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nLab > Latest Changes: associative n-category

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  1.  
    • CommentRowNumber1.
    • CommentAuthorjamievicary
    • CommentTimeDec 7th 2018

    Stub for associative n-categories.

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeDec 7th 2018

    made page name singular, to comply with convention. Hyperlinked some more of the keywords

    I think there ought to be clarification which compositons exactly are associative. I gather all of them?

    diff, v2, current

    • CommentRowNumber3.
    • CommentAuthorTodd_Trimble
    • CommentTimeDec 7th 2018
    • (edited Dec 7th 2018)

    Also I would imagine there is some relevant prehistory here, such as Simpson’s weak units conjecture, and the analysis of this in dimension 3 by Joyal-Kock. I don’t know how this relates to the development described in the article.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeDec 8th 2018

    added pointers to

    diff, v3, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeDec 31st 2018

    added today’s arXiv identifier for

    diff, v4, current

    • CommentRowNumber6.
    • CommentAuthorChristoph Dorn
    • CommentTimeMar 14th 2023

    added pointers

    diff, v6, current

    • CommentRowNumber7.
    • CommentAuthorzskoda
    • CommentTimeMar 14th 2023

    Despite these strictness properties, it is conjectured that every weak n-category is weakly equivalent to an associative n-category with strict units.

    Isn’t this just a version of Simpson’s conjecture ? I would crosslink if I were sure in what conventionsassumption in this issue are.

    • CommentRowNumber8.
    • CommentAuthorChristoph Dorn
    • CommentTimeMar 15th 2023

    @zskoda well, I think it’s related but somewhat orthogonal. Roughly:

    • Simpson’s conjecture: weak identity laws ≃\simeq fully weak structure
    • The conjecture that’s alluded to here: weak exchange laws ≃\simeq fully weak structure (example: Gray categories ≅\cong fully weak tricategories)
    • CommentRowNumber9.
    • CommentAuthorChristoph Dorn
    • CommentTimeMar 15th 2023
    • addressed (much) earlier comments by Todd and Urs.
    • added examples.

    diff, v7, current

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeMar 29th 2023
    • (edited Mar 29th 2023)

    I have adjusted the first sentences to include more adjectives and links illuminating what the concept is about:

    The notion of associative nn-categories (ANCs, Dorn 2018, Reutter & Vicary 2019) is a semi-strict algebraic model for higher category theory based on the geometric shape of globular sets with strictly associative composition: All higher coherence laws in associative nn-categories are strict, except for weak versions of higher exchange laws.

    Notably the fact that “associative nn-categories” are modeled on globular sets was not mentioned before, nor was there a link to Globular nor homotopy.io (which I have added now).

    diff, v8, current

1 to 10 of 10

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