> 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/discrete-category.md).

# Discrete category

> A discrete category is a category whose only morphisms are the identity morphisms.

Given a set `𝐴`, we get a discrete category `𝒞`, in which the objects are the elements of `𝐴` and the morphisms are the identity morphisms, one for each `𝑥 ∈ 𝐴`, which are uniquely determined by the identity axiom. A discrete category is so determined by its objects, which correspond exactly to its identity morphisms.