Timeline for Canonical comparison between $\infty$ and ordinary derived categories
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 9 at 15:04 | comment | added | Yang | I am probably wrong, but I feel like you should use the differential graded nerve of $Ch(A)$ as a dg-category instead of nerve of it just as 1 category during the localization? | |
| Apr 30 at 23:13 | comment | added | Stahl | @PeterHaine Thanks Peter -- that's a very clever fix! Is there any technical reason that I should take $\mathcal{W}$ to be wide, or is this just convention? | |
| Apr 29 at 6:49 | comment | added | Peter Haine | Here's one way to resolve the set-theory issues using an alternative presentation of the localization. Write $\mathrm{B} \colon \mathbf{Cat}_{\infty} \to \mathbf{Spc}$ for the left adjoint to the inclusion. Given an $\infty$-category $\mathcal{C}$ and subcategory $\mathcal{W} \subset \mathcal{C}$, the localization $\mathcal{C}[\mathcal{W}^{-1}]$ is the pushout of the span $\mathrm{B}\mathcal{W} \leftarrow \mathcal{W} \to \mathcal{C}$. (Usually we take $\mathcal{W}$ to be the subcategory containing all objects and with morphisms in a specified class.) | |
| Apr 27 at 23:52 | comment | added | Stahl | @R.vanDobbendeBruyn Thanks for your comment. I am aware of the set-theoretic issues and usual resolution here. There is also the option to view $\mathsf{Ch}(\mathsf{A})$ as a model category whose weak equivalences are $\mathsf{qis},$ and model the localization using one of the usual methods for model categories. I had in mind fixing some universes $\mathcal{U}\in\mathcal{V}$ (and perhaps beyond this if necessary), and taking all categories to be $\mathcal{U}$-small, so that the localizations would be $\mathcal{V}$-small. Would this not also resolve the issue, or is it more nuanced than that? | |
| Apr 27 at 18:07 | comment | added | R. van Dobben de Bruyn | There is an obvious set-theoretic subtlety in the formation of $\coprod_{f \in \mathbf{qis}} [1]$. In the 1-categorical setting, this is often solved by (i) first taking the homotopy category of chain complexes, which is a triangulated category, and (ii) showing that in cases of interest (notably Grothendieck abelian categories), there is then a small set of things left to invert. See for instance §10.3 of Weibel's Introduction to homological algebra. | |
| Apr 27 at 3:22 | history | edited | Stahl | CC BY-SA 4.0 |
added 18 characters in body
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| Apr 27 at 3:10 | history | answered | Stahl | CC BY-SA 4.0 |