---
tag: K7A04ZBD92
kind: concept
name: Equivalence of categories
msc: [18A22, 18A05]
sources:
  - url: https://en.wikipedia.org/wiki/Equivalence_of_categories
same_as:
  wikidata: Q1082992
  wikipedia: Equivalence of categories
---

## Summary

An equivalence of categories is a [[functor|28NQ0KRHD3]] $F \colon \mathcal{C} \to \mathcal{D}$ that has a functor $G \colon \mathcal{D} \to \mathcal{C}$ in the other direction.
The composites $G \circ F$ and $F \circ G$ must be naturally isomorphic to the identity functors.
It is the right notion of "sameness" for categories: it ignores how many copies of each object up to isomorphism a category has.
A functor is an equivalence exactly when it is [[fully faithful|GRG8WEBSH5]] and [[essentially surjective|6ERBSBVF14]], by [[this theorem|WA60RW9E04]].
Every category is equivalent to its [[skeleton|A49QJYG669]].

## Explanations

### L1 human

Two categories are equivalent when they have the same objects up to isomorphism and the same arrows between them.
Isomorphism of categories asks for more: an exact match of objects, which is rarely useful.
For example, the category of finite-dimensional real vector spaces has a huge number of objects.
The category with one object $n$ for each natural number, and with $m \times n$ real matrices as the arrows from $n$ to $m$, has very few.
Yet the two are equivalent, which is why linear maps between finite-dimensional spaces can be studied as matrices.