nLab
(infinity,1)-category theory

Idea

In the pattern of (n,r)-categories the notion of (∞,1)-category is special in that it is precisely the structure that

• admits homotopy theory in that it encodes the notion of higher coherent equivalence;

• admits category theory in terms of universal constructions: limits, adjunctions, Grothendieck constructions, etc.

This entry surveys the category theory of $(\mathrm{\infty }\mathrm{,1})$-categories .

Basic notions

• homotopy category of an (∞,1)-category

• presentation by a category with weak equivalences: Dwyer-Kan localization

• (∞,1)-functor

• (∞,1)-category of (∞,1)-categories

• the archetypical $(\mathrm{\infty }\mathrm{,1})$-category: ∞Grpd

• (∞,1)-category of (∞,1)-presheaves

• opposite quasi-category

• terminal object in a quasi-category

Universal constructions

The universal constructions of category theory generalize, with unique existence of universal morphisms replaced by the requirement of a contractible space of universal morphisms.

• limit in a quasi-category

• adjoint (∞,1)-functor

• (∞,1)-Grothendieck construction