Timeline for What is the relationship between $O_G$-parameterized $\infty$-categories and $\infty$-categories enriched in $Top_G$?
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| Jun 28, 2018 at 7:21 | comment | added | Denis Nardin | @TimCampion Late to the party, but the problem with this approach for compact Lie group has nothing to do with the notion of G-category. In fact a majority of our exposé 2 (J. Shah's thesis) works in this setting. The problems arise when you start to consider monoidal structure, because you have no analogue of finite G-sets. | |
| Jun 13, 2018 at 18:50 | comment | added | Tim Campion | There's something funny, though -- in ordinary category theory, taking the completion of a Segal space corresponds to localizing at the fully faithful, essentially surjective functors. But notoriously, this is tricky to do for internal categories -- for e.g. Lie groupoids, you need to introduce a more flexible notion of internal functor before you can recover some version of "equivalence = fully faithful + essentially surjective". Perhaps such a notion would be necessary to push the study of parameterized categories to do equivariant homotopy theory over general compact Lie groups. | |
| Jun 13, 2018 at 18:46 | comment | added | Tim Campion | @DylanWilson Thanks for the reference! I've definitely seen this idea before that fibered categories correspond to "non-small internal categories" -- I've never really looked into it, but it should have come to mind in connection to this question. Remarkably, Pare-Schumacher are actually working with categories equipped with a specified class of "canonical isomorphisms" (1st paragraph of I.0 and more discussion near the end of II.1.2) -- this seems to correspond precisely to the fact that an internal category without the univalence condition really corresponds to an incomplete Segal space! | |
| Jun 11, 2018 at 16:50 | comment | added | Dylan Wilson | (sorry, should be Johnstone-Pare-Rosebrugh-Schumacher-Wood-Wraith, and the chapter in question is by Pare-Schumacher) | |
| Jun 11, 2018 at 16:49 | comment | added | Dylan Wilson | It looks like some discussion of this in the 1-categorical context is in Johnstone-Pare's book on "indexed categories and their applications" page 25. (Their "indexed categories" are basically the 1-categorical versions of today's "parameterized categories" ) | |
| Jun 11, 2018 at 15:49 | comment | added | Denis Nardin | I think you can get the statement you want for enriched categories from a variant of the proof of theorem 4.4.6 of Gepner and Haugseng enriched ∞-categories paper | |
| Jun 11, 2018 at 15:04 | history | edited | Tim Campion | CC BY-SA 4.0 |
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| Jun 11, 2018 at 14:57 | comment | added | Tim Campion | This leaves me wondering how much of the parameterized category theory developed by Barwick et al can be recast as general internal category theory. | |
| Jun 11, 2018 at 14:51 | history | edited | Tim Campion | CC BY-SA 4.0 |
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| Jun 11, 2018 at 14:38 | comment | added | Tim Campion | All of this works for an arbitrary presheaf category. That is, if $T$ is any small $\infty$-category, then categories fibered over $T$ are the same as categories internal to presheaves on $T$. That is, so long as you agree with my formulation of univalence for internal categories -- which admittedly doesn't necessarily make sense if you want to work internal to an arbitrary finitely-complete category. But at least it makes sense in the presheaf case. I suppose if you wanted it to work for an arbitrary finitely-complete category, you could use an appropriate truncation of $E_\bullet$. | |
| S Jun 11, 2018 at 14:28 | history | answered | Tim Campion | CC BY-SA 4.0 | |
| S Jun 11, 2018 at 14:28 | history | made wiki | Post Made Community Wiki by Tim Campion |