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 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 internal-categories k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure 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 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: simplicial groupoid

Bottom of Page

1 to 12 of 12

  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

  1.