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Let's see: topological infinity-groupoid
No, we don't seem to have that, but you could certainly write it!
I added some thoughts about the definition.
I like Domenico's latest, and started editing the main text.
I adeed to internal infinity-groupoid an alternative definition In terms of simplicial sheaves, applicable in the case that the ambien category is a Grothendieck topos.
It would be great if we could eventually expand this entry with further discussion. Notably about the relation between the two definitions. Notably concerning the notion of morphisms for the definition with internal horn filler conditions.
I think what is really necessary here is a good abstract definition of "geometric oo-stack": a general oo-groupoid modeled on a site S is a oo-stack on S. If it is "geometric", then this is a simplicial object in S, in a certain sense.
Of course, a simplicial object in a topos of sheaves is a simplicial sheaf, not a simplicial presheaf. I'm not sure what you are looking for with the notion of morphisms; the obvious thing is just a map of simplicial objects.
I added a note saying that I think the two definitions are not equivalent, that the one using simplicial (pre)sheaves is stronger. I could be wrong though, please look. One might argue, if this is true, that the simplicial-(pre)sheaves definition is actually the "right" one, I suppose.
Hasn't Lurie defined a "geometric -stack" somewhere in his massive works yet? (-:
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Of course, a simplicial object in a topos of sheaves is a simplicial sheaf, not a simplicial presheaf.
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<p>Sure. i tend to speak of simplicial presheaves anyway, since degreewise sheafification is a Quillen equivalence, so it doesn't matter much, and since saying "simplicial sheaf" always runs the risk of making some people think that we are already referring to an oo-stack condition. For instance Lurie speaks of sheaves when he means oo-stacks.</p>
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I'm not sure what you are looking for with the notion of morphisms; the obvious thing is just a map of simplicial objects.
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<p>I was thinking that we need to think about resolving here, too. The right notion of morphism will in general not be an internal simplicial morphism, but one out of a suitable resolution of the domain.</p>
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Hasn't Lurie defined a "geometric \infty-stack" somewhere in his massive works yet? (-:
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<p>To some extent, yes. but I am not sure of the fully general story. The point to look at is his description of <a href="https://ncatlab.org/nlab/show/Deligne-Mumford+stack">Deligne-Mumford stack</a>s as oo-schemes. It should be true that with the right <a href="https://ncatlab.org/nlab/show/geometry+%28for+structured+%28infinity%2C1%29-toposes%29">geometry (for structured (infinity,1)-toposes)</a> used, geometric oo-stacks are those that are generalized schemes.</p>
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Yes, I know that simplicial presheaves model the same homotopy theory, I was just referring to the equation which isn't true as stated.
The right notion of morphism will in general not be an internal simplicial morphism, but one out of a suitable resolution of the domain.
Or, in other words, an ana--functor?
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I was just referring to the equation <img src="/extensions/vLaTeX/cache/latex_26ad9e93ef9b281e79efffb291b1c5ca.png" title="[\Delta^{op},C] \simeq [S^{op}, SSet]" style="vertical-align: -20%;" class="tex" alt="[\Delta^{op},C] \simeq [S^{op}, SSet]"/> which isn't true as stated.
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<p>Ah, sorry, I was being dense. Right, thanks for fixing this.</p>
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Or, in other words, an ana-<img src="/extensions/vLaTeX/cache/latex_ff52bcaf24f3d4af5a2a50cf5200b74f.png" title="\infty" style="vertical-align: -20%;" class="tex" alt="\infty"/>-functor?
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<p>Exactly. Every Kan complex in <img src="/extensions/vLaTeX/cache/latex_ec1a016d941b959862681904dba6e7b9.png" title=" SSet" style="vertical-align: -20%;" class="tex" alt=" SSet"/> is cofibrant, of course, but not generally every Kan complex in another topos, wrt any reasonable model structure.</p>
<p>As long as our topos is a sheaf topos, I am inclined to say, as you indicated, too, that the <a href="https://ncatlab.org/nlab/show/local+model+structure+on+simplicial+sheaves">local model structure on simplicial sheaves</a> is the right thing to look at, to deal with such questions.</p>
<p>There is also work on model structures on simplicial objects in arbitrary elementary toposes. But I know little about that. igor Bakovic seems to have told me that Marta Bunge has worked on this, but I don't recall the details. Zoran probably knows more.</p>
<p>And when we don't even have a topos, we'd strictly speaking need a theory of internal and/or enriched (oo,1)-categories.</p>
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I added a note saying that I think the two definitions are not equivalent, that the one using simplicial (pre)sheaves is stronger. I could be wrong though, please look.
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<p>Yes, that looks right, to me. I have now added details concerning your remark (see <a href="http://ncatlab.org/nlab/show/internal+infinity-groupoid#Comparison">below your query box</a>): every fibrant object in a global or local model structure on simplicial sheaves is a sheaf that takes values in Kan complexes. (This is a direct consequence of the definition of the global structures (for the projective structure, at least, for the injective one it is less direct) and the nature of left Bousfield localization).</p>
<p>So for every locally/globally fibrant simplicial sheaf <img src="/extensions/vLaTeX/cache/latex_5b33586fe138ebca0c30bd0e3643f6b1.png" title=" X" style="vertical-align: -20%;" class="tex" alt=" X"/> the canonical morphism <img src="/extensions/vLaTeX/cache/latex_083588a53a659424ec3283afe79d3c9e.png" title=" X_n \to X^{\Lambda_k[n]} " style="vertical-align: -20%;" class="tex" alt=" X_n \to X^{\Lambda_k[n]} "/> is an objectwise surjection, hence in particular a stalkwise surjection.</p>
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have now added a fairly detailed Idea section at internal infinity-groupoid.
Also added the floating toc higher geometry - contents and added some items to that toc.
It would be nice if now somebody actually created topological infinity-groupoid and talked about the special properties of that case. To my mind a statement of central importance here is Dugger's result, which says that oo-stacks on Top that are invariant under homotopy are equivalent to topological spaces. I think that statement serves to clarifiy a few cases where topological spaces appear in a Janusian way both as models for oo-groupoids and as something carrying a topology. Whenever this happens, it is useful, I think, to realize them instead as oo-stacks on Top.
Looks good, thanks.
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