Timeline for Homotopy coherent space maps induces homotopy coherent chain complex morphisms
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Jun 16, 2021 at 14:37 | comment | added | Andrea Marino | Let's move this discussion to chat! chat.stackexchange.com/rooms/126541/infinity-chains | |
| Jun 16, 2021 at 14:35 | comment | added | Maxime Ramzi | It is a minor issue, the underlying $\infty$-category of $Ch_{\geq 0}(\mathbb Z)$ embeds fully faithfully in the underlying $\infty$-category of $Ch(\mathbb Z)$, which is stable | |
| Jun 16, 2021 at 14:27 | comment | added | Andrea Marino | About the second approach, the aim of taking chains (for me) is doing homological algebra thereafter, whose higher version is stable infinity categories. For this to happen, my model for chain complexes is the $\infty$-derived category as introduced in Lurie, that is the $\infty$ localization of right bounded chain complexes at quasi isomorphism (the one you cited in first approach). But as far as I can see $sAb \simeq Ch_{\ge 0}(\mathbb{Z})$ is not stable because you can't (de)suspend! Is this a minor issue? | |
| Jun 16, 2021 at 14:26 | comment | added | Maxime Ramzi | Well, the point of this result 1.3.4.20 from Higher algebra is that it is capturing the higher homotopies. In other words, these higher homotopies just exist because you're killing contractible things : and it makes sense, a higher homotopy is just something that looks like $X\times |\Delta^n|\to Y$. So what you want to call the category of spaces is the same thing. For your question, you simply have to show that $Top$ and $sSet$ are Quillen equivalent, which is a famous result due to Quillen | |
| Jun 16, 2021 at 14:20 | comment | added | Andrea Marino | I am not anymore convinced of the two approaches. About the first approach, using the infinity localization of a plain model category, I feel like we are not capturing the higher homotopy contained in the topology of hom-spaces in $Top$. I'd say the category of spaces is rather $N_{coh}(Sing(Top))$, or if you like weak equivalences maybe $RN_{coh}(L^H(Sing(Top), Weq))$. Formally, how do you show that $Top[weq^{-1}]$ is equivalent to the quasicategory of Kan complexes? | |
| Jun 11, 2021 at 11:20 | comment | added | Andrea Marino | Thank you Maxime! Your answers are always precious. The (hammock?) localization is a neat approach that works on the nose for chains. As for simplicial abelian groups, I agree that the functor from spaces to $sAb$ is simplicial, but I am really interested in getting chains at the end. As I remarked in the text, I think the problem is from $sAb$ to chains, and this is tautologically solved if one equips chains with the transferred simplicial structure. It's a pity however that the simplicial structure induced by $\mathbb{Z}[\Delta^n]$ behaves so badly! It should be better remarked in school. | |
| Jun 11, 2021 at 11:16 | vote | accept | Andrea Marino | ||
| Jun 10, 2021 at 12:18 | history | answered | Maxime Ramzi | CC BY-SA 4.0 |