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Paolo Perrone (Jul 01 2020 at 20:29):Hello all! This is the thread of discussion for the talk of Marco Benini, Marco Perin, Alexander Schenkel and Lukas Wolke, "Categorification of algebraic quantum field theories".
Date and time: Friday July 10, 12:35 UTC.
Zoom meeting: https://mit.zoom.us/j/7488874897
YouTube live stream: https://www.youtube.com/watch?v=CNdgmTMMUuw&list=PLCOXjXDLt3pZDHGYOIqtg1m1lLOURjl1Q
Paolo Perrone (Jul 10 2020 at 12:20):Hello. This talk starts in 15 minutes. Mind the "new" Zoom link, https://mit.zoom.us/j/7488874897
Alastair Grant-Stuart (Jul 10 2020 at 13:47):Thanks for the great talk, @Marco Perin. In the Q&A afterwards, you mentioned that "orthogonality" can be used to encode causal disjointness of spacetimes, at least in the context of the category . Is this related to the notion described on nLab's orthogonality page, or is the terminology just coincidental? I notice also that this nLab page links to something called a "factorization system" -- any relation to the prefactorization multicategory of the talk?
Paolo Perrone (Jul 10 2020 at 17:22):Video here!
https://www.youtube.com/watch?v=GEaiSGNPuB4&list=PLCOXjXDLt3pYot9VNdLlZqGajHyZUywdI
Nicolas Blanco (Jul 10 2020 at 18:12):Thanks for the great talk @Marco Perin ! I am not sure if my question makes any sense but let me ask it anyway.
Is there a kind of "Grothendieck construction" that will turn an AQFT into some notion of fibration where the fibre over a space is the corresponding algebra ? If this exists, are their pullbacks and pushforwards and do they have some physical interpretation?
Marco Perin (Jul 11 2020 at 12:01):Thanks @Alastair Grant-Stuart ! Our orthogonality is something different to the one on nlab. We consider pairs of maps to THE SAME target and declare some of them to be special. That’s a further input you have to give from the outside. Hence, it’s just a coincidence that terminology coincides. Unluckily there is quite a lot of clash in terminology! Thanks a lot for the feedback and the nice question!
Marco Perin (Jul 11 2020 at 13:39):Hi @Nicolas Blanco ! Thanks a lot for the feedback. Your question is actually really interesting and intriguing.
The honest answer is that I never thought about it. There are notions of fibrations for multicategories (see http://sqig.math.ist.utl.pt/pub/HermidaC/fib-mul.pdf) and there is a notion of Grothendieck construction for multicategories (see https://ncatlab.org/nlab/show/fibration+of+multicategories), but I do not know what this would produce in our specific case. I will discuss it with my collaborators and will try to come back to you with a reasonable answer.
Nicolas Blanco (Jul 13 2020 at 10:30):Hello @Marco Perin . Thanks for the answer. I had indeed in mind the connection to Hermida's fibration of multicategories.
Under the Grothendieck construction (of categories) any pseudofunctor into Cat gives rise to a category of elements that comes with a pullback or pushforward functor between the fibres (depending on the variance of the pseudofunctor). This can be extended to multicategories where a contravariant pseudofunctor from a multicategory to Cat gives a multicategory of elements where the fibres are related by a pushforward multifunctor (the covariant case is a little more subtle and we really get a family of pullback multifunctors).
It should be possible to get something similar by replacing Cat by Alg ; then the fibres will be algebras and they will be related by pushforward algebra morphisms. I am not sure if I give anything relevant to think in this way though.
As a side note: the Grothendieck construction is a special case of a more general correspondence between functors (not necessarily fibred) and lax normal functor into Dist (the 2-category of distributors) see Displayed categories. In this setting replacing categories by algebras should I think correspond to replace Dist by the 2-category of algebras and bimodules. Still not sure if this add anything useful.
Marco Perin (Jul 13 2020 at 11:33):Hi @Nicolas Blanco . Spot on! That is exactly what I do not know :grinning_face_with_smiling_eyes: I will look into this ( at the moment I do not see something useful coming out, but it is anyhow interesting, thanks!)