Timeline for Simplicial nerve of a topological group
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 3, 2022 at 2:26 | vote | accept | Ken | ||
| Jul 3, 2022 at 2:25 | comment | added | Ken | Aha! I greatly appreciate your patience and insight. | |
| Jul 3, 2022 at 2:21 | comment | added | Dmitri Pavlov | @Ken: Sets of generators can be transported accross Quillen equivalences. Since Cat_Top is Quillen equivalent to sSet_Joyal, it suffices to see that {0→1→⋯→n}, i.e., Δ^n∈sSet_Joyal, generate sSet_Joyal under homotopy colimits. Indeed, any simplicial set X is the homotopy colimit of the canonical diagram Δ/X→sSet that sends (Δ^n→X)↦Δ^n, i.e., takes values in the generators. | |
| Jul 3, 2022 at 2:05 | comment | added | Ken | Thank you very much. Sorry for asking dumb questions, but can you explain why do the categories $\{0\to\cdots\to n\}$ generate $\mathsf{Cat}_{mathsf{Top}}$ (under homotopy colimit, I guess)? | |
| Jul 3, 2022 at 1:26 | comment | added | Dmitri Pavlov | @Ken: I added more information about this statement. | |
| Jul 3, 2022 at 1:25 | history | edited | Dmitri Pavlov | CC BY-SA 4.0 |
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| Jul 3, 2022 at 0:30 | comment | added | Ken | Thank you for your answer. Could you elaborate on your second paragraph? I am not experienced enough to understand the reasoning behind the claim that $N \operatorname{Sing}$ computes the homotopy colimit as stated. What do you mean when you say "$N$ is induced by homotopy cocontinuous functor from simplicial spaces to spaces that sends representable simplices to contractible spaces"? I thought that $N$ had nothing to do with simplicial spaces; rather, it is merely a part of the Quillen equivalence between $\mathsf{sSet}$ and $\mathsf{Cat}_{\mathsf{sSet}}$. | |
| Jul 2, 2022 at 18:24 | history | answered | Dmitri Pavlov | CC BY-SA 4.0 |