00:33:18 Sean Tilson: I think Tom is write and what I described is not really correct. Sorry. 00:33:28 Sean Tilson: goodness… right. 00:39:44 Tom Bachmann: I'm not sure that "derived 1-stacks" make a lot of sense (if I understand this as 1-truncated sheaves on dAff), since typical derived affine schemes are not 1-truncated, so dAff does not embed into this category... 00:41:35 Federico Binda: Well, you can consider functors from SCRng to S_{\leq 1} and you get an embedding from usual stacks 00:42:04 Viktor Kleen: but they won't be 1-truncated objects in the infty-category sense 00:43:41 Tom Bachmann: what does make sense is 1-geometric derived stacks (i.e. the subcategory of sheaves of spaces on SCR admitting a smooth representable map from an SCR). Maybe that's what he meant. Makes more sense actually now taht I think about ti. 00:45:16 Federico Binda: No: they both make sense: given any derived stack F you say that F is an n-stack if F(A) is in S_{\leq n} for every A \in Rng (note that A is not derived here!) 00:45:34 Federico Binda: to check if something is n-stack you test it against usual rings 00:45:55 Tom Bachmann: Why is this a reasonable notion though? 00:49:06 Federico Binda: because that’s the only way you can get reasonable bounds on the homotopy groups of the target: if you evaluate against arbitrary SCRng you can get arbitrary homotopy types. Toen-Vezzosi (and also Lurie) call Artin n-stacks the derived n-stacks (in the previous sense) that are also m-geometric (in the sense of the inductive definition we have seen before) 00:49:29 Tom Bachmann: Interesting! 00:51:53 Ran: So what are derived 0-stacks then according to that definition? 00:52:21 Viktor Kleen: you would need to take the colim in presentable categories, I think 00:52:42 Tom Bachmann: That will give you the Sp(C) again, yes. 00:53:49 Federico Binda: @Ran I mean: you say that F is -1 geometric if F = Spec(A) for some SCRng A. Then inductively… For example a classical Artin Stack in this story is a 1-geometric 1-stack 01:28:56 Tom Bachmann: Reading a bunch of things (instead of following the talk...), I think I have found references that QCoh(X x Y) = QCoh(X) \otimes QCoh(Y) for all qcqs schemes X,Y 01:29:31 Prof Kirsten Wickelgren, Ph.D.: is \otimes some Lurie tensor product? 01:29:34 Tom Bachmann: yes 01:29:49 Tom Bachmann: I guess derived schemes is probably also ok 01:30:08 Tom Bachmann: though not stacks 01:30:10 Prof Kirsten Wickelgren, Ph.D.: Counterexample to = D(A times_C B)? 01:30:36 Tom Bachmann: I don't follow. 01:31:01 Prof Kirsten Wickelgren, Ph.D.: Is there a counterexample to the claim that CQoh(pullback) = D(A otimes^L_C B)? 01:31:53 Prof Kirsten Wickelgren, Ph.D.: If not, derived categories are doing pretty well! 01:33:00 Tom Bachmann: QCoh of any affine scheme is the derived category of that scheme, and pullback of derived affine schemes = derived tensor product. 01:34:16 Prof Kirsten Wickelgren, Ph.D.: Thansk 01:56:06 Viktor Kleen: Thanks! 01:56:13 Can Yaylali: Thanks 01:58:14 Fangzhou Jin: what is the classical relation between f^! and cotangent complexes? 01:58:38 Chirantan Chowdhury: Thank you for the talk !! 01:58:44 Prof Kirsten Wickelgren, Ph.D.: for f smooth, f^! = det L f^* 01:58:49 Federico Binda: See you next week!