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Stream: learning: questions

Topic: ∞-category of lagrangian cobordisms

Josselin Poiret (Jun 25 2020 at 09:53):

Is there anyone familiar enough with the different constructions of cobordism categories here?
I'm currently working on the article A stable infinity-category of Lagrangian cobordisms by Nadler and Tanaka, and there are multiple things that I can't really wrap my head around, especially when comparing with lurie's approach:

• what is the fundamental difference between lurie's and this construction, apart from the lagrangians which are completely accessory here? one uses segal spaces (which i am unfamiliar with) and the other weak kan complexes, so how could I compare them?
• in both cases they use possibly non-compact manifolds, but doesn't that allow for too many cobordisms that eventually make the ∞-category trivial? because i could just take the identity cobordism for my manifold and carve out a morphism to the empty set by intersecting with a good open, and this would be enough to make it isomorphic to the identity, right?
• does anyone know if the time direction in their paper has any vital importance? because to me the cobordisms they build in that paper actually are just manifolds living over standard n-simplexes, but pulled back over the cube via some submersion: their claim that what they are building is indeed a weak kan complex is given without a solid proof (they claim the existence of some deformation retract onto the proper hollow cube), and i'm struggling to write it down since it should involve nontrivial smooth maps of manifolds with corners (which are actually "avoided" in the article) whose formulas i cannot write down. Although with simplicial sets, no one seems to have written the actual deformation retract either

also, my gut feeling tells me that the collar condition could be relaxed up to homotopy, so that we could ignore them entirely, in the same way that transversality can be circumvented, but i haven't found any good literature on the subject: is there any good reference or work on that somewhere?

John Baez (Jun 26 2020 at 05:29):

For how to compare Segal spaces to weak Kan complexes (= quasicategories) and other models of $(\infty,1)$-categories, here's a fundamental reference:

• Julie Bergner, A survey of $(\infty,1)$-categories.