I feel like someone once referred me to a result in Lurie's SAG specifically about the slice topos over an n-truncated object in an ∞-topos (maybe something like such a topos is n-localic, or something like that). Does anyone know what result I'm thinking of?
Does anyone know of a book or lecture notes which includes an explanation of the results in Thom's paper "Quelques propriétés globales des variétés différentiables"? I am interested in understanding Thom's results on representing homology classes of a manifold by submanifolds.
@Michael I'm sure the answer is yes. If you read French, the paper itself is not bad, although I remember when I looked at it my emphasis was on other parts.
@AlexanderCampbell sounds plausible. I guess I'm thinking that classically you might need a Cat-enriched operad. Does that translate into needing something a little fancier $\infty$-categorically?
@TimCampion Yeah, I have the same suspicion, which makes me think the answer to my original question is no. I was hoping we could use Lurie's version of the Day reflection theorem (HA.2.2.1.9).
It's interesting that Lurie states the theorem in the O-monoidal context. Does he really rely on O-monoidality, or does the heart of the proof extend to more general operads?
wait maybe I'm confused and that question doesn't make sense
An O-monoidal category is a cocartesian fibration over O / a colax O-monoidal category is a locally cocartesian fibration over O. So the question seems to be whether in HA 2.2.1.9, we can get away with locally cocartesian rather than cocartesian. Which I think boils down to the same question about HA 2.2.1.11...
Actually wait -- I think this is over-optimistic. Lurie's results about localization are about reflective localizations, but the localization from a model category to its presented $\infty$-category is decidedly not of this form.
Just catching up on the really valuable discussion @AaronMazel-Gee initiated a few days ago, I just want to say that it renews my hope in the MO community to see us talk about these things in such a heartfelt way. For my part, I find many of these challenges to be hard enough normally without a pandemic to exacerbate them!
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@JonathanBeardsley I hope that having such conversations openly can be a step towards the goal Jon expresses that we should always be striving to be a more welcoming space for more people. Are there other things we can be doing in this direction?
Personally, I was already in a mathematical funk at the beginning of last semester. Then I had the amazing experience of being at MSRI with so many amazing people -- some of them in this room now -- and it was really rejuvenating. Then COVID cut things short and I've felt myself regressing for awhile since.
