Timeline for A question on the stability of $\operatorname{Cat}$ in $\operatorname{Cat}_\infty$
Current License: CC BY-SA 4.0
21 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 28, 2020 at 19:26 | answer | added | Viktoriya Ozornova | timeline score: 11 | |
| Oct 27, 2020 at 18:03 | comment | added | PushoutOfCategories | @AndreaGagna Ah! I think that covers some of the cases I'm interested in, and I know a reference for the key fact underlying that argument. I've posted this example as an answer. | |
| Oct 27, 2020 at 12:19 | comment | added | Denis Nardin | @PushoutOfCategories You are right, I was mistaken in saying that Achim's example is of this kind (the leg is a subcategory but not a replete subcategory). | |
| Oct 27, 2020 at 8:50 | comment | added | Andrea Gagna | A sufficient condition, that I've learnt from Viktoriya Ozornova and Martina Rovelli, is that the functor $F$ is a Dwyer map. The proof is not trivial. | |
| Oct 26, 2020 at 23:26 | history | edited | PushoutOfCategories | CC BY-SA 4.0 |
added 227 characters in body
|
| Oct 25, 2020 at 11:30 | comment | added | PushoutOfCategories | I found the hypothesis that $F$ be a monomorphism plausible, since it feels like whatever construction freely generates an $\infty$-category should be doing so by simply adding new points to the hom-sets, and shouldn't require any cells identifying morphisms. But I don't understand cofibrations well enough in simplicial categories to work it out there, I don't have enough proficiency with inner anodyne extensions to work it out in quasi-categories, and I haven't yet been able to work out the localization from simplicial objects to category objects in $Gpd_\infty$ to show it there. | |
| Oct 25, 2020 at 10:02 | comment | added | PushoutOfCategories | Oic, I had miswritten the monomorphism condition; the preimages of isomorphisms have to exist in each relevant homset, not just once globally. | |
| Oct 25, 2020 at 10:01 | history | edited | PushoutOfCategories | CC BY-SA 4.0 |
added 27 characters in body
|
| Oct 25, 2020 at 9:33 | comment | added | PushoutOfCategories | ... but maybe I misunderstand. The two descriptions of of "every $\infty$-category is a localization of a category" I know are (1) pushing out along a disjoint union of copies of the map from $\Delta^1$ to the interval groupoid (which fails the monomorphism condition), and (2) pushout along the map from a subcategory to its $\infty$-groupoidification (which fails the 1-category condition). If there is another description that fits the condition of the post, it's not coming to mind. | |
| Oct 25, 2020 at 9:24 | comment | added | PushoutOfCategories | @DenisNardin Actually, the example of AchimKrause isn't of that form. While every $\infty$-category is the localization of a poset, the map from a disjoint union of $\Delta^1$ into that poset is not a monomorphism. E.g. the map $\{0 \leq 1 \} \amalg \{1' \leq 2\} \to \{0 \leq 1 \leq 2\}$ is not a monomorphism in $Cat_\infty$ because the domain doesn't have a map $1 \to 1'$ whose image is the identity on $1$. Similarly, the map from $\Delta^1$ to its localization is not a monomorphism either (since it lacks a map $1 \to 0$). | |
| Oct 25, 2020 at 9:20 | history | edited | PushoutOfCategories | CC BY-SA 4.0 |
added 66 characters in body
|
| Oct 25, 2020 at 9:14 | history | edited | PushoutOfCategories | CC BY-SA 4.0 |
added 512 characters in body
|
| Oct 25, 2020 at 9:09 | comment | added | Denis Nardin | @PushoutOfCategories "F is a monomorphism in $Cat_∞$" is precisely the condition of being (equivalent to) the inclusion of a replete subcategory. Unfortunately Achim's example shows that this is of course not nearly enough (every localization can be realized as a pushout along a replete subcategory!). | |
| Oct 25, 2020 at 9:04 | comment | added | PushoutOfCategories | @DenisNardin I was thinking about computing the pushout in the canonical model structure on Cat which is why I was focusing on the "injective on objects" condition. IIRC, the second of the two conditions I list the proposition "$F$ is a monomorphism in $Cat_\infty$" restricted to the case $F$ is injective on objects. So the equivalence-respecting condition would be to consider $F$ being a monomorphism in $Cat_\infty$. | |
| Oct 25, 2020 at 7:37 | comment | added | Achim Krause | For groupoids, something akin to your first condition should indeed work, namely that the functors are injective on $\pi_0$ and faithful. This should reduce to the corresponding statement for classifying spaces of groups. For general categories, condition 1 is definitely not enough. You can consider a pushout where the upper right corner is an arbitrary category, the upper left corner is a disjoint union of multiple $\Delta^1$, and the l.l. corner is the localisation of the u.l. corner. Then the pushout is a localisation of the category you started with, and not generally a $1$-category. | |
| Oct 25, 2020 at 6:39 | comment | added | Denis Nardin | Note that being injective on objects or on morphisms is not a condition stable under equivalence, and so it is unlikely to be of help. Maybe asking for one leg to be a replete inclusion will work, but I'm honestly skeptical there's a sensible condition for this. | |
| Oct 25, 2020 at 3:51 | history | edited | LSpice | CC BY-SA 4.0 |
Minor proofreading and TeXing
|
| Oct 25, 2020 at 2:19 | history | edited | PushoutOfCategories | CC BY-SA 4.0 |
added 564 characters in body
|
| Oct 25, 2020 at 2:07 | comment | added | David Roberts♦ | If F is an equivalence then I imagine the pushout is still a 1-category, but this is too restrictive. | |
| Oct 25, 2020 at 2:02 | review | First posts | |||
| Oct 25, 2020 at 3:51 | |||||
| Oct 25, 2020 at 1:59 | history | asked | PushoutOfCategories | CC BY-SA 4.0 |