---
tag: 3A1NMGJKJ3
kind: concept
name: Higher category theory
msc: [18N65, 18N60, 18N20]
sources:
  - url: https://en.wikipedia.org/wiki/Higher_category_theory
same_as:
  wikidata: Q5758318
  wikipedia: Higher category theory
---

## Summary

Higher category theory studies structures with morphisms between objects, 2-morphisms between morphisms, 3-morphisms between 2-morphisms, and so on.
Its basic notions are $n$-categories and $\infty$-categories.
They come in strict forms such as [[2-categories|CQACXG10JZ]] and in weak forms such as [[bicategories|TAHV24ED74]], where composition is associative only up to higher morphisms.
The theory of [[(∞,1)-categories|YFY1FWA1MY]], modelled for example by [[quasi-categories|9328WEVJEK]], is its most developed part.
It is now a standard tool in homotopy theory and algebraic geometry.
A guiding idea is the [[homotopy hypothesis|2E83D9ACDY]], which relates $\infty$-groupoids to topological spaces.

## Explanations

### L1 human

In an ordinary category, two arrows between the same objects are either equal or not.
In many examples, though, there are ways to compare arrows, such as homotopies between continuous maps or natural transformations between functors.
Higher category theory treats these comparisons as arrows of a higher level, and keeps going to comparisons of comparisons.
For example, the paths in a space, the homotopies between paths, the homotopies between homotopies, and so on, form an $\infty$-groupoid.