χle Ormsby
Anyone know of sources in differential geometry / smooth manifolds using correspondences? I'm imagining spans of smooth manifolds X ← Z → Y with Z → Y a submersion (so that composition-by-pullback is well-defined).
Mar 6, 2023 at 10:27pmWeb
Show threadCharlotte Kirchhoff-LukatMar 6, 2023
@MotivicKyle Correspondences as morphisms are a thing in #symplectic geometry; Lagrangian correspondences. You can use these to construct a category of smooth symplectic manifolds (of arbitrary dimension).
However, they wouldn't be submersions on each side, merely so-called reductions -- so may be more general than what you want.
I like http://arxiv.org/abs/1012.0105 as an introduction; it includes a little treatment of the smooth case without symplectic structure, too.
A note on the Wehrheim-Woodward category

Wehrheim and Woodward have shown how to embed all the canonical relations between symplectic manifolds into a category in which the composition is the usual one when transversality and embedding assumptions are satisfied. A morphism in their category is an equivalence class of composable sequences of canonical relations, with composition given by concatenation. In this note, we show that every such morphism is represented by a sequence consisting of just two relations, one of them a reduction and the other a coreduction.

arXiv.org
Show threadCharlotte Kirchhoff-LukatMar 6, 2023
@MotivicKyle Something different: There are various (related) notions of Morita equivalence for geometric structures on smooth manifolds that consist of such a span of smooth manifolds with submersions on each side.
These include Morita equivalences of Poisson manifolds (e.g. https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-142/issue-3/Morita-equivalence-of-Poisson-manifolds/cmp/1104248717.pdf) and of singular foliations (https://arxiv.org/abs/1803.00896)
Show threadχle OrmsbyMar 7, 2023
@charlottekl Thanks for the references! This is exactly the sort of thing I was looking for. I suppose the setup I was imagining is a way to force strong transversality of all the (putatively) composable smooth relations under consideration. (Weinstein doesn't seem committed to a strictly categorical viewpoint on compositionality in that paper and I imagine there are good reasons for that.)
Show threadCharlotte Kirchhoff-LukatMar 7, 2023
@MotivicKyle The history of the #symplectic #category in this form, the Wehrheim-Woodward category, is roughly as follows:
People wanted to construct a symplectic category with more general morphisms than symplectomorphisms (which only exist between diffeomorphic manifolds). The idea of using Lagrangian correspondences is due to Weinstein and his observation that symplectomorphisms are examples. (Weinstein's philosophy: "Everything is a Lagrangian.")
This, of course, does not work due to smooth Lagrangian correspondences not always composing smoothly.
The Wehrheim-Woodward construction avoids this transversality issue entirely by taking as morphisms paths of relations modulo composing individual relations in the sequence that can be smoothly composed -- as far as I am aware, nobody has found a way of making a nice category with only smoothly composable relations.
But now this note by Weinstein shows that, actually, every morphism in the WW category has a representative path consisting of *at most* two non-composable Lagrangian relations, resulting in the span picture. I was somehow surprised by that!