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Nathanael Arkor (Sep 21 2020 at 21:28):Where is there a proof in the literature that there is a 2-monad for finite (co)limit completion? (As opposed to a pseudomonad.) I've seen assertions in several papers, but the strictness seems subtle, and it would be good to know where to find it explicitly.
(It would be even better if there was a reference for -(co)completion, for some suitable class of (co)limits .)
fosco (Sep 21 2020 at 21:42):A fairly explicit construction of the finite colimit completion of is given in "The symmetric topos", where Bunge-Carboni give a two-step recipe to build first the (co)product and then the (co)equaliser completion of . But maybe there are better references where the completion is performed in a single step. They seem to refer to Kock's original definition of KZ-monad.
This said, is the finite colimit completion KZ-monad really strictifiable to a 2-monad? I find this surprising: up to size issues, the cocompletion under all colimits of (so, the coequaliser and small-coproduct completion of ) is genuinely pseudo-. Please, tell us more!
Nathanael Arkor (Sep 21 2020 at 21:48):Right, the construction in Bunge–Carboni is pseudomonadic, though they note that several authors have pointed out that it can be made 2-monadic. The free cocompletion is actually the reason I ask: I want to understand whether the arguments for finite colimits apply for small colimits, or where the obstruction is, if not.
Nathanael Arkor (Sep 21 2020 at 21:50):An example of a source that mentions that finite limits are 2-monadic without proof is Two-dimensional monad theory.
John Baez (Sep 22 2020 at 04:16):I was just asking Todd Trimble this question! I would like to know the answer!
fosco (Sep 22 2020 at 08:38):I see; very interesting question indeed. Let me know!
This comment is a shot in the dark, because I probably remember too little about it, but try have a look at @Paolo Perrone MIT talk https://www.youtube.com/watch?v=wTjdEzFGuOg
There's a 2-monad on Cat whose algebras are cocomplete categories with a choice of colimits: if this 2-monad is strict, it is believable this works as a strictification of the presheaf construction. But probably (as always) this messes up your algebra morphisms: now cocontinuous functors are "pointed" in that they have to preserve the choice of colimits you made.
Paolo Perrone (Sep 22 2020 at 15:01):In my talk, (see minute 8:30), I actually use pseudomonads. While I don't delve into higher coherence issues in the talk, they are there (you'll see in the paper, in preparation).
Afaik, I don't know how to "strictify" those monads, and I'm not even sure it's possible. I'd be happy to see a reference where it's explicitly done.
Dan Doel (Sep 22 2020 at 15:04):Do the papers definitely say strict, and it's not some kind of situation where people used to say "2-monad" to (sometimes) mean the same thing as what is now called "pseudomonad"?
John Baez (Sep 22 2020 at 16:07):I think people like Hyland and Power sometimes use 2-monads for categories with certain specified choices of colimits along with 'pseudomorphisms' between their algebras, to avoid the problem Fosco mentions.
John Baez (Sep 22 2020 at 16:08):A simpler example is this: I think there's a 2-monad on Cat whose algebras are strict monoidal categories and whose morphisms are strict monoidal functors... but whose pseudomorphisms are strong monoidal functors. So the pseudomorphisms are useful.
Martti Karvonen (Sep 22 2020 at 18:37):John Baez said:
A simpler example is this: I think there's a 2-monad on Cat whose algebras are strict monoidal categories and whose morphisms are strict monoidal functors... but whose pseudomorphisms are strong monoidal functors. So the pseudomorphisms are useful.
Yes, and the pseudoalgebras are unbiased monoidal categories rather than the usual monoidal categories (which afaik are the strict algebras of an altogether different monad), so at times strict algebras + pseudomorphisms is more convenient than going fully strict/fully pseudo.
Nathanael Arkor (Sep 22 2020 at 19:34):Dan Doel said:
Do the papers definitely say strict, and it's not some kind of situation where people used to say "2-monad" to (sometimes) mean the same thing as what is now called "pseudomonad"?
Yes, they're definitely Cat-enriched monads. For example, in Two-dimensional monad theory, they explicitly say they're leaving the pseudo setting to a later paper.
Nathanael Arkor (Sep 22 2020 at 19:36):John Baez said:
I think people like Hyland and Power sometimes use 2-monads for categories with certain specified choices of colimits along with 'pseudomorphisms' between their algebras, to avoid the problem Fosco mentions.
This is the case here: Lex is a category of algebras and pseudomorphisms for a 2-monad on Cat, for instance.
Nathanael Arkor (Sep 22 2020 at 19:59):(Although the 2-monad is finitary, so we can take the flexible replacement.)
John Baez (Sep 22 2020 at 20:06):Where is "here"? Two-Dimensional Monad Theory?
Nathanael Arkor (Sep 22 2020 at 20:08):"Here" as in "for the 2-monad for categories with finite limits", though it appears as an example in Two-dimensional monad theory, yes.
John Baez (Sep 22 2020 at 20:12):I suddenly realized I'd like to use this version of Lex, or actually Rex, in a paper of mine.
Reid Barton (Sep 22 2020 at 20:15):Is the 2-monad for categories with finite colimits called T-Rex?
John Baez (Sep 23 2020 at 00:01):That reminds me of my own bad joke: "What's the opposite of Tyrannosaurus rex?"
Nathanael Arkor (Sep 23 2020 at 00:04):A T-Lex? :big_smile:
John Baez (Sep 23 2020 at 00:17):Right! Tyrannosaurus lex.
Nathanael Arkor (Sep 23 2020 at 17:13):John Bourke gave references for this question. Kelly–Lack's paper On the monadicity of categories with chosen colimits establishes this result in a very general setting (including for small cocompletion).