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Stream: theory: category theory

Topic: "Inverse" of free

Ralph Sarkis (Jun 11 2021 at 20:13):

The free monad on a functor $F$ is $M$ such that $\mathrm{EM}(M) \cong \mathrm{Alg}(F)$. Given a monad $M$, what would you call an endofunctor $F$ such that $M$ is the free monad on $F$?

John Baez (Jun 12 2021 at 01:46):

I don't know. You probably know this already, but since you said "the free monad on a functor $F$ is...", and amateurs might read too much into that, I feel like adding: the free monad on a functor may or may not exist.

Nathanael Arkor (Jun 12 2021 at 09:52):

Also, a free monad doesn't necessarily satisfy the property that its category of algebras coincides with the category of algebras for the original endofunctor: this is the stronger property of being algebraically-free.

Ralph Sarkis (Jun 12 2021 at 10:52):

Oh, I thought it was the other way around (although it was clearly mentioned in the nLab). Thanks

Mike Shulman (Jun 12 2021 at 14:32):

As for the original question, one could call it a "generator" or "basis" for $M$.

Ralph Sarkis (Jun 12 2021 at 15:00):

Thanks :blush:

Eigil Rischel (Jun 22 2021 at 09:46):

I guess "basis" has the advantage of weakly connoting the fact that there are no "equations", i.e that $M$ is really free on it, and not just generated by it

dusko (Jun 24 2021 at 20:55):

there is this strange phenomenon that ross street's "formal theory of monads" developed the general theory, and then there is a sequence of later papers spelling out what might be viewed as special cases and direct consequences. i guess it is because ross's original paper is a lot of work to read. but it is perhaps not good that we are speaking about "the" category of monads. if street's paper is completely forgotten, someone someday will have to work it all out again. BTW, it is much easier to read in string diagrams. someone in need of practice in generating lots of string diagrams fast could do a favor to us all by translating the crucial sections of that paper from the command-line algebra into pictures.

incidentally, should the category theory community not endorse "mathematics of bad drawings" and promote an easy speech format of our natural discourse medium of diagrams?

Joe Moeller (Jun 24 2021 at 21:04):

Ya know, I'm constantly translating papers I'm reading into string diagrams. What should be done with these? It's not original research. Blog posts? Maybe.

Cole Comfort (Jun 24 2021 at 21:17):

Joe Moeller said:

Ya know, I'm constantly translating papers I'm reading into string diagrams. What should be done with these? It's not original research. Blog posts? Maybe.

Is it gauche to put a string diagrammatic treatment of someone elses research on the arxiv if credit is given? Blog posts are more temporary.

Joe Moeller (Jun 24 2021 at 21:20):

Right, this is my issue with blog posts. I remember once telling someone some of their proofs in their new paper could've been done with strings, and they seemed slightly insulted. I'm not sure.

Mike Shulman (Jun 24 2021 at 21:21):

What about TAC Expositions?

Joe Moeller (Jun 24 2021 at 21:23):

At the moment I'm writing up something about some diagrams that I think are useful, but I'm not proving anything new. I was thinking I'd put it on the arxiv but not attempt to publish. Does this make sense to do?

Joe Moeller (Jun 24 2021 at 21:24):

Mike Shulman said:

What about TAC Expositions?

That does seem like a good idea. There's nothing there yet, so I can't tell if "redone with strings" is sufficiently novel of a presentation, but I guess if you're suggesting it, it could be.

Mike Shulman (Jun 24 2021 at 22:15):

Well, we'll never discover what counts as "sufficiently novel" for TAC Expositions unless we submit stuff there and find out. And I expect "we" (meaning the people who get asked to referee for it) will play a large part in determining what counts. (-:

Mike Shulman (Jun 24 2021 at 22:16):

I'm not necessarily saying I think "redone with strings" would be sufficiently novel, but it's certainly not obvious to me that it wouldn't be.

Mike Shulman (Jun 24 2021 at 22:17):

Especially if the exposition were clear and helpful, perhaps adding some context and motivation to the original paper(s), and the examples were well-chosen to display the usefulness of the notation. I.e. it was a good expository paper.

Mike Shulman (Jun 24 2021 at 22:18):

In any case, I think this sort of thing is certainly worth arXiving, if nothing else.

dusko (Jun 25 2021 at 02:18):

we should find a way. e.g., if anyone is interested to take a bite from a section or two of the "formal theory of monads", we could ask ross street whether TAC Expositions is the right venue. he was with TAC from day one.

dusko (Jun 25 2021 at 02:32):

((and he knows that the diagrams can make a difference between communicable and not. eg he knows that i had been sitting on the nucleus and the street monad construction for 4 years, trying to latex the string diagrams for a readable 2-categorical version, before i realized that the iso-2-cells version has all of the same theorems and gave up on the 2-cells because latexing the string diagrams would take more time than proving the results... i think there will be more and more such situations. thoughts?))

Ralph Sarkis (Jun 25 2021 at 02:46):

I am extremely down!

dusko (Jun 25 2021 at 02:56):

Ralph Sarkis said:

The free monad on a functor $F$ is $M$ such that $\mathrm{EM}(M) \cong \mathrm{Alg}(F)$. Given a monad $M$, what would you call an endofunctor $F$ such that $M$ is the free monad on $F$?

what examples do you have in mind? the posetal case of this question would be: what monotone endomorphisms $F$ generate closure operators $M$ whose fixpoints are the subfixpoints of $F$. kelly's paper shows that there is a brutal force answer to such questions. but it is like asking what are the free ordered topological groups over discrete sets. category theory arose from the fact that in real life, such things are ordered objects in the category of topological groups, which are the group objects in the category of topological spaces. monads naturally arise from adjunctions, not directly from endofunctors. and adjunctions naturally arise from profunctors, not from functors. so there is a mismatch.

if the goal is to generate the monad $M$ as the algebraic theory over a signature of operations presented by a functor $F$ (i extrapolate this from your earlier sketch of quotient monads; this $M$ would then be a quotient of the equation-free monad $\overline{F}$... sorry if this extrapolation is wrong; but IF that is what you are trying to do) --- THEN the $F$ needs to be restricted to polynomial functors. i think that is presented in detail in manes' book "algebraic theories". it mostly goes back to frend linton's thesis and eilenberg.

dusko (Jun 25 2021 at 03:00):

Ralph Sarkis said:

I am extremely down!

i'll ask.

Ralph Sarkis (Jun 25 2021 at 03:31):

The two questions are not related at the abstract level (at least in my mind and for now). I will come back to explain where this question comes from when I am done fighting with arXiv to compile my preprint.

Teaser: we ended up calling our construction the semifree monad.

Morgan Rogers (he/him) (Jun 25 2021 at 08:03):

Mike Shulman said:

Well, we'll never discover what counts as "sufficiently novel" for TAC Expositions unless we submit stuff there and find out. And I expect "we" (meaning the people who get asked to referee for it) will play a large part in determining what counts. (-:

I submitted a paper to TAC expositions and was told it contained too much original material, so I think it's a safe thing to try!

Ralph Sarkis (Jun 25 2021 at 14:59):

Ralph Sarkis said:

I will come back to explain where this question comes from when I am done fighting with arXiv to compile my preprint.

The preprint is on HAL.

Ralph Sarkis (Jun 25 2021 at 14:59):

The relevant results for this thread are Theorem 3.3 and 3.4. We show that for any monad $M$ on a category with coproducts, there is a monad $M^{\mathrm{s}}$ such that the category of semialgebras for $M$ (EM algebras but we drop the unit axiom) is isomorphic to the category of algebras for $M^{\mathrm{s}}$.

We call $M^{\mathrm{s}}$ the semifree monad on $M$.

Note that when we construct the algebraically free monad on an endofunctor, we add two axioms to the algebras. In our case, we remove one axiom, that is why I wanted some terminology for the "inverse" of algebraically free. Since there was no widely used term, we stuck with our original idea of semifree monad.