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    • CommentRowNumber1.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 4th 2018

    I put the opening material into an Idea section, and replaced several ’?’s with ∞.

    diff, v20, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeFeb 19th 2020
    • (edited Feb 20th 2020)

    In diuscussion with laymen I noticed that this old entry didn’t explain very much at all. I have now expanded the Idea-section as follows:


    Grpd\infty Grpd is the (∞,1)-category of ∞-groupoids, i.e. of (∞,0)-categories. This is the archetypical (∞,1)-topos, the home of classical homotopy theory.

    Equivalently this means all of the following:

    1. Grpd\infty Grpd is the simplicial localization of the category Top k{}_k of (weakly Hausdorff) locally compact topological spaces at the weak homotopy equivalences. As such it is the ∞-category-enhancement of the classical homotopy category: τ 0(Grpd)\tau_0(\infty Grpd) \simeq Ho(Top), itself presented by the classical model structure on topological spaces: GrpdL wheTop k\infty Grpd \simeq L_{whe} Top_k.

    2. Grpd\infty Grpd is the simplicial localization of the category sSet of simplicial sets at the simplicial weak homotopy equivalences. As such it is the ∞-category-enhancement of the classical homotopy category: τ 0(Grpd)\tau_0(\infty Grpd) \simeq Ho(sSet), itself presented by the classical model structure on simplicial sets: GrpdL whesSet\infty Grpd \simeq L_{whe} sSet.

      Hence, as a Kan-complex enriched category (a fibrant object in the model structure on sSet-categories) Grpd\infty Grpd is the full sSet enriched-subcategory in sSet on those that are Kan complexes.

    3. Grpd\infty Grpd is the full sub-(∞,1)-category of (∞,1)Cat on those (∞,1)-categories that are ∞-groupoids.


    diff, v22, current

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