> 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/03-key-concepts/homeset.md).

# Homeset

<https://en.wikipedia.org/wiki/Hom-set>

In a category `𝒞`, the collection of all morphisms between two objects `A` to `B`, is the *hom-set* between `A` and `B`, is most often denoted by `Hom(A,B)`.

Other notations: Homᶜ(A,B), Arrᶜ(A,B), Arr(A,B), 𝒞(A,B).

The term hom-set is a misnomer because the collection of morphisms is not required to be a set. A category where `Hom(A,B)` is indeed a set, for all objects `A` and `B`, is called *locally small*.

The domain and codomain are in fact part of the definition of morphisms. For example, in the category of sets, where morphisms are functions, two functions may be identical in terms of the sets of ordered pairs - they may have the same range, while having different codomains. However, from the viewpoint of category theory, such functions are different. Therefore, many authors require that the hom-classes `Hom(A,B)` be *disjoint* (set/class/collection). In practice, this is not a problem because if the disjointness does not hold, it can be assured by appending the domain and codomain, e.g. as `(Hom(A,B), A, B)` such that an ordered triple represents homset of morphisms.