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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?

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    $\begingroup$ Yes, there is an enriched Yoneda lemma: see this question, or this nLab page. $\endgroup$
    – varkor
    Commented Nov 26, 2020 at 15:01
  • $\begingroup$ 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. $\endgroup$
    – Jake Wetlock
    Commented Nov 26, 2020 at 15:14
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    $\begingroup$ 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. $\endgroup$
    – Simon Henry
    Commented Nov 26, 2020 at 16:08
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    $\begingroup$ 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. $\endgroup$
    – Mike Shulman
    Commented Nov 26, 2020 at 17:26
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    $\begingroup$ @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 :) $\endgroup$
    – Jake Wetlock
    Commented Nov 26, 2020 at 19:01

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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.

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