---
tag: CQACXG10JZ
kind: concept
name: 2-category
msc: [18N10]
sources:
  - url: https://en.wikipedia.org/wiki/2-category
same_as:
  wikidata: Q4596935
  wikipedia: 2-category
---

## Summary

A (strict) 2-category has objects, morphisms between objects, and 2-morphisms between parallel morphisms.
The 2-morphisms compose in two directions, vertically and horizontally, and the two compositions satisfy the interchange law.
The main example is the 2-category of small [[categories|A2DE66937D]], [[functors|28NQ0KRHD3]] and [[natural transformations|KYCPCC57FT]].
Formally, a 2-category is a category [[enriched|BPH9SJ30Y0]] over the category of small categories.

## Explanations

### L1 human

In a 2-category there are not only arrows between objects, but also arrows between arrows.
Picture objects as points, morphisms as paths between points, and 2-morphisms as surfaces filling the space between two paths with the same ends.
Surfaces can be glued along a common path (vertical composition) or side by side at a common point (horizontal composition).
Categories, functors and natural transformations behave exactly like this: a natural transformation goes from one functor to another.