $V$ has limits of size the set of objects (of isomorphism class actually) of C seem to be enough for this. It does not seems usefull to assume that $C$ or $V$ are locally small. Commented Nov 26, 2020 at 16:08
• I don't understand the question. Is it "why does the nLab page on the enriched Yoneda lemma assume that the monoidal category is locally small?" If so then I think Simon is right that it's not necessary; probably whoever wrote the page was just throwing in a sufficient set of assumptions rather than thinking about which were actually necessary. Commented Nov 26, 2020 at 17:26
• @Mike: Yes, this is precisely the question. I want to know if the assumption is necessary or not. If it is not, I would be much happier :) Commented Nov 26, 2020 at 19:01

## 1 Answer

The Yoneda lemma is a purely formal result that does not require any size assumptions. For any closed symmetric monoidal category $$\mathbf{V}$$, any $$\mathbf{V}$$-category $$C$$, any object $$A\in C$$, and any functor $$F:C\to \mathbf{V}$$, there is an isomorphism

$$[C,\mathbf{V}](よ^A,F) \cong F(A).$$

Here $$よ^A$$ denotes the hom-functor $$C(A,-)$$ and $$[C,\mathbf{V}]$$ denotes the $$\mathbf{V}$$-enriched hom-category. It is true that one needs $$\mathbf{V}$$ to have limits of the size of $$C$$ in order for $$[C,\mathbf{V}]$$ to exist as a $$\mathbf{V}$$-category, but even if this fails, the statement is true and provable in the following sense: if we write down the diagram whose limit would, if it existed, be the LHS, then the RHS is a limit of that diagram.

It is even possible to formulate and prove versions of the Yoneda lemma that do not require $$\mathbf{V}$$ to be closed or symmetric, and even that allow it to be a multicategory rather than a monoidal category. See, for instance, Lemma 5.29 of my paper enriched indexed categories, or proposition 8.2 of my paper with Richard Garner, enriched categories as a free cocompletion for the bicategorical case.

• where does the notation $よ$ come from? Is it a Japanese character? Commented Nov 27, 2020 at 1:36
• Commented Nov 27, 2020 at 4:27