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nLab > Latest Changes: Batanin omega-category

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeNov 13th 2012

    I fixed some of the references at Batanin omega-category.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeNov 13th 2012
    • (edited Nov 13th 2012)

    Here is a question:

    around corollary 9.4 of

    • Stephen Lack, A Quillen model structure for Gray-categories (arXiv:1001.2366)

    it is shown for the special case of Gray-categories that the weak ω\omega-functors of

    are actually the resolved morphisms with respect of a canonical model category structure, out of cofibrant resolutions into fibrant objects.

    Is an analog of this known for general Batanin ω\omega-categories? Is there a canonical model structure on Batanin ω\omega-categories such that the weak ω\omega-functors represent the correct derived hom-space?

    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeNov 13th 2012

    AFAIK that is not known.

    • CommentRowNumber4.
    • CommentAuthorJohn Baez
    • CommentTimeSep 18th 2023

    Added link to Kachour’s paper and corrected title.

    diff, v18, current

    • CommentRowNumber5.
    • CommentAuthorJohn Baez
    • CommentTimeSep 19th 2023

    I added a reference to some papers by Cheng and Leinster on Batanin (and Trimble) infinity-categories.

    Is there any good reason why this entry is called “Batanin omega-category” while the one on another approach is called “Grothendieck-Maltsiniotis infinity-category”? Maybe people used to use “omega-category” to mean “weak infinity-category” - I forget. That may old-fashioned. But either way, it seems we should use the same term both for the Batanin infinity-categories and the Grothendieck-Maltsiniotis infinity-categories.

    diff, v19, current

    • CommentRowNumber6.
    • CommentAuthorzskoda
    • CommentTimeSep 19th 2023
    • (edited Sep 19th 2023)

    5: This is the self-consistent technical terminology in Batanin’s paper, rather than newspaper’s description or an informal discussion. If one is to change it, one has to do this systematically within the formal framework. Batanin says

    A weak ω\omega-category is a globular set together with the structure of algebra over a universal contractible ω\omega-operad.

    Hence, only if one disagrees with calling this kind of higher operads ω\omega-operads then it makes sense to disgree with ω\omega-category. So how would you call the kind of higher operads from Batanin’s paper and in Batanin’s context as opposed to other kinds from literature?

    Of course, you are right in the sense that Malstiniotis-Grothendieck notion is in essence a variant/sister notion, but is it formally defined in the same setup ?

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeSep 21st 2023

    And maybe it does make sense to keep referring to “omega”-structure unless and until it is proven that they are indeed equivalent to the corresponding infinity-structures.

    (Not that “omega” is particularly suggestive of that distinction (or of anything of relevance, for that matter?!) but at least it is a different term than “infinity” that prevents confusion while the hoped-for equivalences have not been established… yet.)

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