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



By a homotopical category authors tend to mean something like a relative category/category with weak equivalences, possibly satisfying further axioms (notably two-out-of-six, as in Dwyer, Hirschhorn, Kan & Smith 2004), but in any case a 1-category equipped with a class of morphisms to be called weak equivalences and satisfying some extra properties.

The terminology is that of a concept with an attitude: One is interested in the localization/homotopy category with respect to the weak equivalences, or rather in the simplicial localization, hence in the ( , 1 ) (\infty,1) -category and hence “the homotopy theory” presented by this data.

(An earlier proposal by Grandis 1992, 1994 to say “homotopical categories” for 1-categories equipped instead with extra structure of homotopies/2-morphisms subject (only) to horizontal composition does not seem to have caught on.)

Accordingly, a functor between the underlying categories of homotopical categories which preserves the weak equivalences is called a homotopical functor.


Most authors seem to agree that a homotopical category is at least a strict 1-category equipped with a sub-class of morphisms to be called the weak equivalences, closed under composition and containing all identity morphisms, hence forming a wide subcategory (a relative category).

The authors Dwyer, Hirschhorn, Kan & Smith (2004) require that the weak equivalences satisfy the 2-out-of-6-property (this includes all model categories, see here) and give a detailed discussion. (Notice that — with the assumption that all identity morphisms are among the weak equivalences — 2-out-of-6 implies the 2-out-of-3 property required for categories with weak equivalences.)

This definition of “homotopical category” has found more followers, e.g. Szumiło (2014), Hekking (2017).

But some authors [Bergner (2014), Riehl (2019)] use the term “homotopical category” more vaguely, apparently thinking at least of categories with weak equivalences but focusing on examples that do satisfy also the two-out-of-six property (without mentioning this property).

On the other hand, Arndt (2015) seems to use “homotopical categories” as a synonym for “relative categories” and Szumiło (2019) seems to use it as synonymous with “category with weak equivalences”, see also Bergner (2019).


An early notion of “homotopical categories” as 1-categories equipped with homotopies subject to horizontal composition (hence extra structure mess than but in the direction of 2-category-structure):

The notion of “homotopical categories” as relative categories with the requirement that the weak equivalences satisfy the 2-out-of-6 property:

Authors following this terminology:

On the other hand, “homotopical categories” is used as synonymous with “category with weak equivalences” in:

Usage of “homotopical categories” understood with more relaxed or unspecified axioms on the weak equivalences but focusing on examples which are homotopical in the above sense:

Usage of “homotopical categories” as, apparently, synonymous with relative categories:

see also:

  • Julie Bergner, MAA review (2019) [web] of: Denis-Charles Cisinski‘s Higher Categories and Homotopical Algebra

    “the two subjects [homotopy theory and category theory] have come together in a deep way in the development of what one might call higher homotopical categories. The idea is to consider something like a category, but whose morphisms from one object to another form a topological space, rather than simply a set, and for which composition might only be defined up to homotopy. Such a structure turns out to have several other interpretations: a certain kind of higher category for which various higher morphisms are invertible (often called an (∞,1)-category or simply ∞-category), or even as a category with weak equivalences in the sense of abstract homotopy theory.”

Last revised on July 25, 2023 at 06:46:11. See the history of this page for a list of all contributions to it.