Timeline for Is the geometric realization of simplicial functors interesting?
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| Nov 19 at 6:38 | comment | added | Tim Porter | Another area is given by the generalisations of the Riemann-Hilbert theorem. Some of the recent work on this use simplicial methods quite extensively. My own favorite in both that area and the Pursuing Stacks area is the study and use of the homotopy coherent nerve and its dg and A-infty analogues due to Lurie, Faonte and others. | |
| Nov 19 at 6:34 | comment | added | Tim Porter | Perhaps the correspondence between G-sets and covering groupoids / spaces which is at the heart of Galois theory and of the SGA1 approach to the fundamental group(oid) of schemes is an example of a theory (not of simplicial sets) that bridges from the algebraic to the geometric. If so the whole of the Pursuing Stacks project starting with the notes of Grothendieck could be where to look as they are the infinity categorical analogue of (parts of) Galois theory. | |
| Nov 19 at 3:47 | comment | added | Andrea Marino | I am looking for simplicial functors at the intersection of these two categories. Nothing fancy, just examples to play around with my lemma and see if it is significant or trivial in meaningful contexts. As an example of what I mean, the classifying space of a group is the nerve of a particular category, but its cohomology has interesting implications in the study of G-bundles. If this distinction is apparent, and the 'algebraic simplicial sets' always have interesting cohomology/homotopy groups, I'd be more than happy to know! Hope I made myself clearer now! Thanks :) | |
| Nov 19 at 3:41 | comment | added | Andrea Marino | Maybe I am wrong, but I think of simplicial sets being used for two different purposes: algebraic and geometrical. Algebraically they can be used e.g. as a non-commutative version of chain complexes, or to provide resolutions in higher context. The nerve of a category provides a nice algebraic example. Geometrical simplicial sets, instead, are mainly employed to give a combinatorial basis to the (co)homology of their geometric realizations. [cont'd] | |
| Nov 13 at 6:28 | comment | added | Tim Porter | For your second comment, any homotopy coherent diagram can be rectified to give an equivalent commutative diagram. That is more-or-less wht Vogt's theorem states. Note that Cordier's definition of homotopy coherent diagram involves a simplicial resolution of the domain of the functor defining the diagram. Playing around with that resolution is great fun and is quite insightful for what is going on. | |
| Nov 13 at 6:24 | comment | added | Tim Porter | In reply to your first Comment, I am not 100% what you mean by geometry when applied to geometric realisations. There is a combinatorial geometry which can come from a resolution. I enjoyed reading Loday's Homotopical Syzygies, in Une d ́egustation topologique: Homotopy theory in the Swiss Alps, volume 265 of Contemporary Mathematics, 99 – 127, which looks at both the idea of resolution and of the geometric structures involved,also M. Kapranov and M. Saito, 1999, Hidden Stasheff polytopes in algebraic K-theory and in the space of Morse functions, in , volume 227 of Contemporary Mathematics.. | |
| Nov 12 at 20:30 | comment | added | Andrea Marino | On a side note, I see a possible source of misunderstanding in my question, as there are two "higher" involved. First is replacing objects with simplicial resolutions that want to capture higher data. Second is replacing diagrams with homotopy coherent diagrams. I am interested in examples of strict diagrams of simplicial objects, so there is only the first "higher" involved. These are the 'input' of the lemma I came up with. The output of has also the second higher involved, but it's not quite relevant. Sorry if that was misleading. | |
| Nov 12 at 20:24 | comment | added | Andrea Marino | Thanks. Two examples that Breen's notes made me think about are the classifying space and the nerve of a category. These are maybe different from the initial spirit I had in mind, since the domain is large category and not a geometric object (e.g. poset of open sets), but it can still be a nice source of examples. In the first case the geometric realization is relevant as e.g. the cohomology is interesting. As far as I understand, you seem to confirm that the combinatorics of these 'simplicial resolutions' is more interesting than its geometry, and so the realization has limited interest. | |
| Nov 12 at 7:38 | history | answered | Tim Porter | CC BY-SA 4.0 |