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nLab > Latest Changes: walking isomorphism

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  1. Little page to focus on this important notion, as opposed to the general remarks at walking structure.

    v1, current

    • CommentRowNumber2.
    • CommentAuthorYuxi Liu
    • CommentTimeJul 4th 2020

    explained what “walking” means

    diff, v3, current

    • CommentRowNumber3.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 4th 2020
    • (edited Jul 4th 2020)

    This should be tied in or amalgamated with interval groupoid, which is a redirect for (and section of) interval category.

  4. Yes, I agree. I will attempt something now, and then others can refine if needed.

  5. Adding redirect for ’interval groupoid’, and adding remark that the walking isomorphism is the free groupoid on the walking arrow, which was previously at interval category.

    diff, v4, current

  6. Noting various 2-categories which categorify it. I will add the walking 2-isomorphism later, which I am intending to use at Lack fibration; run out of time for the moment!

    diff, v5, current

    • CommentRowNumber7.
    • CommentAuthorRichard Williamson
    • CommentTimeJul 5th 2020
    • (edited Jul 5th 2020)

    Actually what I wish to use at Lack fibration is something with trivial boundary, i.e. only one object and only identity 1-arrows, and exactly two non-identity 2-arrows which are the inverses of a 2-isomorphism (I think the horizontal composites are dictated by the interchange law to be the identity then). This is not really the walking 2-isomorphism, in that it does not represent 2-isomorphisms. Let me know if you have a suggestion for alternative terminology!

    • CommentRowNumber8.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJul 5th 2020

    Strengthened the proposition to an explicit bijection between functors and isomorphisms.

    diff, v6, current

    • CommentRowNumber9.
    • CommentAuthorHurkyl
    • CommentTimeSep 4th 2020

    I have a question about the nerve of the walking isomorphism.

    Let J(n)⊆N(J)J(n) \subseteq N(J) be the subsimplicial set that is the image of the nondegenerate nn-simplex starting from vertex 00.

    I’m pretty sure that for n≥3n \geq 3, J(n)→N(J)J(n) \to N(J) is supposed to be a categorical equivalence of simplicial sets (i.e. a weak equivalence in the Joyal model structure for quasi-categories), and I know that the inclusion J(n)→J(n+1)J(n) \to J(n+1) is left anodyne, being the pushout of Λ 0 n+1→Δ n+1\Lambda^{n+1}_0 \to \Delta^{n+1}. (and therefore J(3)→JJ(3) \to J is left anodyne)

    Are the inclusions J(n)→J(n+1)J(n) \to J(n+1) inner anodyne? I thought I’ve worked out once that they are but I can’t reconstruct the argument, and the only references I can find at the moment on related issues are in terms of an analysis of the construction via left anodyne maps.

    • CommentRowNumber10.
    • CommentAuthorAlexanderCampbell
    • CommentTimeSep 4th 2020
    • (edited Sep 4th 2020)

    I can’t see how to prove that J(n)→J(n+1)J(n) \to J(n+1) is inner anodyne. However, observe that, if n≥3n \geq 3, it does have the left lifting property with respect to all inner fibrations between quasi-categories, by Joyal’s lifting theorem. Since JJ is a quasi-category, it follows that J(n)→JJ(n) \to J is inner anodyne, for n≥3n \geq 3.

    • CommentRowNumber11.
    • CommentAuthorHurkyl
    • CommentTimeSep 5th 2020
    • (edited Sep 5th 2020)

    Now that I’ve written out my question, I (think I) was able to remember my argument, and where my memory about the conclusion went wrong. Basically, it involved taking the pushout along a monomorphism of a splitting of an inner anodyne map, so I was only constructing trivial cofibrations in the Joyal model structure, not inner anodyne maps.

    • CommentRowNumber12.
    • CommentAuthorHurkyl
    • CommentTimeSep 8th 2020

    Included the fact that the interval groupoid is the full subcategory classifier of Cat and qCat, and also the full subspace classifier of sSet.

    diff, v8, current

    • CommentRowNumber13.
    • CommentAuthorDavidRoberts
    • CommentTimeSep 8th 2020

    @Richard #7

    Actually what I wish to use at Lack fibration is something with trivial boundary, i.e. only one object and only identity 1-arrows, and exactly two non-identity 2-arrows which are the inverses of a 2-isomorphism

    Isn’t this B 2ℤ/3\mathbf{B}^2\mathbb{Z}/3? It has a unique object, a unique 1-arrow, and three 2-arrows. You can think of the latter as being {−1,0,+1}\{-1,0,+1\} under addition, if you like.

    • CommentRowNumber14.
    • CommentAuthormattecapu
    • CommentTimeApr 24th 2023

    improved the first two lines a bit

    diff, v9, current

    • CommentRowNumber15.
    • CommentAuthormattecapu
    • CommentTimeApr 24th 2023

    gave a more precise definition in terms of generators and relations

    diff, v9, current

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeApr 24th 2023

    Further adjusted wording and typesetting of the first couple of paragraphs.

    diff, v10, current

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeApr 25th 2023

    Added a reference:

    A generalization of the notion of the interval groupoid to simplicial groupoids is considered in

    and plays a key role in the discussion of the model structure on simplicial groupoids, see there.

    diff, v11, current

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