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 and in weak forms such as bicategories, where composition is associative only up to higher morphisms. The theory of (∞,1)-categories, modelled for example by quasi-categories, is its most developed part. It is now a standard tool in homotopy theory and algebraic geometry. A guiding idea is the homotopy hypothesis, which relates $\infty$-groupoids to topological spaces.