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Paolo Perrone (Jun 09 2020 at 00:18):Hey all! Here's the official thread of discussion for my talk, "Kan extensions are partial colimits".
Date and time: Thursday June 11th, 12 noon EDT
Zoom meeting: https://mit.zoom.us/j/280120646 (Meeting ID: 280 120 646)
Youtube live stream: https://youtu.be/s9v1q3tdsrc
Paolo Perrone (Jun 11 2020 at 15:59):Hello all! We start in 1 minute.
Emily Riehl (Jun 11 2020 at 16:12):There are (at least) two ways to turn the category of 1-categories into an (∞,1)-category (modeled as a quasi-category). One thing you can do is treat it as a 1-category and take it's usual nerve. Or you can treat it as a (2,1)-category and take its 2-categorical nerve.
The difference is whether the 2-simplices witness a strict composition relation of functors or witness a natural isomorphism between functors.
It seems likely to me that the bar construction for a pseudo-algebra A for a pseudomonad T on Cat defines a simplicial object Δ^οp -> Cat (in the sense of (∞,1)-category theory) where Cat is considered as a (2,1)-category not as a 1-category.
Martti Karvonen (Jun 11 2020 at 17:02):I'm wondering if one could also get a handle on the matter (or perhaps sidestep it, if the -approach is seen as the primary one) by taking a flexible/cofibrant replacement of the monad so that pseudoalgebras become strict algebras.
Paolo Perrone (Jun 11 2020 at 17:41):Emily Riehl said:
There are (at least) two ways to turn the category of 1-categories into an (∞,1)-category (modeled as a quasi-category). One thing you can do is treat it as a 1-category and take it's usual nerve. Or you can treat it as a (2,1)-category and take its 2-categorical nerve.
The difference is whether the 2-simplices witness a strict composition relation of functors or witness a natural isomorphism between functors.
It seems likely to me that the bar construction for a pseudo-algebra A for a pseudomonad T on Cat defines a simplicial object Δ^οp -> Cat (in the sense of (∞,1)-category theory) where Cat is considered as a (2,1)-category not as a 1-category.
Yep. Thank you! That sounds indeed what's happening there. But I have to look into that more deeply.
fosco (Jun 11 2020 at 17:54):Emily Riehl said:
There are (at least) two ways to turn the category of 1-categories into an (∞,1)-category (modeled as a quasi-category). One thing you can do is treat it as a 1-category and take it's usual nerve. Or you can treat it as a (2,1)-category and take its 2-categorical nerve.
The difference is whether the 2-simplices witness a strict composition relation of functors or witness a natural isomorphism between functors.
It seems likely to me that the bar construction for a pseudo-algebra A for a pseudomonad T on Cat defines a simplicial object Δ^οp -> Cat (in the sense of (∞,1)-category theory) where Cat is considered as a (2,1)-category not as a 1-category.
Is this in some way addressing also the question on what is a simplicial object where the simplicial identities hold up to coherent iso?
fosco (Jun 11 2020 at 17:56):If not, I would try to compare it to "a pseudofunctor , where codomain is categories, pseudofunctors, and pseudonatural transformations"
fosco (Jun 11 2020 at 17:57):and domain is... some enhancement of Delta? :slight_smile:
Nathanael Arkor (Jun 11 2020 at 18:04):Could "some enhancement of " be given as some restriction of the free monoidal bicategory containing a monoidal category? Or is this weakening in the wrong sense?
Paolo Perrone (Jun 11 2020 at 20:57):Hey all! Here's the video.
https://youtu.be/wTjdEzFGuOg
Paolo Perrone (Jan 13 2021 at 06:49):For all who are interested, the preprint is out!
https://arxiv.org/abs/2101.04531
Nathanael Arkor (Jan 13 2021 at 16:12):This looks very interesting! A few initial observations after skimming through:
Paolo Perrone (Jan 14 2021 at 08:49):Nathanael Arkor said:
- There are two occurrences of the typo "preshaves".
Of course a preshave is just the dual to an aftershave.
Paolo Perrone (Jan 14 2021 at 10:30):To answer the other questions:
- There's another approach to pseudomonads for free cocompletion, found in Kelly–Lack's On the monadicity of categories with chosen colimits. In particular, they show that there is a 2-monad (rather than just a pseudomonad) for small cocompletion. I'm not sure whether it's directly relevantly for your purposes, as it may be that working with the presheaves explicitly is simpler, but it could be worth mentioning.
Yes, true. I found that working with presheaves gives a much more explicit characterization, but we probably should mention that strict 2-monad. (I think it an explicit picture for the strict monad may have to do with set-theoretical subtleties.)
- Given your conjecture about the algebras for , do you expect it to have the structure of a weak 3-monad, rather than just a pseudomonad?
I suspect it's a strict 3-monad, but maybe it's just me being scared of weak 3-monads :grimacing: (In my construction, Diag is strict.)