4:53 AM
i'd like to make a plug for a project i'm involved in, which i may have mentioned here before: a double conference. the first double conference is currently underway, on "higher algebra & mathematical physics" -- see here and here . there have been some really neat talks so far, and all videos are being posted to the PI page (scroll down and then click on the "videos" tab).
@JonathanBeardsley right. if all you want is a statement at the level of $\infty$-categories, then as i think has been mentioned before, all you need to do is trace through the image of the full subcategory on the objects [0] and [1] (to see if the morphisms "s" and "t" get swapped or not)
@HarryGindi i don't think the specific functor is very important, really all that matters is that it's a reedy-cofibrant replacement of the standard inclusion $\Delta \to Cat \to sCat$
2 hours later…
7:33 AM
@PraphullaKoushik Since we talked about suitable top-level tags for questions about stacks (and gerbes), I'll add that stacks are explicitly listed as one of the topics in the ag.algebraic-geometry tag-info.
However, it seems from @DenisNardin's response and @DavidRoberts' edit/retag, that your questions are probably closer to dg.differential-geometry.
(Sorry for bothering you with the pings, but since this user explicitly mentioned that they were unsure which tags to use and you have interacted with them here in chat or on the main site, I thought you might have a reasonable advice.)
Of course, if this is too big digression from the topic of this room, the discussion about this can continue in MO editors' lounge - tagging is one of the main topics there.
Hey, I have two very basic questions about localizations of $\infty$-categories:
Let $C$ be a presentable $\infty$-category
1. What extra words do I need to put in to make sure that the following is precisely true: "There's a bijective correspondence between idempotent monads on $C$ and reflective localizations of $C$" (in particula I'm worried about what kind of monads am I allowed to use)
2. Suppose that $C$ is moreover pointed. Then (modulo set theoretical issues which I prefer to ignore for this point) any reflective localization of $C$ gives a coreflective localization. You take the…
Let $C$ be a presentable $\infty$-category
1. What extra words do I need to put in to make sure that the following is precisely true: "There's a bijective correspondence between idempotent monads on $C$ and reflective localizations of $C$" (in particula I'm worried about what kind of monads am I allowed to use)
2. Suppose that $C$ is moreover pointed. Then (modulo set theoretical issues which I prefer to ignore for this point) any reflective localization of $C$ gives a coreflective localization. You take the…
3 hours later…
10:27 AM
1 hour later…
11:27 AM
When C is an Eilenberg-Zilber Reedy Cat (this is equivalent to Regular Skeletal in the sense of Cisinski, which is what I am actually using) admitting a terminal object, I can equip Psh(ΘC) with an 'enriched Joyal model structure' and sPsh(C)-Cat has the enriched model structure coming from the injective model structure for simplicial presheaves
In that situation, I have sketched a proof in my private notes that the enriched HC-realization/nerve pair is a Quillen pair
I might be able to show that it's a QEQ, but I'm not that interested in showing it in the full generality.
Anyway, I'm working now on showing that this Quillen adjunction remains a Quillen adjunction when you do Bousfield localization on both sides
It's straightforward but I haven't written it up yet. I am not that interested in developing it in full generality. The case I have in mind is the case where C=Θ=Θ_ω. I also have a definition of the explicit ω-unstraightening and ω-straightening functors in this case, but I can't quite prove anything about them yet because there is some greatly-needed combinatorial work in understanding the lax join/lax slice for cellular sets combinatorially
The missing combinatorial thing is: If X is an ω-quasicat and x is a vertex of X, then the projection map
X_{/_{lax}x} -> X
is an ω-cartesian fibration
X_{/_{lax}x} -> X
is an ω-cartesian fibration
12:41 PM
7 hours later…
8:00 PM
8:38 PM
@SaalHardali The natural map X -> π_0(X) is a reflective localization on based spaces (it's the nullification of the map S^1_+ -> S^0).
The kernel of this map is the class of connected spaces, and the coreflective localization is the natural inclusion X_0 -> X of the basepoint component.
2 hours later…
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