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nLab > Latest Changes: model structure on simplicial groupoids

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeApr 18th 2023

    weirdly, this page never linked to simplicial groupoid. I am adjusting a little…

    diff, v8, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeApr 18th 2023

    added missing pointer to:

    diff, v8, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeApr 20th 2023

    Added (here) the objervation that acyclic fibrations of simplicial groupoids are surjective on objects (with a lazy proof)

    diff, v13, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeApr 23rd 2023

    added statement of the characterization of the cofibrations

    diff, v14, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeApr 23rd 2023

    added remark (here) on the relation to the model structure on simplicial groups

    diff, v14, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeApr 24th 2023

    Mostly as a reminder to myself: Want to add a remark that the model structure on simplicial groupoids is cartesian monoidal.

    Namely the preservation of (acyclic) injections under pushout-product is essentially inherited from sSet QusSet_{Qu}, and only that freeness condition on the codomain needs to be argued. But it should be exactly the pushout property of the domain which should ensure that its image in the codomain does not exceed the given basis morphisms.

    Something like this. Will make a note later, need to go offline now.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeApr 25th 2023
    • (edited Apr 25th 2023)

    added pointer to:

    [edit: Hm, the definition of the functor “U GU_G” on p. 4 is lacking statement of the free generation.]

    diff, v18, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeApr 25th 2023

    added (here) statement of cofibrant generation and right properness

    diff, v18, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeMay 12th 2023

    added missing cross-link with the model structure on sSet-categories

    and added (here) statement of the evident Quillen adjunction

    sSet-Grpd DK⊥ Qu↪ι⟵FsSet-Cat Berg. sSet\text{-}Grpd_{DK} \underoverset {\underset{\iota}{\hookrightarrow}} {\overset{F}{\longleftarrow}} {\;\;\; \bot_{\mathrm{Qu}} \;\;\;} sSet\text{-}Cat_{Berg} \,.

    diff, v23, current

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeMay 14th 2023
    • (edited May 14th 2023)

    added (here) mentioning of the functor

    sGrp ↪ sSet-Grpd 𝒢 ↦ B𝒢 \array{ sGrp &\hookrightarrow& sSet\text{-}Grpd \\ \mathcal{G} &\mapsto& \mathbf{B}\mathcal{G} }

    and its homotopical properties.

    diff, v24, current

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeMay 29th 2023
    • (edited May 29th 2023)

    I have made explicit (here) a corollary of Minichiello, Rivera & Zeinalian (2023), namely that the composite

    sSet-Grpd DK⊥ Qu↪ι⟵FsSet-Cat⊥⟶N⟵ℭsSet KQ sSet\text{-}Grpd_{DK} \underoverset {\underset{\iota}{\hookrightarrow}} {\overset{F}{\longleftarrow}} {\;\;\; \bot_{\mathrm{Qu}} \;\;\;} sSet\text{-}Cat \underoverset {\underset{N}{\longrightarrow}} {\overset{\mathfrak{C}}{\longleftarrow}} {\;\;\; \bot \;\;\;} sSet_{KQ}

    is a Quillen adjunction with respect to the Kan-Quillen model structure on sSetsSet.

    diff, v26, current

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeMay 31st 2023

    added pointer to:

    diff, v28, current

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeJun 1st 2023

    added (here) the observation that every object of sSet-Grpd DKsSet\text{-}Grpd_{DK} has a bifibrant resolution by a skeleton.

    diff, v29, current

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeOct 31st 2023

    added previously missing clause in the definition of free maps (here)

    diff, v31, current

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeOct 31st 2023

    Added statement and proof (under “Examples”, here) that for a GG-set WW we have a Quillen adjunction

    sSet-Grpd /BG⊥ Qu⟵Map(W⫽G,−) BG⟶(W⫽G)× BG(−)sSet-Grpd /BG. sSet\text{-}Grpd_{/\mathbf{B}G} \underoverset { \underset{ Map\big( W \sslash G, - \big)_{\mathbf{B}G} }{\longleftarrow} } { \overset{ (W \sslash G) \times_{\mathbf{B}G} (-) }{\longrightarrow} } { \bot_{\mathrlap{Qu}} } sSet\text{-}Grpd_{/\mathbf{B}G} \,.

    diff, v33, current

1 to 15 of 15

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