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I found the discusssion at internal infinity-groupoid was missing some perspectives
I made the material originally there into one subsection called
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.
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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.
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<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>
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