Timeline for Sheaves over simplicial sets
Current License: CC BY-SA 2.5
20 events
| when toggle format | what | by | license | comment | |
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| Jul 14, 2019 at 19:45 | comment | added | Nikolai Mnev | sheeaves are well defined on simplicial sets which are semi-simplicial sets with singular simplexes trivially added. It is a site with grothendic topolyn generated just by nonsingular simplices. Then sheaves are the same as local systems on simplicial complexes. for the most applications it is enough. | |
| Mar 18, 2010 at 14:07 | vote | accept | Mikael Vejdemo-Johansson | ||
| Mar 18, 2010 at 6:23 | answer | added | Torsten Ekedahl | timeline score: 10 | |
| Mar 18, 2010 at 5:34 | history | edited | Mikael Vejdemo-Johansson | CC BY-SA 2.5 |
Motivating acceptance of answer.
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| Mar 18, 2010 at 5:32 | vote | accept | Mikael Vejdemo-Johansson | ||
| Mar 18, 2010 at 14:07 | |||||
| Mar 18, 2010 at 5:26 | comment | added | Mikael Vejdemo-Johansson | Yeah, we're talking past each other. | |
| Mar 18, 2010 at 5:17 | comment | added | Harry Gindi | Because if all that you have is vertices and degeneracy/face maps, you can only talk about degenerate simplices. That's the whole reason why simplicial sets are so fascinating. Simplicial complexes are determined by those properties (appropriately translated), but simplicial sets are much more powerful. | |
| Mar 18, 2010 at 5:14 | comment | added | Mikael Vejdemo-Johansson | @fpqc How is a simplicial set not determined by the family of objects and degeneracy and face maps? That's what I think I'm talking about... | |
| Mar 18, 2010 at 5:14 | comment | added | Harry Gindi | Every simplicial set is a contravariant functor from finite nonempty ordinals into sets. I think we're talking past each other. | |
| Mar 18, 2010 at 5:12 | comment | added | Harry Gindi | don't know what you mean by diagram. The definition of a diagram that I am familiar with is a functor from an appropriately shaped index category (or equivalently a map from an appropriate diagram scheme). Do you mean the associated quiver? Either way, the whole point here is that nearly none of the information of a simplicial set is contained in a quiver, because simplicial sets are literally higher dimensional. | |
| Mar 18, 2010 at 5:11 | comment | added | Mikael Vejdemo-Johansson | @fpqc I mean a functor from an appropriately shaped index category. Such as, for the simplicial set, the simplicial category that the simplicial set is a functor from. | |
| Mar 18, 2010 at 5:11 | comment | added | Mikael Vejdemo-Johansson | @Reid I think it might be. Or, say, pick a vector space V, and define a sheaf F by $F(S) = V$ if $S\subset U$ and $F(S) = 0$ otherwise; possibly even involving several vector spaces, and introducing specific linear maps to deal with certain restrictions. I'm hoping to arrive at the point where I can associate interesting interpretation to global sections over the sheaf I end up considering. | |
| Mar 18, 2010 at 5:06 | comment | added | Mikael Vejdemo-Johansson | A sheaf is a contravariant functor into sets. Yes, quite, from an appropriate Heyting algebra - most often the Heyting algebra of open subsets of a given topological space. This is why I end up thinking about simplicial sets - using the nerve functor, they're an interesting way to get a topology out of a category; and I'm hoping to find interesting structures by considering sheaves on that topology. And, in order to stay combinatorial, I'm hoping to be able to construct at least a subclass of those sheaves as structures defined as functors from the diagram of the simplicial set. | |
| Mar 18, 2010 at 5:05 | comment | added | Reid Barton | @Mikael: What I mean is: take any open subset $U$ of the geometric realization (there are uncountably many) and define a sheaf $F$ by $F(S) = *$ if $S \subset U$, $F(S) = \emptyset$ otherwise. Is this an example of the kind of object you are after? | |
| Mar 18, 2010 at 4:55 | comment | added | Harry Gindi | Here's the problem that I see: First of all, a sheaf over a simplicial set does not exactly make sense using the regular definition of a sheaf and on a general simplicial set. A sheaf is a contravariant functor into sets. Another problem here is that the definition of a sheaf is relative to the grothendieck topology on the underlying site. Certainly, every presheaf is a sheaf in the chaotic grothendieck topology on a category. In general as well, simplicial sets do not in general have diagrams that describe the whole structure. This is what I meant by "in general, this is not well-defined. | |
| Mar 18, 2010 at 4:44 | comment | added | Mikael Vejdemo-Johansson | I think I might have meant sheaves over the geometric realization, but I'm not certain. Certainly, I don't really want to consider the entire category, but rather I'm interested in finding out ways to deal with concrete sheaves. | |
| Mar 18, 2010 at 4:42 | history | edited | Mikael Vejdemo-Johansson | CC BY-SA 2.5 |
Expanded question.
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| Mar 18, 2010 at 4:29 | comment | added | Reid Barton | Just to check, do you literally mean "sheaves over the geometric realization" (which is a very big category, and depends on the homeomorphism type of the realization) or something more like locally constant sheaf/local system which is more homotopy-theoretic in nature? (I know I often say simply "sheaf" for the latter concept, so maybe others do too) | |
| Mar 18, 2010 at 3:55 | answer | added | Harry Gindi | timeline score: 1 | |
| Mar 18, 2010 at 3:52 | history | asked | Mikael Vejdemo-Johansson | CC BY-SA 2.5 |