Timeline for Is the $\infty$-category of presentable $\infty$-categories presentable?
Current License: CC BY-SA 3.0
20 events
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| Jul 11, 2015 at 11:52 | comment | added | David White | Thanks for making the edit. This is all much clearer now to me at least. Previously it seemed like the two answers were at odds, but no longer. | |
| Jul 10, 2015 at 19:01 | history | edited | Yonatan Harpaz | CC BY-SA 3.0 |
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| Jul 10, 2015 at 18:58 | comment | added | Yonatan Harpaz | This definition is not correct, it should be the (not full) subcategory spanned by $\kappa$-compactly generated presentable $\infty$-categories and functors which preserve $\kappa$-small objects. This is the definition to which Proposition 5.5.7.10 applies to and that is indeed a (non-full) subcategory which is small. The argument appearing in the answer above still applies to show that $Pr^L$ is just large, and not very large. | |
| Jul 10, 2015 at 18:44 | comment | added | Ilan Barnea | @YonatanHarpaz In your answer above you define $Pr^L_{\kappa}$ to be the full subcategory of $Pr^L$ spanned by $\kappa$-compactly generated presentable $\infty$-categories. Then by Proposition 5.5.7.10 in HTT we know that if $\kappa > \omega$ the $\infty$-category $Pr^L_{\kappa}$ is equivalent to the $\infty$-category of small $\infty$-categories admitting $\kappa$-small colimits. This last category is locally small so it looks to me that $Pr^L$ is also locally small | |
| Jul 10, 2015 at 17:55 | comment | added | Yonatan Harpaz | @IlanBarnea, colimit preserving functors $C \longrightarrow D$ are in bijection with the collection all functors $C_{\chi} \longrightarrow D$, which is big. This also made me realize that I need to correct the definition of $Pr^L_{\kappa}$ in the answser above. | |
| Jul 10, 2015 at 2:07 | comment | added | Ilan Barnea | @YonatanHarpaz Let $C$ and $D$ be presentable $\infty$-categories. There exists some regular (small) cardinal $\chi$ such that $C\cong Ind_{\chi}C_{\chi}$ and $D\cong Ind_{\chi}D_{\chi}$, where $C_{\chi}$ and $D_{\chi}$ are essentially small and admit $\chi$-colimits. Then the set (0-simplicies) of colimit preserving functors $C\to D$ is bijective to the set of $\chi$-colimit preserving functors $C_{\chi}\to D_{\chi}$, which is small. Is there something wrong with this argument? | |
| Jul 5, 2015 at 21:38 | comment | added | Eric Wofsey | @YonatanHarpaz: Ah, OK, sorry, I was misreading your definitions. | |
| Jul 5, 2015 at 20:46 | comment | added | Yonatan Harpaz | -> Now the category of large categories is very large (its object set cannot belong to $V_{\tau}$, but it does belong to $V_{\lambda}$), and will become just large (but not small), when I change my universe to $V_{\tau}$. | |
| Jul 5, 2015 at 20:46 | comment | added | Yonatan Harpaz | @EricWofsey, Let $\lambda > \tau > \kappa$ be strongly inaccessible cardinals and let $V_{\kappa}, V_{\tau}, V_{\lambda}$ be the associated universes. If $V_{\kappa}$ is my universe then I call elements in $V_{\kappa}$ "small sets", elements of $V_{\tau}$ "large sets" and elements of $V_{\lambda}$ "very large sets". If I change my universe to $\tau$ it means I now refer to sets in $V_{\tau}$ as "small" and to sets in $V_{\lambda}$ as "large" -> | |
| Jul 5, 2015 at 19:22 | history | edited | David White | CC BY-SA 3.0 |
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| Jul 4, 2015 at 23:42 | comment | added | Zhen Lin | @YonatanHarpaz I would rather think of the $(\infty, 2)$-category of $(\infty, 1)$-categories as the archetypical $(\infty, 2)$-topos, just as the 1-category of 0-categories is the archetypical 1-topos. | |
| Jul 4, 2015 at 21:08 | history | edited | Yonatan Harpaz | CC BY-SA 3.0 |
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| Jul 4, 2015 at 21:00 | comment | added | Yonatan Harpaz | It just occurred to me that $Pr^L$ is not presentable simply because it is not locally small (the space of functors between two presentable $\infty$-categories is the maximal $\infty$-groupoid of a presentable $\infty$-category, and as such is large). However, when considered as an $(\infty,2)$-category, the mapping categories in $Pr^L$ between every two objects are presentable, and hence controlled by a small amount of data. This might be considered as evidence that $Pr^L$ should be something like $(\infty,2)$-presentable. | |
| Jul 4, 2015 at 20:52 | comment | added | Yonatan Harpaz | I think there is just a terminology gap here. To avoid confusion, one should not (as I did) use the language of "large" and "small" and the language of universes at the same time. By a large $\infty$-category I just meant a small $\infty$-category of the next universe. Consequently, the $\infty$-category of possibly large $\infty$-categories is what would be the $\infty$-category of small $\infty$-categories in the next universe, and hence not small in the next universe. These kind of objects is what I called "very large". | |
| Jul 4, 2015 at 20:34 | comment | added | Eric Wofsey | Sure. I just wanted to correct your assertion that a very large category might still be large in a larger universe. | |
| Jul 4, 2015 at 20:32 | comment | added | Yonatan Harpaz | @EricWofsey, you're right, but after you made this enlargement it will not be interesting anymore to ask if the original $\infty$-category is presentable. Instead, you should stop when it is just large, and ask if it is presentable then. | |
| Jul 4, 2015 at 20:24 | comment | added | Eric Wofsey | Even "very large" categories become small when you pass to a larger universe. For instance, if $V_\kappa$ is a universe, then the collection of all "large" categories in $V_\kappa$ has cardinality only $2^\kappa$, which is far smaller than any inaccessible larger than $\kappa$. | |
| Jul 4, 2015 at 20:19 | comment | added | Theo Johnson-Freyd | ... are well-studied in some areas of computer science, because they allow for coinductive, rather than inductive, reasoning. A more down-to-earth reason not to be afraid of Cantor's paradox is the well-known theorem that the homotopy category of topological spaces is not concretizable (because every non-empty object has a proper class, not a set, of subobjects). | |
| Jul 4, 2015 at 20:16 | comment | added | Theo Johnson-Freyd | I disagree: in principle the answer should be yes. Indeed, $Pr^L$ contains all limits and colimits (the strict version of this statement is due to Greg Bird in his unpublished '76 thesis, if my memory is correct), and feels like it is generated under colimits by some basic building blocks. The reason Cantor's paradox does not apply is that we are in the homotopical world, not the set-theoretic world. Indeed, there are good homotopical models (although now I am speaking outside my expertise) in which all functors, and in particular the "power set" functor, have fixed points. These worlds ... | |
| Jul 4, 2015 at 19:55 | history | answered | Yonatan Harpaz | CC BY-SA 3.0 |