Enriched category has to be locally small

Question

I am reading material on enriched categories. The requirements for the category over which one enriches (be it $\textbf{Grp}$, $\textbf{Vect}_k$ are very clear, that they need to be monoidal and so). But what are the conditions on the category $\mathscr{C}$ that is being enriched? Some references say that one replace the hom-sets by hom-objects, which may imply that $\mathscr{C}$ is at least locally small. Is this correct, and why?

Answer 1

The comments from EBP and Berci are correct - I just want to give some more detail and an example. I quote EBP's comment here, since it's important:

When considering an enriched category, this is not necessarily done by enriching an ordinary category which already exists. We are simply defining an enriched category to be a class of objects together with hom objects for each pair of objects.

A locally small category is one in which the hom-sets are indeed sets: there is a set of morphisms between any pair of objects.

In an enriched category, one replaces these sets with an object in some other category, $\mathscr{V}$. If $\mathscr{V}$ has the form of "a set with some extra structure", such as the examples you mentioned, then it could make sense to think of the enriched category as "locally small", in the sense that you can still think of its hom-sets as being sets of morphisms. As Berci noted in the comments this can be made formal by specifying a faithful functor from $\mathscr{V}$ to $\mathsf{Set}$.

However, that need not be the case at all. A good example is a Lawvere metric space, which is a category enriched in the monoidal poset $([0,\infty],\ge,+)$. Then the thing that's assigned to each pair of objects isn't a set of morphisms but just a number, which can be interpreted as a distance. (It turns out that these "distances" are guaranteed to obey the triangle inequality, although they relax some of the other axioms of a metric space.) A good gentle introduction to Lawvere metric spaces and similar poset-enriched categories is found in chapter 2 of Spivak and Fong's Seven Sketches in Compositionality.

In the case of a Lawvere metric space it doesn't make much sense to ask whether the category is locally small, because its hom-objects are neither sets nor proper classes but something else entirely. So I would say that counts as a counterexample - enriched categories don't necessarily have to be locally small.