Timeline for compact objects in model categories and $(\infty,1)$-categories
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 26, 2012 at 16:25 | vote | accept | Mike Shulman | ||
| Apr 26, 2012 at 16:25 | comment | added | Mike Shulman | Okay, thanks. I'd like to see more details, but I guess if it isn't written down anywhere, I'll have to work them out myself. | |
| Apr 26, 2012 at 3:03 | comment | added | Jacob Lurie | I'm afraid I don't know references. | |
| Apr 26, 2012 at 3:02 | comment | added | Jacob Lurie | The reverse implication seems a little trickier. One way to prove it is to first argue that it depends only on the Quillen equivalence class of $\mathcal{C}$ (in other words, only on the underlying $\infty$-category of $\mathcal{C}$). This lets you assume that $\mathcal{C}$ is a Bousfield localization of simplicial presheaves on some small simplicial category. From here it's not hard to reduce to the case where $\mathcal{C}$ is the category of simplicial sets, in which case the result is true for any uncountable $\kappa$. | |
| Apr 26, 2012 at 2:55 | comment | added | Jacob Lurie | One direction is fairly straightforward: if $\mathcal{C}$ is combinatorial, then there exists a cardinal $\kappa$ such that weak equivalences in $\mathcal{C}$ are closed under $\kappa$-filtered colimits. Then $\kappa$-filtered colimits are also homotopy colimits, so the functor $F$ from $\mathcal{C}$ to its underlying $\infty$-category preserves $\kappa$-filtered colimits, and in particular is accessible. It then follows that $F$ preserves $\tau$-compact objects for $\tau$ sufficiently large. | |
| Apr 25, 2012 at 21:14 | comment | added | Mike Shulman | Also, my next question is about an analogous statement for relatively $\kappa$-compact morphisms. I hope it would be possible to deduce that from the absolute version, but maybe you know a reference for the relative case too? | |
| Apr 25, 2012 at 21:06 | comment | added | Mike Shulman | Thanks! I'm happy to assume that $\kappa$ is sufficiently large. Where can I find a proof of your first statement? | |
| Apr 25, 2012 at 18:17 | history | answered | Jacob Lurie | CC BY-SA 3.0 |