nLab homotopical category


Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology



Paths and cylinders

Homotopy groups

Basic facts


(,1)(\infty,1)-Category theory



A homotopical category is a structure used in homotopy theory, related to but more flexible than a model category.


A homotopical category is a category with a distinguished class of morphisms (called ‘weak equivalences’) satisfying the following conditions:

  • Every identity map is a weak equivalence.

  • It has the 2-out-of-6-property: if morphisms hgh \circ g and gfg \circ f are weak equivalences, then so are ff, gg, hh and hgfh \circ g \circ f.


Simplicial localization

Every homotopical category CC “presents” or “models” an (infinity,1)-category LCL C, a simplicially enriched category called the simplicial localization of CC, which is in some sense the universal solution to inverting the weak equivalence up to higher categorical morphisms.


This definition is in page 23 of

with the main development of the concept starting in subsection 33 on page 96.

Survey with an eye towards (∞,1)-categories:

Last revised on December 19, 2021 at 13:17:15. See the history of this page for a list of all contributions to it.