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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeJan 8th 2010
   • (edited Jan 8th 2010)
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   Author: Urs
   Format: MarkdownI 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.

   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.

2. • CommentRowNumber2.
   • CommentAuthordomenico_fiorenza
   • CommentTimeJan 9th 2010
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   Author: domenico_fiorenza
   Format: Htmlsomehow 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 <latex>\pi_0</latex> is a single element set and whose <latex>\pi_1</latex> is G, that is <latex>\pi_0(G_{discr})</latex>; higher <latex>\pi_i</latex> vanish. moreover this groupoid is strict. so one could reintroduce the topology of G by looking at the whole oo-Poincare' groupoid of G, <latex>\Pi_\infty(X)</latex>. 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 <latex>\pi_0(G)</latex>; 1-morphisms are elements of G (i.e. we are loooking at <latex>\pi_0(G)</latex> as a G-set), 2-morphisms are paths in G, and so on. but <latex>\pi_0(G)</latex> is still a group, so we can deloop it to obtain a single object oo-groupoid, with <latex>\pi_0(G)</latex> 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, <latex>G_e</latex>, which is the set of morphisms of <latex>[e]\in \pi_0(G)</latex>. 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.

   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.
3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeJan 9th 2010
   • (edited Jan 9th 2010)
   • PermaLink
   Author: Urs
   Format: Markdown<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> Well, maybe one has to be careful here. 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: 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. 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. 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. Now, the deloooping procedure from simplicial presheaves to simplicial presheaves is obvious: it takes the presheaf <latex> G : U \mapsto Hom_{Top}(U,G) =: C(U,G)</latex> to the presheaf <latex> \mathbf{B}G : U \mapsto </latex> the nerve of the groupoid version of the group <latex>C(U,G)</latex>. 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 <latex>\mathbf{B}G</latex> on <latex>Top</latex> again itself as a topological space? We expect to find the classifying space <latex> \mathcal{B}G \in Top</latex>. 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>.

   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="/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="/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="/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="/extensions/vLaTeX/cache/latex_6fe27bb58bba0732c47fa241197ff5fc.png" title="\mathbf{B}G" style="vertical-align: -20%;" class="tex" alt="\mathbf{B}G"/> on <img src="/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="/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>
4. • CommentRowNumber4.
   • CommentAuthordomenico_fiorenza
   • CommentTimeJan 9th 2010
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   Author: domenico_fiorenza
   Format: HtmlThanks 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 <latex>\mathbf{B}G</latex> is <latex>\mathcal{B}G</latex> 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.

   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.