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    • CommentRowNumber1.
    • CommentAuthorTim_Porter
    • CommentTimeDec 17th 2018
    • (edited Dec 17th 2018)

    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.)

    diff, v15, current

    • CommentRowNumber2.
    • CommentAuthorMarc
    • CommentTimeDec 19th 2018

    edited Definition 2 (following DHK 33.1) so that Remark 3 does not appear circular.

    diff, v17, current

    • CommentRowNumber3.
    • CommentAuthorMarc
    • CommentTimeDec 19th 2018

    streamlined Remark 3.1 with rephrased Definition 2

    diff, v17, current

    • CommentRowNumber4.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJul 17th 2023

    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.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeJul 24th 2023
    • (edited Jul 25th 2023)

    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

    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:

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

    (though maybe she also means to use the term as synonymous with “(,1)(\infty,1)-category”; at least resellers advertise her book on (,1)(\infty,1)-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.

    diff, v27, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJul 25th 2023
    • (edited Jul 25th 2023)

    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.”

    diff, v28, current