> For the complete documentation index, see [llms.txt](https://mandober.gitbook.io/math-debrief/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://mandober.gitbook.io/math-debrief/450-category-theory/10-categories/locally-small-category.md).

# Locally small category

A category is locally small if between any pair of objects there is only a set's worth of morphisms.

It is traditional to write `𝒞(X,Y)` or `Hom(X,Y)` for the set of morphisms from `X` to `Y` in a locally small category `𝒞`.

The set of arrows between a pair of fixed objects in a locally small category is typically called a *hom-set*, whether or not it is a set of homomorphisms of any particular kind.

Because the notation is so convenient, it is also adopted for the collection of morphisms between a fixed pair of objects in a category that is not necessarily locally small.