nLab homotopical category

1. Idea

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 $(\infty,1)$-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.

2. Definition

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).

3. Related concepts

• relative category

• category of fibrant objects

• cofibration category

• Waldhausen category

• model category

• calculus of fractions

• resolution

4. References

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):

• Marco Grandis, On the categorical foundations of homological and homotopical algebra, Cahiers de Topologie et Géométrie Différentielle Catégoriques 33 2 (1992) 135-175 [numdam:CTGDC_1992__33_2_135_0]

• Marco Grandis, Homotopical algebra in homotopical categories, Applied Categorical Structures 2 (1994) 351–406 [doi:10.1007/BF00873039]

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

• William Dwyer, Philip Hirschhorn, Daniel Kan, Jeff Smith, p. 23 & pp. 96 of: Homotopy Limit Functors on Model Categories and Homotopical Categories, Mathematical Surveys and Monographs 113 AMS (2004) [ISBN: 978-1-4704-1340-8, pdf]

Authors following this terminology:

• Karol Szumiło, Two Models for the Homotopy Theory of Cocomplete Homotopy Theories, Bonn (2014) [arXiv:1411.0303, hdl:20.500.11811/6136]

• Jeroen Hekking, Segal Objects in Homotopical Categories & K-theory of Proto-exact Categories, Leiden (2017) [pdf]

• Chris Kapulkin, Peter LeFanu Lumsdaine, Homotopical inverse diagrams in categories with attributes, Journal of Pure and Applied Algebra 225 4 (2021) 106563 [doi:10.1016/j.jpaa.2020.106563]

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

• Karol Szumiło, slide 10 of: Internal Languages of Higher Categories (2019) [pdf]

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:

• Julie Bergner, An introduction to homotopical categories, lecture at MSRI (2014) [part 1:YT, 2:YT]

• Emily Riehl, Homotopical categories: from model categories to (∞,1)-categories [arXiv:1904.00886] in: Andrew J. Blumberg, Teena Gerhardt, Michael A. Hill (eds,) Stable categories and structured ring spectra, MSRI Book Series, Cambridge University Press (2022) [ISBN:9781009123297]

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

• Peter Arndt, Homotopical categories of logics, in: The Road to Universal Logic, Studies in Universal Logic Birkhäuser (2015) [pdf, doi:10.1007/978-3-319-10193-4_2]

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.”