---
tag: A49QJYG669
kind: concept
name: Skeleton of a category
msc: [18A05, 18A22]
sources:
  - url: https://en.wikipedia.org/wiki/Skeleton_(category_theory)
same_as:
  wikidata: Q2291918
  wikipedia: Skeleton (category theory)
---

## Summary

A skeleton of a [[category|A2DE66937D]] $\mathcal{C}$ is a [[full subcategory|V3BFGKBJF7]] that contains exactly one object from each isomorphism class of objects of $\mathcal{C}$.
Under the axiom of choice every category has a skeleton, and the inclusion of the skeleton is an [[equivalence of categories|K7A04ZBD92]].
So a skeleton is a smallest model of the category up to equivalence, and two categories are equivalent exactly when their skeletons are isomorphic.

## Explanations

### L1 human

Many objects of a category are copies of each other up to isomorphism.
A skeleton keeps one representative of each kind and throws the copies away, while keeping all arrows between the representatives.
For finite sets, the sets $\set{1, \dots, n}$, one for each $n \ge 0$, form a skeleton.
Nothing essential is lost, because every finite set is in bijection with exactly one of them.