Not signed in (Sign In)

Start a new discussion

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 bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-theory cohesive-homotopy-type-theory cohomology colimits combinatorics 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 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 homology homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory kan lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology natural nforum nlab nonassociative 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 string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topological topology topos topos-theory 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).
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMar 26th 2010

    edited decalage a bit

    there was the statement that  Dec Y \to Y is a "fibration". I made that Kan fibration. Is that right?

    • CommentRowNumber2.
    • CommentAuthorTim_Porter
    • CommentTimeMar 27th 2010

    It certainly is a Kan fibration if Kis a simplicial group since it is clearly an epimorphism.

    In general, I think it is fairly easy to prove. IDEA: Suppose you have a (n,k) horn in Dec K so a sequence of faces with a gap. Map that down to K where you are given a filler for the horn and hence know the gap and what fills things there. The filler is a n+2 simplex in K so gives an n+1 simplex in Dec K. I think that will be the filler up top. There was a moment when I had some doubts about the compatibility with respect to the last face, but in fact you are GIVEN a filler for the horn down the bottom so I don't think now there is a problem.

    Does that look right?

    • CommentRowNumber3.
    • CommentAuthorHarry Gindi
    • CommentTimeMar 27th 2010

    You should add the link to PJ Ehlers's thesis here as well, which discusses all of this in detail (and has become pretty much my standard source for all of this simplicial homotopy theory ) stuff.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeMar 29th 2010
    • (edited Apr 26th 2012)

    It should be true that when  X_0 = * that the decalage  \mathrm{Dec} X of  X is the pullback along  X^{\Delta^1} \stackrel{d_1}{\to} X of  \Delta^0 \to  X . Do we have a discussion of this in the literature anywhere?

    [ edit: this is of course not true: the decalage gives a smaller model for $X^I \times_X X_0$ ]

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeMar 29th 2010

    I added a query box to decalage on this point

    • CommentRowNumber6.
    • CommentAuthorHarry Gindi
    • CommentTimeMar 29th 2010
    • (edited Mar 30th 2010)

    Edit:

    @Urs: Yes, you actually can describe it that way. In Ehlers's thesis, he defines the internal hom in terms of the Décalage, because the latter is easier to describe explicitly.

    • CommentRowNumber7.
    • CommentAuthorTim_Porter
    • CommentTimeMar 30th 2010

    beware, which 'internal hom'. The join is the tensor ofor an internal hom on augmented sSet, but this is not thee same as the usual internal hom, so 'an internal hom' would be safer wording.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeMar 30th 2010
    • (edited Apr 26th 2012)

    [ old stupid remark removed ]

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeJul 3rd 2021
    • (edited Jul 3rd 2021)

    I moved the example of simplicial classifying spaces from the last paragraph of the Idea-section to a first subsection of the Examples-section (now here).

    Incidentally, it used to say (and still says):

    A central application is the special case where X=W¯GX = \overline{W} G is the simplicial classifying space of a simplicial group GG (see at simplicial principal bundle). In this case Dec 0W¯GDec_0 \overline{W} G, called WGW G, is a standard model for the universal simplicial principal bundle.

    But that seems wrong to me, if we stick to standard conventions. I have added this followup remark:

    Or rather, with the conventions used at simplicial classifying spaces (which are those of Goerss & Jardine, p. 269) we have WG=Dec 0(W¯G)W G \,=\, Dec^0(\overline{W}G) (shifting and forgetting the first, i.e. 0th, face-and degeneracy maps.)

    diff, v39, current

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeJul 3rd 2021
    • (edited Jul 3rd 2021)

    It feels like there ought to be a slick way to use décalage to produce a simplicial model for the relation [B,B𝒢]𝒢 ad𝒢\big[ \mathbf{B}\mathbb{Z}, \mathbf{B}\mathcal{G} \big] \;\simeq\; \mathcal{G} \!\sslash_{\!\!ad}\! \mathcal{G}, for any \infty-group (simplicial group) 𝒢\mathcal{G}. But if so, I don’t see it yet. Does anyone know?

Add your comments
  • Please log in or leave your comment as a "guest post". If commenting as a "guest", please include your name in the message as a courtesy. Note: only certain categories allow guest posts.
  • To produce a hyperlink to an nLab entry, simply put double square brackets around its name, e.g. [[category]]. To use (La)TeX mathematics in your post, make sure Markdown+Itex is selected below and put your mathematics between dollar signs as usual. Only a subset of the usual TeX math commands are accepted: see here for a list.

  • (Help)