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nLab > Latest Changes: model structure on reduced simplicial sets

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeFeb 21st 2011
    • (edited Feb 21st 2011)

    have created model structure on reduced simplicial sets

    (I thought I had a vague memory that this or something similarly titled already existed, but apparently it didn’t).

    Also added a little bit more detail on the Quillen equivalence with simplicial groups here and there, notably in the last section at groupoid object in an (infinity,1)-category

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeApr 13th 2012

    the model structure on reduced simplicial sets: is is cofibrantly generated? Does the projective model structure on functors with values in it exist?

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeApr 13th 2012

    Never mind, Danny kindly points out that it follows easily from prop. A.2.6.13 of HTT.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeFeb 28th 2017

    I have added to model structure on reduced simplicial sets statement and (immediate) proof that reduced suspension/looping (co-)restricts to a Quillen endo-adjunction on reduced simplicial sets (here)

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeMar 1st 2017

    Some questions:

    Is the model structure on reduced simplicial sets still right proper?

    And how about 2-reduced simplicial sets and nn-reduced simplicial sets? Will they still carry the analogous model structures? And will they be right proper?

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJul 4th 2021
    • (edited Jul 4th 2021)

    added (here) statement of Goerss&Jardine V Cor. 6.9:

    Let f:X⟶Yf \colon X \longrightarrow Y be a fibration in the model structure on reduced simplicial sets such that both XX and YY are Kan complexes. Then ff is a Kan fibration precisely if it induces a surjection on the first simplicial homotopy group π 1(f):π 1(X)↠π 1(Y)\pi_1(f) \colon \pi_1(X) \twoheadrightarrow \pi_1(Y).

    diff, v13, current

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeJul 4th 2021

    added (here) the following statement:

    Let 𝒢 1→ϕ𝒢 2\mathcal{G}_1 \xrightarrow{\phi} \mathcal{G}_2 be a homomorphism of simplicial groups which is a Kan fibration. Then the induced morphism of simplicial classifying spaces W¯𝒢 1→W¯(ϕ)W¯𝒢 2\overline{W}\mathcal{G}_1 \xrightarrow{ \overline{W}(\phi)} \overline{W}\mathcal{G}_2 is a Kan fibration if and only if π 0(ϕ):π 0(𝒢 1)↠π 0(𝒢 1)\pi_0(\phi) \colon \pi_0(\mathcal{G}_1) \twoheadrightarrow{\;} \pi_0(\mathcal{G}_1) is a surjection.

    diff, v13, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeJul 4th 2021

    added (here) the following statement and its proof:

    Let 𝒢 1→ϕ𝒢 2\mathcal{G}_1 \xrightarrow{\phi} \mathcal{G}_2 be a homomorphism of simplicial groups which is a Kan fibration. Then the induced morphism of simplicial classifying spaces W¯𝒢 1→W¯(ϕ)W¯𝒢 2\overline{W}\mathcal{G}_1 \xrightarrow{ \overline{W}(\phi)} \overline{W}\mathcal{G}_2 is a Kan fibration if and only if π 0(ϕ):π 0(𝒢 1)↠π 0(𝒢 1)\pi_0(\phi) \colon \pi_0(\mathcal{G}_1) \twoheadrightarrow{\;} \pi_0(\mathcal{G}_1) is a surjection.

    diff, v13, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeMay 16th 2023

    added pointer to

    • Nigel Burke, §2 of: Homotopy Theory of Monoids and Group Completion, PhD thesis, Cambridge (2021) [cam:1810/325358, pdf]

    for discussion of the Joyal-type (in fact transferred) model structure on reduced simplicial sets, modelling quasi-categories with a single object.

    diff, v16, current

    • CommentRowNumber10.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 16th 2023

    Added a DOI hyperlink (not sure if cam.ac.uk URL is still necessary, since the DOI redirects to it); any URL that has the word “handle” in it almost certainly has a DOI, see here: https://www.doi.org/the-identifier/resources/factsheets/doi-system-and-the-handle-system.

    diff, v17, current

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeMay 16th 2023

    Right, thanks.

1 to 11 of 11

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