Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration finite foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf sheaves simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).

nLab > Latest Changes: walking isomorphism

Bottom of Page

1 to 17 of 17

  1.  

  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

1 to 17 of 17

  1.