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 comma 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 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).
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMay 28th 2010

    polished and expanded somewhat the entry groupoid object in an (infinity,1)-category

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJan 23rd 2011

    added to groupoid object in an (infinity,1)-category a subsecton with a remark on the notion of (,1)(\infty,1)-quotients / homotopy quotients.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeMar 29th 2011

    finally added the central theorem about delooping in an \infty-topos: to a new section Delooping

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeApr 13th 2012
    • (edited Apr 13th 2012)

    I have added to the References at groupoid object in an (∞,1)-category the items

    • Julia Bergner,

      Adding inverses to diagrams encoding algebraic structures, Homology, Homotopy and Applications 10 (2008), no. 2, 149–174. (arXiv:0610291)

      Adding inverses to diagrams II: Invertible homotopy theories are spaces, Homology, Homotopy and Applications, Vol. 10 (2008), No. 2, pp.175-193. (web, arXiv:0710.2254)

    wherein model category presentations of the \infty-categories of groupoid objects in Grpd\infty Grpd are discussed.

    It seems sort of straightforward to generalize this to model categories presenting \infty-categories of groupoid objects in more general presentable \infty-categories / \infty-toposes. The groupoidal version of Segal space objects in model structures of simplicial (pre)sheaves.

    But: has it been written out? Is anyone aware of something?

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeApr 13th 2012
    • (edited Apr 13th 2012)

    In fact, even for the ambient \infty-category being Grpd\infty Grpd, a more thorough model theory presentation of the theory of groupoid objects would be desireable. The two articles mentioned above focus mostly on models for the actual objects of that \infty-category. It’s section 3 of the second article that gives a genuine model for groupoid objects in Grpd\infty Grpd (“invertible Segal spaces”). It would for instance be nice to have a Quillen equivalence form there to a model structure for effective epimorphisms in Grpd\infty Grpd. Things like that. I am wondering if this has been done in citable form somewhere, or if it still needs to be written out.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeDec 3rd 2012
    • (edited Dec 3rd 2012)

    I have added to groupoid object in an (infinity,1)-category a section Equivalent characterizations with some of those equivalent characterizations.

    Or rather, for the moment I have mostly concentrated on adding a little remark on how to translate from the “cone-style” conditions as they appear in HTT to the equivalent “powering-style”-conditions, as they appear in “I2CATGC”.

    • CommentRowNumber7.
    • CommentAuthorColin Tan
    • CommentTimeJul 21st 2014
    • (edited Jul 21st 2014)

    There is a certain ambiguity of denoting the (oo,1)-categorial equivalence from pointed connected object in a Grothendieck (oo,1)-topos to group objects in that topos by Ω\Omega, namely see proposition 7 at groupoid object in an (oo,1)-category. This is because, for X a pointed connected object, the notation could mean either a pointed object internal to the said topos or a group object internal to the said topos.

    I was troubled by this ambiguity when writing up a proof at suspension object that, internal to a Grothendieck (oo,1)-topos, suspending is equivalent smashing with the classfying space of the integers. There, for G a group object, I use the notation ΩBG\Omega {\mathbf{B}} G to denote a pointed object. Taking the adjunction (ΩB)(\Omega \vdash {\mathbf{B}}) in the current notation per se, this notation ΩBG\Omega {\mathbf{B}} G ought to (by abstract nonsense) refer to a group object equivalent to GG.

    One suggestion, if we follow Lurie and take the “complete Segal-space style” presentation of a group object as a simplicial object satisfying some conditions, then is to write the categorial equivalence as Cˇ(*X):PointedConnGrp\check{C}(*\to X):{\mathrm{PointedConn}}\to {\mathrm{Grp}}, which sends a pointed connected object XX to the (underlying simplicial set) of the Cech nerve of the based map *X*\to X from the terminal object.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeJul 21st 2014
    • (edited Jul 21st 2014)

    This is about whether to leave the forgetful functor from groups to pointed objects notationally implicit.

    By the way, over at suspension object you are using notation in both ways. First you say that ΩB\Omega \mathbf{B} lands in pointed objects, but in the first proof you use Ω\Omega as landing in group objects.

    My suggestion is: say locally, eg in the entry on suspension, ecplicitly what the conventions are. There won’t ever be consistent conventions across all nLab entries.

    Indeed, since you write F for the free group functor, it would be most ntural to call the forgetful functor just U, as usual, and not Omega B.

    • CommentRowNumber9.
    • CommentAuthorTim_Porter
    • CommentTimeJul 21st 2014
    • (edited Jul 21st 2014)

    Urs: the link in #8 does not work as you made a typo. suspension object

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeJul 21st 2014

    Fixed.

    • CommentRowNumber11.
    • CommentAuthorColin Tan
    • CommentTimeJul 21st 2014
    • (edited Jul 21st 2014)

    Thanks Urs, for pointing out the inconsistencies I made.

    There are really two right adjoint functors from group objects in question here. One is, following your suggestion, the functor U:GrpPointedU:{\mathrm{Grp}}\to {\mathrm{Pointed}} which sends a group GG (regarded as a simplicial object) to G 1G_1 (note that G 1G_1 could possibly not be 0-connected). Its left adjoint is F:PointedGrpF:{\mathrm{Pointed}}\to {\mathrm{Grp}} which sends a pointed object XX to the (underlying simplicial object) of the Cech nerve of *ΣX*\to \Sigma X. The other is the functor B:GrpPointedConnected{\mathbf{B}}:{\mathrm{Grp}}\to{\mathrm{PointedConnected}} which sends a group GG to the colimit of GG, regarded as a diagram over the simplex category. (The Lab calls this functor B{\mathbf{B}}; In the 0-truncated case, this functor is usually called the nerve; Lurie calls it geometric realization |||-|.) Its left adjoint PointedConnectedGrp{\mathrm{PointedConnected}}\to{\mathrm{Grp}} sends a pointed connected object XX to the (underlying simplicial object) of the Cech nerve of *X*\to X. It is really the interplay of these two functors that proves the concretization of the suspension functor.

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeJul 21st 2014

    Yes, and I am really fond of wring B\mathbf{B} and not writing “nerve” or “geometric realization” because the latter two are conceptually misleading or at best highly ambiguous, no matter how standard they may be.

    • CommentRowNumber13.
    • CommentAuthorColin Tan
    • CommentTimeAug 2nd 2014

    Is there a reason for your use of the letter B\mathbf{B}?

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeAug 10th 2014
    • (edited Aug 10th 2014)

    Sure, for GG a topological group regarded as an infinity-group via its underlying homotopy type, then the super-traditional classifying space construction BGB G is the delooping of GG in the \infty-topos Grpd\infty Grpd. The boldface B\mathbf{B} is to denote delooping in any other \infty-topos. The boldface is to be suggestive of “delooping remembering additional structure” (fat structure). More precisely, if H\mathbf{H} is a cohesive \infty-topos then under suitable conditions the functor Π:HGrpd\Pi : \mathbf{H}\longrightarrow \infty Grpd takes BG\mathbf{B}G to BGB G.