Timeline for 1-connected infinity groupoids, groupoids and 1-connected spaces
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
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| Jan 16, 2020 at 15:41 | comment | added | Andrea Marino | Regarding part 1, I was probably thinking the following. Even if $\infty-Grpd$ is not a model category, it still make sense to consider its derived category to be the essential image of $\infty-Grpd \to sSet \to D(sSet)$. Geometric realization induces a functor $D(\infty-Grpd) \to D(Top)$ which is fully faithful (as it is a restriction), and it also essentially sujective, because $D(Sing) : D(Top) \to D(sSet)$ factors through $D(\infty-Grpd)$, so that $X \simeq |Sing(X)|$ in the derived category. Can one make sense of $D(\infty-Grpd)$ in a more elegant way? | |
| Jan 15, 2020 at 11:27 | comment | added | Andrea Marino | Ah well, thank you :) | |
| Jan 14, 2020 at 23:40 | comment | added | Dmitri Pavlov | @AndreaMarino: What Denis said. The 2-coskeletization of a 1-truncated simplicial set is indeed weakly equivalent to the original simplicial set. | |
| Jan 14, 2020 at 16:45 | comment | added | Denis Nardin | @AndreaMarino That's what's commonly known as 1-truncated, not 1-connected (that'd be $\pi_0=\pi_1=\ast$) | |
| Jan 14, 2020 at 16:42 | history | edited | Andrea Marino | CC BY-SA 4.0 |
edited title
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| Jan 14, 2020 at 16:41 | comment | added | Andrea Marino | Yes,Indeed your first comment answers me. I knew about the classical Quillen equivalence, and it seemed straightforward to me that it restricts to Kan complexes (as sing has image in Kan complexes ). But as you pointed out the problem is that they do not have colimits. 1 connected here means that the homotopy groups of the geometric realization are trivial for n greater than 2. Not sure if the 2-coskeletization of a 1-comnected sset is weakly equivalent to himself, which would conclude. | |
| Jan 14, 2020 at 15:53 | comment | added | Dmitri Pavlov | I am curious though as to what source on model categories does not mention the Quillen equivalence between sSet and Top. | |
| Jan 14, 2020 at 15:50 | comment | added | Dmitri Pavlov | If 1-connected really means 1-truncated (e.g., 2-coskeletal simplicial sets), then indeed the fundamental groupoid functor and the nerve functor induce a Quillen equivalence between 2-coskeletal simplicial sets and groupoids. This answers (2). | |
| Jan 14, 2020 at 15:49 | comment | added | Dmitri Pavlov | Kan complexes do not have finite (co)limits, so cannot have a model structure. Simplicial sets do have a model structure, whose fibrant objects are Kan complexes. Furthermore, |−| and Sing form a Quillen equivalence. This answers (1). | |
| Jan 14, 2020 at 9:07 | history | undeleted | Andrea Marino | ||
| Jan 14, 2020 at 9:07 | history | deleted | Andrea Marino | via Vote | |
| Jan 14, 2020 at 8:56 | history | asked | Andrea Marino | CC BY-SA 4.0 |