Timeline for (Homotopy) colimit and manifold
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
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| Apr 19, 2022 at 1:07 | comment | added | chriswest | I went through these references and I think that is what I need. Thanks again! Never thought it could get this complicated though, I was thinking may be a diagram would do it lol | |
| Apr 17, 2022 at 20:18 | comment | added | Dmitri Pavlov | @chriswest: One criterion is given at mathoverflow.net/questions/38575/colimits-of-manifolds: the colimit of a proper etale Lie groupoid with trivial isotropy groups is a manifold. This subsumes the example with Čech nerves of open covers. Some references can be found in the nLab article etale groupoid. | |
| Apr 17, 2022 at 9:04 | comment | added | chriswest | By the way do you have any references for the above construction? Especially formal theorems or lemmas regarding the manifoldness of the obtained space. | |
| Apr 17, 2022 at 5:54 | comment | added | chriswest | Thank you so much! That is very helpful. | |
| Apr 17, 2022 at 4:14 | comment | added | Dmitri Pavlov | @chriswest: I would say your description matches the (re)construction of a manifold out of the Čech nerve of an open cover, or, more generally, out of a hypercover. For the latter, in a reasonable generality, given a projectively cofibrant simplicial presheaf whose simplicial structure maps are local diffeomorphisms, the homotopy colimit of this diagram is (in general) an etale stack over the site of smooth manifolds, which under further assumptions can be shown to be a manifold. | |
| Apr 17, 2022 at 3:10 | comment | added | chriswest | Let me describe it a bit more precise. Suppose I have several nodes in a system, each has a space of state parameterized by different parameters. So, to do calculus on the total state space of the entire system, it must be a manifold. But, first, the nodes are interconnected in a specific way, which forms a topology. The topology is too arbitrary to be a manifold for itself. Second, if gluing of spaces is considered, the state spaces of nodes must be glued “along the topology”. So is there any way to consider both the underlying topology and the associated spaces to form a smooth manifold? | |
| Apr 17, 2022 at 2:51 | comment | added | chriswest | Thanks! I got it, for this question. But that really gets me back to where it all begins...Since, ultimately, I want to get a smooth manifold to do calculus. However, the topology of a network, which can be modeled as a directed graph, or an acyclic category, or simplicial complex often does not naturally satisfy the conditions required for it to be a manifold (for example, the “link of every vertex being equivalent to a sphere” condition for simplicial complexes). So is there any other way to construct a smooth manifold from an arbitrary topology and the spaces associated with the vertices? | |
| Apr 16, 2022 at 16:03 | comment | added | Dmitri Pavlov | @chriswest: The homotopy colimit of a diagram of contractible spaces is always weakly equivalent to the nerve of the indexing category. Your question is thus equivalent to asking under what conditions the nerve of a category is homotopy equivalent to a manifold. As far as I am aware, no specific criteria are known for nerves, other than the generic criteria for arbitrary homotopy types. See this answer for more information: mathoverflow.net/questions/278901/… | |
| Apr 16, 2022 at 7:18 | comment | added | chriswest | Ok. Then if what I really want to obtain is a manifold, is there any way I can do this from homotopy colimit? For example, under what conditions the homotopy colimit can indeed be a manifold? | |
| Apr 16, 2022 at 6:03 | comment | added | Dmitri Pavlov | @chriswest: Any homotopy type can be obtained as the homotopy colimit of a diagram of contractible spaces, indexed by an ordinary category. Thus, the answer to your question is negative because not every homotopy type is homotopy equivalent to a manifold. | |
| Apr 16, 2022 at 0:49 | comment | added | chriswest | That is helpful. But I think my question is a little bit different (or may be I still haven’t get the point). What I have now is a diagram of contractible spaces and its homotopy colimit, does that mean it is automatically homotopy equivalent to a smooth manifold? You mention that any smooth manifold is homotopy equivalent to the homotopy colimit of a diagram of contractible spaces indexed by a 1-category, my question is actually whether the inverse of this statement holds. | |
| Apr 15, 2022 at 16:44 | history | answered | Dmitri Pavlov | CC BY-SA 4.0 |