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Stream: theory: category theory

Topic: Iterated Plethories

Tim Campion (Mar 18 2023 at 17:30):

For $C$ a locally presentable category, let $Pleth(C)$ denote the category of left-adjoint monads on $C$ (dual to the category of righ-adjoint comonads). Then $Pleth(C)$ is again locally presentable. Here are a few examples:

• $Pleth(Set) = Mon$, the category of monoids

• $Pleth(Ab) = Ring$, the category of unital associative rings

• $Pleth(CRing) = Pleth$, the category of plethories (hence the name)

I don't really know a lot of other examples, but one thing that strikes me about the examples we do have here is that $Pleth = Pleth(CRing)$ is _almost_ $Pleth(Pleth(Ab))$, except for the commutativity that shows up.

Question: Do we have $Pleth(Pleth(Ab)) = Pleth$ anyway? That is, do we have $Pleth(Ring) = Pleth(CRing)$?

I think the answer is probably "no", but there's some possibility that the answer is actually "yes" -- this would be analogous to the fact that addition in a semiring is always commutative -- "if some monoid operation has another operation distributing over it, then that monoid operation is in fact commutative".

Simon Burton (Mar 20 2023 at 17:08):

This is reminding me of the more general setup of Mon(C), the category of monoid objects in a monoidal category. If C is symmetric monoidal then so is Mon(C), but if C is braided monoidal, then Mon(C) is only monoidal. And (is this folklore?) this categorifies; with Mon(C) taking away one dimension of commutativity, unless C is symmetric, which is the stable case. Eg. the bicategory of pseudomonoids in a sylleptic monoidal bicategory, will be braided.

John Baez (Mar 21 2023 at 00:21):

I think that last fact is "folklore". At least I don't remember seeing a proof.

John Baez (Mar 21 2023 at 00:22):

The iterated plethory idea is different, I believe... but it may follow some similar patterns.

John Baez (Mar 21 2023 at 00:25):

By the way, my paper with @Joe Moeller and @Todd Trimble studies 2-plethories, meaning 'categorified' plethories.

Todd Trimble (Mar 21 2023 at 10:56):

Tim Campion said:

For $C$ a locally presentable category, let $Pleth(C)$ denote the category of left-adjoint monads on $C$ (dual to the category of righ-adjoint comonads). Then $Pleth(C)$ is again locally presentable. Here are a few examples:

• $Pleth(Set) = Mon$, the category of monoids

• $Pleth(Ab) = Ring$, the category of unital associative rings

• $Pleth(CRing) = Pleth$, the category of plethories (hence the name)

I don't really know a lot of other examples, but one thing that strikes me about the examples we do have here is that $Pleth = Pleth(CRing)$ is _almost_ $Pleth(Pleth(Ab))$, except for the commutativity that shows up.

Question: Do we have $Pleth(Pleth(Ab)) = Pleth$ anyway? That is, do we have $Pleth(Ring) = Pleth(CRing)$?

I think the answer is probably "no", but there's some possibility that the answer is actually "yes" -- this would be analogous to the fact that addition in a semiring is always commutative -- "if some monoid operation has another operation distributing over it, then that monoid operation is in fact commutative".

James Borger was talking about this over here, and the answer is 'no'. This sort of thing is addressed in the book by Bergman and Hausknecht, Cogroups and Co-rings in the Categories of Associative Rings. I haven't looked at their arguments myself, but the upshot seems to be that there are precious few plethories on noncommutative rings, something like rings equipped with an endomorphism or with an anti-endomorphism are the only nontrivial examples.

One of the easier "classical" plethories to think about is commutative rings equipped with a derivation. So, while I haven't done it myself, it might pay to see why this same thought won't work for noncommutative rings.