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nLab > Latest Changes: walking equivalence

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  1. Quick page, analogous to walking isomorphism.

    v1, current

    • CommentRowNumber2.
    • CommentAuthorYuxi Liu
    • CommentTimeJul 4th 2020

    explained what “walking” means

    diff, v2, current

    • CommentRowNumber3.
    • CommentAuthorDavidRoberts
    • CommentTimeJul 4th 2020

    Just because I’m marking exams and can’t think about this right now: might it be worth doing the walking adjoint equivalence? It would presumably be a quotient of Adj, and there might be an interesting relationship between the walking equivalence and the walking adjoint equivalence, seeing at equivalences can (generally?) be upgraded to an adjoint equivalence (see Theorem 3.3 as numbered in the current version), at the cost of perhaps changing part of the data.

  4. Thanks for chiming in! Absolutely! My recent edits are mainly building up to adding some material to the new page Lack fibration and some material on canonical model structures on higher categories; I think the walking adjoint equivalence will be needed for this.

    • CommentRowNumber5.
    • CommentAuthorMike Shulman
    • CommentTimeJul 4th 2020

    Corrected the definition – there is more than one arrow 0→10\to 1 and 1→01\to 0 (e.g. i∘i −1∘ii\circ i^{-1}\circ i), and the definition as stated didn’t imply that ι 0\iota_0 and ι 1\iota_1 are isomorphisms. Also, the representing functor FF is not unique.

    diff, v3, current

    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeJul 4th 2020

    I actually think that “the walking equivalence” should be an adjoint equivalence. I can’t think of any context in which one would want to use the walking non-adjoint equivalence. However, for the moment I refrained from making that change myself.

  7. Thanks for the corrections! Writing very fast, as I have to due to circumstances, apologies!

    Yes, if we could wait until I’m finished with the stuff I plan to add to Lack fibration before changing things with regard to walking equivalence vs walking adjoint equivalence, that would be great.

    • CommentRowNumber8.
    • CommentAuthorDavidRoberts
    • CommentTimeJul 4th 2020

    The walking equivalence is definitely worth a mention, though, even if the ’correct’ notion is the walking adjoint equivalence. Even if to point out it only works in some cases. Also, there’s connection to the HoTT definition of equivalence, where the information is packaged differently (an a priori distinct left and a right quasi-inverse etc), or some of the Riehl–Verity ideas around coherent adjunctions.

  9. Yes, I agree, we should definitely mention it somehow. I guess Mike was thinking about what should be the default meaning of the term ’walking equivalence’; let’s disuss that a little further down the road!

    Back to #5: I agree that as I stated it the representing functor is not unique, but I think we should strengthen the hypotheses a little to make it unique, at least up-to-something. I’ll have a think myself about how to do this a little later if nobody else gets to it; but if anyone has a preference for how to state it, please just go ahead,

    • CommentRowNumber10.
    • CommentAuthorDavidRoberts
    • CommentTimeJul 5th 2020

    Deleted the following remark as redundant, after the updating of the definition, which now states ι 0\iota_0 and ι 1\iota_1 are 2-isomorphisms up front.

    The 2-arrows ι 0:i −1∘i→id(0)\iota_{0}: i^{-1} \circ i \rightarrow id(0) and ι 1:i∘i −1→id(1)\iota_{1}: i \circ i^{-1} \rightarrow id(1) are 2-isomorphisms, whose inverses are the 2-arrows ι 0 −1\iota_{0}^{-1} and ι 1 −1\iota_{1}^{-1}.

    diff, v5, current

    • CommentRowNumber11.
    • CommentAuthorYuxi Liu
    • CommentTimeJul 5th 2020

    explained the representation

    diff, v6, current

    • CommentRowNumber12.
    • CommentAuthorRichard Williamson
    • CommentTimeJul 5th 2020
    • (edited Jul 5th 2020)

    Regarding #5 and #10: I do feel that the definition is kind of cheating now! The purpose of the definition as I initially gave it was to give an explicit-as-possible description. What I wrote was not far from correct, just missing a couple of additional clarifications. We could give both phrasings, of course.

    • CommentRowNumber13.
    • CommentAuthorDavidRoberts
    • CommentTimeJul 5th 2020

    Re #12 yes, that seems good.

    • CommentRowNumber14.
    • CommentAuthorMike Shulman
    • CommentTimeJul 5th 2020

    The walking non-adjoint equivalence is worth mentioning, but I don’t think it deserves its own page. The alternative definitions mentioned in #8 are all different constructions of the walking adjoint equivalence.

    Re #9, the representing functor is by construction unique if we specify to begin with the entire (non-adjoint) equivalence data (a morphism, its quasi-inverse, and two witnessing 2-isomorphisms). If the original data is just a morphism with the property of being an equivalence, then the representing functor is not unique even up to anything I can think of – but if we used the walking adjoint equivalence then it would be unique up to unique isomorphism.

  15. Tweaked the definition a bit to express it slightly more formally as a quotient of a free strict 2-category. Strengthened the hypotheses of the proposition about representability so that the representing functor is unique. Added the walking semi-strict equivalence. Observed that the walking semi-strict equivalence defines an interval which can be equipped with all the structures of my thesis, and hence that we obtain a ’canonical model structure’ on the category of strict 2-categories in which every object is both fibrant and cofibrant, where the fibrations and cofibrations are ’Hurewicz’ fibrations and cofibrations respectively with respect to the interval defined by the free-standing semi-strict equivalence. I will add more on this to a different page later.

    diff, v7, current

    • CommentRowNumber16.
    • CommentAuthorMike Shulman
    • CommentTimeJul 13th 2020

    This isn’t right either. Your F ≤1F_{\le 1} contains composites like i −1∘ii^{-1}\circ i, but its underlying reflexive globular set doesn’t remember that these are in fact composites. So your FF contains the “i −1∘ii^{-1}\circ i” from F ≤1F_{\le 1} as a generator, but also a new 1-cell i −1∘ii^{-1}\circ i that is actually the composite of the generators i −1i^{-1} and ii in FF. And so on.

    • CommentRowNumber17.
    • CommentAuthorMike Shulman
    • CommentTimeJul 13th 2020

    I think a more standard name for your “semi-strict equivalence” would be a “retract equivalence” (the categorical version of a deformation retract).

    Can you describe explicitly the cofibrations, fibrations, and weak equivalences in this model structure?

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