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-type-theory cohomology colimits combinatorics complex-geometry computable-mathematics computer-science constructive cosmology definitions 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 homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory kan lie-theory limit 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 newpage nlab nonassociative noncommutative noncommutative-geometry number-theory object 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 subobject superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory 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
    • CommentTimeJan 8th 2010
    • (edited Jan 8th 2010)

    I found the discusssion at internal infinity-groupoid was missing some perspectives

    I made the material originally there into one subsection called

    • Kan complexes in an ordinary category

    and added two more subsections

    • Kan complexes in an (oo,1)-category

    • Internal strict oo-groupoids .

    The first of the two currently just points to the other relevant entry, which is groupoid object in an (infinity,1)-category, the second one is currently empty.

    But I also added a few paragraphs in an Idea section preceeding everything, that is supposed to indicate how things fit together.

  1. somehow in connection with this, I was thinking to BG for a (nice) topological group G.

    considering just the delooping BG encodes the group structure of G, but forgets its topology. this is equivalent to look at G as to a discrete group. to overcome this, let us have a closer look at the delooping BG: it is a single object groupoid with G as set of morphisms, so we can think of it as an oo-groupoid whose is a single element set and whose is G, that is ; higher vanish. moreover this groupoid is strict.

    so one could reintroduce the topology of G by looking at the whole oo-Poincare' groupoid of G, . objects are elements of G, 1-morphisms are paths in G, and so on. This oo-groupoid can be delooped twice! first delooping produces the groupoid whose objects are elements in ; 1-morphisms are elements of G (i.e. we are loooking at as a G-set), 2-morphisms are paths in G, and so on. but is still a group, so we can deloop it to obtain a single object oo-groupoid, with as 1-morphisms, G as 2-morphisms, paths in G as 3-morphisms, and so on.

    this oo-groupoid knows about the homotopy type of G. moreover it knows about G, which is the set of 1-morphisms, and of the connected component of the identity, , which is the set of morphisms of .

    in this it is crucial that the twice delooped oo-groupoid of G is strict up to 1-morphisms, so one still has a groupoid if one forgets about all higher morphisms (i.e. only keeps identities). so, in some way I know is there but is still foggy and unclear to me, subsequent higher strictifications should produce the Withehead tower of G.
    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJan 9th 2010
    • (edited Jan 9th 2010)
    This comment is invalid XHTML+MathML+SVG; displaying source. <div> <blockquote> considering just the delooping BG encodes the group structure of G, but forgets its topology. this is equivalent to look at G as to a discrete group. </blockquote> <p>Well, maybe one has to be careful here.</p> <p>It is situations like this that I think are best thought of in the oo-topos of stacks on Top. I mentioned that recently elsewhere:</p> <p>since G is a topological group, when we start deppoing it topological spaces play a double role: on the one hand G as a discrete group corresponds to a groupoid and hence a topological space, on the other hand G carries a topology itself.</p> <p>It is important to disentangle these two topological aspects carefully. And I think this is conceptually best done by thinking of G as the presheaf on Top that it represents, then regard this as a simplicially discrete simplicial presheaf on Top.</p> <p>The fact that this then is a presheaf on Top encodes the topology of G, the fact that it is a sikmplicially discrete presheaf encodes the fact that G here is categorically discrete.</p> <p>Now, the deloooping procedure from simplicial presheaves to simplicial presheaves is obvious: it takes the presheaf <img src="https://nforum.ncatlab.org/extensions//vLaTeX/cache/latex_d268151a767cda3a7f42904accdaa203.png" title=" G : U \mapsto Hom_{Top}(U,G) =: C(U,G)" style="vertical-align: -20%;" class="tex" alt=" G : U \mapsto Hom_{Top}(U,G) =: C(U,G)"/> to the presheaf <img src="https://nforum.ncatlab.org/extensions//vLaTeX/cache/latex_8eb96f3c341ac8a81a54f7d01fe38a2a.png" title=" \mathbf{B}G : U \mapsto " style="vertical-align: -20%;" class="tex" alt=" \mathbf{B}G : U \mapsto "/> the nerve of the groupoid version of the group <img src="https://nforum.ncatlab.org/extensions//vLaTeX/cache/latex_067baa9872c3d7815a1006556cbc6fbe.png" title="C(U,G)" style="vertical-align: -20%;" class="tex" alt="C(U,G)"/>.</p> <p>This is just the general prescription. Now all the subtlety that, I think, your message is about, is: how do we think of the simplicial presheaf <img src="https://nforum.ncatlab.org/extensions//vLaTeX/cache/latex_6fe27bb58bba0732c47fa241197ff5fc.png" title="\mathbf{B}G" style="vertical-align: -20%;" class="tex" alt="\mathbf{B}G"/> on <img src="https://nforum.ncatlab.org/extensions//vLaTeX/cache/latex_d135fddf839f5858264c53bd39d0a7fc.png" title="Top" style="vertical-align: -20%;" class="tex" alt="Top"/> again itself as a topological space? We expect to find the classifying space <img src="https://nforum.ncatlab.org/extensions//vLaTeX/cache/latex_53966bca4583ffb179e655c1af457713.png" title=" \mathcal{B}G \in Top" style="vertical-align: -20%;" class="tex" alt=" \mathcal{B}G \in Top"/>.</p> <p>For fully understanding this question, I think Dugger's notes are indespensible, or at least very useful. He discusses in great detail how simplicial presheaves on Top geometrically realize to just topological spaces. See the notes prsesented on his website "Sheaves and homotopy theory" <a href="http://ncatlab.org/nlab/files/cech.pdf">pdf</a>.</p> </div>
  2. Thanks for the reference, I didn't know it!

    as far as concerns BG, it was not my intention to be drastic in saying that one cannot make a correct constructon of BG for a topological group G (and by the way, the question whether the topological realization of the presheaf is is worth investigating), but to point out that starting with the "wrong" construction, one is naturally led to the n-connected "covers" of G by the "strictify the double delooping groupoid" procedure. I guess this is well known, but I'm unaware of a reference, so I pointed that out.
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)