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nLab > Latest Changes: simplicial groupoid

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeAug 9th 2011
    • (edited Aug 9th 2011)

    added looping/delooping statements to model structure on simplicial groupoids and model structure on presheaves of simplicial groupoids.

    Question: Suppose we take the catgeory of genuine simplicial groupoids Grpd ΔGrpd^\Delta (no restriction on the simplicial set of objects). Does W¯:Grpd Δ→sSet\bar W : Grpd^\Delta \to sSet send degreewise fibrations to fibrations?

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeApr 18th 2023

    fixed glitches in text and references and added cross-link with model structure on simplicial groupoids

    diff, v17, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeApr 18th 2023

    The definition section here was pretty useless and misleading. I have deleted an and rewritten something from scratch (here)

    diff, v17, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeApr 25th 2023

    I have made explicit (here) how (sSet-CatsSet\text{-}Cat and hence) sSet-GrpdsSet\text{-}Grpd is a cartesian closed category.

    This must be a classical fact, but I have not seen it made explicit anywhere.

    diff, v19, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeApr 28th 2023

    for completeness, I have added a section “Relation to simplicial groups” (here), culminating in the statement that – assuming the axiom of choice — every simplicial groupoids is enriched-homotopy-equivalent to a disjoint union of delooped simplicial groups.

    diff, v21, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeApr 28th 2023
    • (edited Apr 28th 2023)

    added the remark (here) that, therefore, simplicial presheaves on simplicial groupoids are equivalently products of simplicial group actions

    diff, v22, current

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeApr 29th 2023

    finally added (here)

    that for 𝒳∈sSet-Grpd\mathcal{X} \in sSet\text{-}Grpd and C\mathbf{C} a combinatorial simplicial model category, we have a Quillen equivalence

    sFunc(𝒳,C) proj⊥ ≃⟶ι *⟵ι !∏i∈π 0(𝒳)sFunc(B(𝒳(x i,x i)),C) proj=∏i∈π 0(𝒳)(𝒳(x i,x i))Act(C) Borel. sFunc\big( \mathcal{X} ,\, \mathbf{C} \big)_{proj} \underoverset {\underset{\iota^\ast}{\longrightarrow}} {\overset{\iota_!}{\longleftarrow}} {\;\; \bot_{\simeq} \;\;} \underset{i \in \pi_0(\mathcal{X})}{\prod} sFunc\Big( \mathbf{B}\big(\mathcal{X}(x_i,x_i)\big) ,\, \mathbf{C} \Big)_{proj} \;=\; \underset{i \in \pi_0(\mathcal{X})}{\prod} \big(\mathcal{X}(x_i,x_i)\big) Act(\mathbf{C})_{Borel} \,.

    diff, v25, current

    • CommentRowNumber8.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 26th 2023
    • (edited May 26th 2023)

    Re #4: it’s a general fact that if EE is a finitely complete cartesian closed category, then the same is true for the category Cat(E)Cat(E) of internal categories in EE. Same goes for Gpd(E)Gpd(E). I have a proof sitting on my nLab page, which I can tidy up and create a page for on nLab main. (Oh, #4 is referring to the enriched case, not internal. Well, I’ll see about working that in as well.)

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeMay 26th 2023

    I still wanted to enhance the statement to make explicit the cartesian *enriched* monoidal structure. (Generally, the notion of enriched monoidal category seems to have been neglected by the classical authors. Of course here in the cartesian case it is not a big deal, but still.)

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeMay 31st 2023

    added pointer to:

    diff, v32, current

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeMay 31st 2023

    edited the argument for cartesian closure (here)

    diff, v33, current

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeJun 1st 2023

    added a sentence to the proposition here making explicit the conclusion that every sSetsSet-groupoid deformation retracts onto a skeletal sSetsSet-groupoid.

    diff, v34, current

1 to 12 of 12

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