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Stream: theory: category theory

Topic: pseudoalgebras of pseudomonads

Matteo Capucci (he/him) (Dec 11 2021 at 22:53):

In 'Two-dimensional monad theory', Kelly, Power and Blackwell write:
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Matteo Capucci (he/him) (Dec 11 2021 at 22:53):

I can't find the 'later article' they refer to, do you know where I can find the result about equivalence of pseudo-T-algebras and strict T'-algebras?

Zhen Lin Low (Dec 11 2021 at 23:04):

I remember hearing about this result too, from Mike Shulman. I'm guessing the proof uses a kind of 2-monadicity theorem. Maybe it's in the same article (and they forgot to update the introduction).

Mike Shulman (Dec 11 2021 at 23:33):

This result follows from the construction of the pseudomorphism classifier for algebras of a 2-monad, constructed in BKP, and from the fact that (finitary/accessible) 2-monads on a (lfp/locally presentable) 2-category are themselves the algebras of a different 2-monad, which was proven by Lack in On the monadicity of finitary monads, together with the observation that a pseudo $T$-algebra structure on $A$ is the same as a pseudomorphism of monads $T\to {\rm End}(A)$.

Mike Shulman (Dec 11 2021 at 23:35):

I believe BKP were referring to their intent to prove the monadicity of finitary monads themselves, but ran into difficulties and were only able to prove it was of "descent type" -- this is the Kelly-Power paper Adjunctions whose counits are coequalizers, and presentations of finitary enriched monads.

Mike Shulman (Dec 11 2021 at 23:36):

(That description was a bit telegraphic; let me know if you want a longer one.)

Mike Shulman (Dec 11 2021 at 23:36):

I learned this from Steve Lack's lectures at Chicago that eventually became his paper A 2-categories companion, but I'm not sure if its stated explicitly in there -- I couldn't find it right now.

Matteo Capucci (he/him) (Dec 11 2021 at 23:38):

Thanks a lot for the references

Matteo Capucci (he/him) (Dec 11 2021 at 23:38):

Mike Shulman said:

a pseudo $T$-algebra structure on $A$ is the same as a pseudomorphism of monads $T\to {\rm End}(A)$.

Mike, could you expand on this?

Matteo Capucci (he/him) (Dec 11 2021 at 23:41):

Anyway, what I'm mostly interested in is recycling the results in the aforementioned BKP paper to talk about pseudoalgebras of a pseudomonad. In particular I'd like to state non-strict defs but prove things as if they were strict. I know explicitly how my pseudomonad strictifies to a corresponding 2-monad, and I believe the pseudoalgebras of this 2-monad turn out to actually be all strict.

Matteo Capucci (he/him) (Dec 11 2021 at 23:45):

I guess I can be a little less mysterious: the pseudomonad I'm interested in is $\mathcal M \times - : Cat \to Cat$ for $\mathcal M$ non-strict monoidal (a bit more generally, replace it with pseudomonoid in a monoidal 2-category). This of course strictifies to a 2-monad by MacLane's coherence theorem. Then a pseudoalgebra for the first strictifies uniquely to a strict algebra of the second, since pseudoalgebras for the first are equivalently strong monoidal functors $\mathcal M \to [A,A]$ which are then mapped to strict monoidal functors $\mathcal M^{st} \to [A,A]$ bc $[A,A]$ is always strict.

Mike Shulman (Dec 12 2021 at 00:04):

You've basically sketched the way the general case goes! Just replace "monoidal category" by "2-monad" and everything else is the same...

Matteo Capucci (he/him) (Dec 12 2021 at 12:36):

:thinking: Oh, I see... But what is a 'pseudomorphism of monads $T \to End(A)$'? I don't see how $End(A)$ is a monad

Matteo Capucci (he/him) (Dec 12 2021 at 12:40):

Also, a bit unrelated question: I seem to only find references for pseudomonads in Gray categories. Is it just a more general version of pseudomonads on 2-categories? I'd say ye bc, apparently, if $\mathbb A$ is Gray and $K:\mathbb A$, then $\mathbb A(K,K)$ is a 2-category. But then I'm not sure composition 'works' in the same way. Gray categories have composition supported by 'cubical functors', I am a bit lost in this.
For reference, I'm looking at Marmolejo's 'Distributive laws for pseudomonads'.

Zhen Lin Low (Dec 12 2021 at 13:43):

Matteo Capucci (he/him) said:

:thinking: Oh, I see... But what is a 'pseudomorphism of monads $T \to End(A)$'? I don't see how $End(A)$ is a monad

In ordinary monad theory there is a gadget called the endomorphism monad, which has the analogous universal property. The name comes from the fact that the the underlying endofunctor is given by $X \mapsto A^{\textrm{Hom} (X, A)}$, which if you squint and set $X = 1$ looks a bit like $A^A$.

Matteo Capucci (he/him) (Dec 14 2021 at 10:39):

Oooh, that's neat