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 finite 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 k-theory 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 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 sheaves 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).
  1. On the nLab page on n-fold categories one finds the following statement:

    Even though an n-fold category is a strict version of an n-category in that all n composition operations are strictly unital and associative and strictly commute with each other, still n-fold groupoids model all homotopy n-types. See homotopy hypothesis.

    But at homotopy hypothesis there is no mention of n-fold groupoids, and one is pointed instead to n-groupoids. So I figure that in some sense n-groupoids and n-fold groupoids are equivalent notions. But which is a precise statement?

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeOct 20th 2013
    • (edited Oct 20th 2013)

    Hi Domenico,

    that page should be further clarified (but I don’t have time for it now).

    Strict n-fold groupoids are equivalent to strict n-groupoids if the horizontal, vertical etc. 1-groupoids are all the same, and if all the expected cubical identity cells (thin cells) are present. Otherwise strict n-fold groupoids are more general, and it is this generality that makes them model more homotopy types than strict n-groupoids.

    Specifically this result is traditionally formulated for connected homotopy types and “cat-n-groups”. That page has a few more relevant citations.

    But let me ask: what is it you are after? Depending on what it is, we might have better hints for how to proceed.

    • CommentRowNumber3.
    • CommentAuthorTim_Porter
    • CommentTimeOct 20th 2013

    But at homotopy hypothesis there is no mention of n-fold groupoids,

    … there is now! The point is that cat-n-groups are nn-fold categories internal to the category of groups, hence are n+1n+1-fold groupoids. I have added some links and a bit more to various pages, but really this could do with more.

  2. Hi Tim,

    thanks!

    But let me ask: what is it you are after? Depending on what it is, we might have better hints for how to proceed.

    I’m interested in the realization of a nn-homotopy type as an nn-fold groupoid. In particular, given a nice topological space XX, I’d like to have a nn-fold groupoid presentation of its Poincare’ nn-groupoid Π n(X)\Pi_n(X).

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeOct 20th 2013

    Domenico, In that case I’d go for either the cubical set of singular cubes, regarded in the model structure on cubical sets, or if making the nn-fold-ness is crucial for your purpose, then I’d consider the n-fold complete Segal space of singular nn-cubes.

    • CommentRowNumber6.
    • CommentAuthorTim_Porter
    • CommentTimeOct 20th 2013
    • (edited Oct 20th 2013)

    An alternative is to work with a simplicially enriched setting. In that case (and I will assume that the homotopy type is connected so as to cheat and talk of a simplicial group), there are explicit (and very simple) formulae for the corresponding crossed n-cube, and simple relations between the models for different n. The advantage with this is that n-simplicial groups collapse nicely back to simplicial groups (using the codiagonal). That aspect is very nicely put in the parallel theory developed by Bullejos, Cegarra, and Duskin. You can fold up the n-simplicial group into a simplicial group or take its classifying space. (A lot is done on this by Cegarra and various collaborators.)

    The idea of a n-fold presentation of a Poincaré n-groupoid may correspond to the collapsing process that I mention above.

    • CommentRowNumber7.
    • CommentAuthorronniegpd
    • CommentTimeJun 24th 2015
    On my preprint page
    http://pages.bangor.ac.uk/~mas010/brownpr.html
    I have a presentation I gave at CT2015 on June 17 entitled
    "A philosophy of modelling and computing homotopy types".
    The ideas here are rather different from most others, so comparison and evaluation is a good idea.

    I thought I had put the abstract on this n-forum but have not yet found it!

    Ronnie Brown
    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeJun 24th 2015
    • (edited Jun 24th 2015)

    I thought I had put the abstract on this n-forum but have not yet found it!

    You had put it here: http://nforum.ncatlab.org/discussion/177/homotopy-ntype/?Focus=53719#Comment_53719