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Xuanrui Qi (Apr 20 2023 at 08:23):Hi all! I recently just started reading about differentiable stacks, and while I currently don't yet know much, they look fantastic. I'm especially interested by the possibility to get equivariance "for free" by taking quotient stacks. If I'm understanding it correctly, it seems possible to do some sort of higher analogue of SDG in differentiable stacks, perhaps via Coquand et al.'s work on stack semantics. I wonder if there are any references that go in this direction, or is my understanding wrong?
John Baez (Apr 20 2023 at 16:04):If you read [[differentiable stack]] in nLab, it points out that these are a special case of [[smooth infinity-groupoids]], and that article says:
There is a refinement of smooth ∞-groupoids to synthetic differential ∞-groupoids. See SynthDiff∞Grpd for more on that.
So, this is pointing you toward an "infinitely higher" version of synthetic differential geometry.
Mike Shulman (Apr 20 2023 at 16:04):Yep, this is the direction of [[differential cohesive homotopy type theory]]. It's still quite nascent, but I think it has exciting possibilities.
John Baez (Apr 20 2023 at 16:07):Maybe there are more papers on synthetic differential geometry at the level of stacks, as opposed to infinity-groupoids? I wasn't trying to force @Xuanrui Qi in the direction of infinity - that's just what the nLab tends to do.
Xuanrui Qi (Apr 20 2023 at 16:10):Yes I'm aware of differential cohesive HoTT, but I'm perhaps not yet ready to go directly to infinity yet. Also since I'm mainly thinking about equivariance right now, it seems that the stack level is more appropriate?
Mike Shulman (Apr 20 2023 at 16:11):Oh, by "stack" you meant "1-stack"? Sorry, my mind is almost always in the -world nowadays by default... (-:O
Mike Shulman (Apr 20 2023 at 16:13):I don't actually know of any work on SDG that uses only 1-stacks. There's plenty of it using plain old sheaves (0-stacks), and the references on the page I linked to for the -case. Of course the 1-stacks sit inside the -stacks, so you can often just ignore the higher stuff if you don't like it.
Xuanrui Qi (Apr 20 2023 at 16:55):Yes, stacks in the algebraic geometric sense :-) I'm not morally opposed to (and quite curious about) -stacks, of course, but I'm not sure how would one take quotients of them.
Mike Shulman (Apr 20 2023 at 16:59):I'm tempted to say "the same way one takes quotients of 1-stacks". (-: How do you think about quotients of 1-stacks?
Xuanrui Qi (Apr 20 2023 at 17:36):I'm not familiar with the details at all, but I know you take the quotient wrt some action of a group, so equivariant bundles and all that. In the case I can just do something like that?
Xuanrui Qi (Apr 20 2023 at 17:37):Actually I just found this: https://arxiv.org/pdf/2112.13654.pdf. Looks like one can indeed do the same!