00:20:37 Prof Kirsten Wickelgren, Ph.D.: I’m trying to understand the two types of squares Fangzhou, Marc and Heng are comparing: in an infinity category, we can take a pullback. That’s one of the types of squares, right? What’s it called and what’s the other?
00:22:37 Marc Noel Levine: In a model categy homotopy pullback is pullback after replacing one map with an equivalent fibration. I assume there is a similar story for infinity categories
00:23:24 Sean Tilson: Isn’t that what is meant by homotopy pullback in an infinity category?
00:24:03 Chirantan Chowdhury: As far as I know, an infinity pullback is a homotopy pullback in the corresponding model category associated to the infinity category. There should be only one such notion in an infinity category.
00:37:57 Sean Tilson: Is it obvious that the cotangent complex always has such a small resolution?
00:38:18 Pavel Sechin: i guess, that's the definition of quasi-smooth
00:38:27 Sean Tilson: Thank you!
00:39:28 Sean Tilson: it is loops.
00:56:18 Pavel Sechin: backing up the question of Kirsten from the beginning, just some confusion on my side: what is the non-derived product of derived schemes? a simplicial ring of an affine derived scheme is only defined up to equivalence, why should we be allowed to make a construction which does not preserve quasi-isomorphisms, e.g. taking the non-derived tensor product?
00:58:40 Sean Tilson: II would agree that we are not allowed to.
01:01:26 Can Yaylali: I would also agree, but as far as I understood every product is the homotopy product and then we pass to the homotopy category, where it is then well defined, right ?
01:02:18 Viktor Kleen: That might not be the case if things are not flat
01:02:24 Marc Noel Levine: I don’t think every product is a homotopy product. This is not true in the infty cat of Kan complexes.
01:04:08 Can Yaylali: I meant the products of derived schemes we consider
01:04:34 Marc Noel Levine: I see, sorry I misunderstood
01:07:16 Sean Tilson: wait, aren’t all kan complexes fibrant?
01:07:31 Marc Noel Levine: but not all maps are fibrations
01:08:22 Sean Tilson: but the map from a kan complex to a point is a vibration and so then when you take the pullback you get the product, no? Or is the homotopy pullback in this situation computing something different from the homotopy product?
01:08:41 Sean Tilson: it doesn’t matter though, sorry for the distraction.
01:08:45 Marc Noel Levine: for a map to a point, everything is fine
01:09:31 Sean Tilson: I guess I thought that implied that every product of kan complexes was a homotopy product. maybe I misunderstood.
01:09:52 Marc Noel Levine: I think this is correct
01:37:38 Sean Tilson: That was helpful, thanks!
02:00:03 Pavel Sechin: thanks!
02:00:10 Can Yaylali: thanks
02:00:11 Chirantan Chowdhury: Thanks !!
02:00:13 Ran Azouri: Thank you!
02:00:14 Alessandro D´Angelo: Thank you!
02:00:18 Prof Kirsten Wickelgren, Ph.D.: Thank you!