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THe old version had ’paragraph 33’ of DHKS as a reference. I changed it to page 23, which gives the definition. (Paragraphs are not numbered as such in DHKS, although one could refer to the subsections as paragraphs which seems, to me, a bit confisuing. Subsection 33 is on the topic but starts on page 96.)
The second reference (Riehl) uses the term “homotopical category” in a different sense from Dwyer–Hirschhorn–Kan–Smith. As far as I can see, there is no material in that reference that is specific to DHKS homotopical categories, i.e., relative categories that satisfy the 2-out-of-6 property.
I have re-worked the text of the entry and added more references. Being currently on family vacation with drastically limited resources for such matters, here is what I found:
The earliest usage of “homotopical categories” that I have seen so far is
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]
who means 1-categories equipped with the extra structure of homotopies/2-morphisms subject (only) to horizontal composition (this he calls “h-categories”) and then furthermore required to have some simple homotopy co/limits with respect to this structure.
The exact definition by Dwyer et al. is picked up for instance in
but the requirement for 2-out-of-6 seems to again be discarded in
Besides Emily Riehl, another author who says “homotopical category” in an unspecified way but apparently thinking mainly of model categories (and hence of special cases of homotopical categories in the sense of Dwyer et al. 2004) is Julie Bergner in:
(though maybe she also means to use the term as synonymous with “-category”; at least resellers advertise her book on -categories as “an introductory treatment to the homotopy theory of homotopical categories”).
Then there is
where the “homotopical categories” appearing (only?) in the title seem to be meant in the generality of relative categories that the main text is about.
Bergner’s terminological attitude towards “homotopical categories” is more explicit in:
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.”
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