Timeline for What are some "good" examples of Kan simplicial manifolds?
Current License: CC BY-SA 4.0
15 events
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| Dec 24, 2020 at 20:37 | comment | added | Adittya Chaudhuri | I got it. Thank you Sir very much. | |
| Dec 24, 2020 at 20:37 | history | edited | Dmitri Pavlov | CC BY-SA 4.0 |
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| Dec 24, 2020 at 20:35 | comment | added | Dmitri Pavlov | @AdittyaChaudhuri: Yes. This also follows directly from the fact that surjective homomorphisms of Lie groups are submersions. Since any simplicial Lie group is a Kan complex, the matching maps are surjective homomorphisms of Lie groups, hence surjective submersions. | |
| Dec 24, 2020 at 20:33 | comment | added | Adittya Chaudhuri | Thank you very much Sir for the reference. So, are you saying that a simplicial Lie group is naturally a Kan simplicial manifold because of example 6 in your answer? Or am I misunderstanding something? | |
| Dec 24, 2020 at 20:30 | history | edited | Dmitri Pavlov | CC BY-SA 4.0 |
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| Dec 24, 2020 at 20:28 | comment | added | Dmitri Pavlov | @AdittyaChaudhuri: Simplicial groups are a special case of Example 6. See Carrasco and Cegarro, Group-theoretic algebraic models for homotopy types. | |
| Dec 24, 2020 at 20:27 | comment | added | Adittya Chaudhuri | Thank you very much Sir. | |
| Dec 24, 2020 at 20:23 | comment | added | Dmitri Pavlov | @AdittyaChaudhuri: This is Corollary 2.2 in arxiv.org/abs/math/0609420v6 | |
| Dec 24, 2020 at 20:07 | comment | added | Adittya Chaudhuri | Sir, we know that every simplicial group is a Kan complex. Is it true that every simplicial Lie group is a Kan fibrant simplicial manifold? (If its true, then it can be an another example) | |
| Dec 24, 2020 at 19:53 | comment | added | Adittya Chaudhuri | Thank you Sir. Can you please suggest some literatures where I can get some details of the statement "This follows from the fact that Hom(Λ^m_j, X) can be presented as the domain of a finite sequence of base changes of surjective submersions" in your last comment? | |
| Dec 24, 2020 at 19:50 | comment | added | Adittya Chaudhuri | Sir, I edited "wedge" symbol to "capital Lambda" symbol in my question. | |
| Dec 24, 2020 at 19:34 | comment | added | Dmitri Pavlov | @AdittyaChaudhuri: Yes, Hom(Λ^m_j, X) is a smooth manifold for any Kan simplicial manifold X, in particular, it is a smooth manifold in all examples given above. (Do note that Λ is the capital Greek letter lambda, not the wedge sign ∧.) This follows from the fact that Hom(Λ^m_j, X) can be presented as the domain of a finite sequence of base changes of surjective submersions. Since surjective submersions of smooth manifolds are closed under base changes along arbitrary smooth maps, this establishes the claim. | |
| Dec 24, 2020 at 19:26 | comment | added | Adittya Chaudhuri | Thank you very much Sir for the answer. Sir, does that mean that in each of these examples $Hom(\wedge_{j}^m,X)$ is a smooth manifold? (Since we need the restriction maps $Hom(\Delta^{m},X) \rightarrow Hom(\wedge^{m}_{j}, X)$ to be surjective submersions for all $m \in \mathbb{N} \cup \lbrace 0 \rbrace $ and $0 \leq j \leq m$ by definition of Kan simplicial manifolds. Is there any canonical smooth manifold structure on the Hom sets $Hom(\wedge_{j}^m,X)$? (Here $X$ is the Kan simplicial manifold) | |
| Dec 24, 2020 at 19:17 | vote | accept | Adittya Chaudhuri | ||
| Dec 24, 2020 at 19:06 | history | answered | Dmitri Pavlov | CC BY-SA 4.0 |