Category Theory
Zulip Server
Archive

You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.

Stream: theory: category theory

Topic: "Relatively free" algebra structure

Amar Hadzihasanovic (May 25 2020 at 12:30):

I'm interested in the following situation and couldn't find anything by googling, I hope someone knows a reference :)
Consider a category $C$ with two monads $T, S$ on it (and no kind of compatibility -- no distributive law, etc). There's a category $\{T,S\}\mathrm{Alg}$ of objects with independent $T$-algebra and $S$-algebra structures, and morphisms that preserve both. There are forgetful functors $U_T: \{T,S\}\mathrm{Alg} \to T\mathrm{Alg}$ and $U_S: \{T,S\}\mathrm{Alg} \to S\mathrm{Alg}$ discarding one of the two structures.
I am interested in conditions for, say, $U_T$ to have a left adjoint -- the idea being that one can universally add an $S$-algebra structure on the carrier of a $T$-algebra, which does not need to be compatible with said $T$-algebra, but does not “discard” it either...

Reid Barton (May 25 2020 at 13:03):

Is it obvious that $\{T,S\}\mathrm{Alg}$ has an initial object (assuming that $C$ does, but not that $T$ and $S$ preserve it)?
I think you probably want some local presentability/accessibility hypotheses; not sure exactly how much you need.

Reid Barton (May 25 2020 at 13:13):

I guess this will also be related to coproducts of monads.

Amar Hadzihasanovic (May 25 2020 at 13:27):

Reid Barton said:

I guess this will also be related to coproducts of monads.

Ah yes, it looks like “indipendent $T$ and $S$-algebra structures” are the same as “algebra structures for the coproduct of $T$ and $S$”! I had never heard of the coproduct of monads. Great, thanks, that gives me somewhere to start from.

Soichiro Fujii (May 27 2020 at 09:02):

There is a general existence theorem for colimits of monads in Section 27 of Kelly’s A unified treatment of transfinite constructions for free algebras, free monoids, colimits, associated sheaves, and so on, though I find this paper hard to read.

Amar Hadzihasanovic (May 27 2020 at 10:20):

Thanks Soichiro, I had indeed found that paper by Kelly and yes, it seems that noone has been able to make much of that section, until computer scientists became interested in computing some simpler cases in this century. Since then there's been a paper by Hyland, Plotkin and Power in 2002 on coproducts of free monads, one by Ghani and Uustalu in 2004 with a simple formula for the coproduct of ideal monads (which 'separate' as T = Id + T', with T' being the 'non-unit' part), and then a few generalisations of this formula, the most comprehensive being this preprint by Adámek.

The basic idea seems to be the following: if the “image” of the unit of a monad can be 'separated' from the rest of it, so that TX decomposes as “X plus things with a non-trivial outer T-layer” (possibly in weaker ways than the case of ideal monads), and the same is true of S, then (T+S)X is going to be made of X, plus “things with (nontrivial) T-layers over S-layers over T-layers, etc”, plus “things with (nontrivial) S-layers over T-layers over S-layers, etc”.

Amar Hadzihasanovic (May 27 2020 at 10:24):

And this can be proved to exist by the usual fixpoint theorems, provided that you can give the “nontrivial part of T/ nontrivial part of S” enough structure (except for the case of ideal monads, they will not be endofunctors on the same category as T and S).

Amar Hadzihasanovic (May 27 2020 at 10:28):

Anyway, to return to my original question, if $S+T$ exists, because we have coproduct inclusions $S, T \to S+T$ in the category of monads on $C$, we can apply the general theorems on the existence of left adjoints to the 'relative forgetful functors' $N\mathrm{Alg} \to M\mathrm{Alg}$ corresponding to monad morphisms $M \to N$.

Mike Shulman (May 28 2020 at 17:26):

@Amar Hadzihasanovic Has anyone written down more explicit formulas for pushouts of monads?

Amar Hadzihasanovic (May 28 2020 at 19:16):

Not that I've seen, I'm afraid. Adámek mentions that absolute coequalisers of monads are created by the forgetful functor to the category of endofunctors, which perhaps makes some pushouts more computable in combination with the coproduct formulas. But none of the articles I've seen treat other cases explicitly.