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Stream: deprecated: mathematics

Topic: Coalgebras for non-operadic monads

Alexander Gietelink Oldenziel (Dec 20 2020 at 17:09):

Given an operad $O(n)$ in a sym. mon. closed category $C$ we can study their algebras, given by operad morphisms $O(n) \to End(X)(n)$ for $$X\in $C$$. Here $End(X)(n)$ is the operad $Hom(X^n, X)$.
A fundamental theorem states that $O(n)$ has an associated monad $M$ such that the algebras coincide: $Alg(M)=Alg(O(n))$.

One apparent benefit of looking at operads instead of the induced monad is that we can also study coalgebras over the operad $O(n)$.
These are given by operad morphisms $O(n) \to CoEnd(X)(n)$ given by $Hom(X, X^n)$.
[do not confuse the coalgebras of $O(n)$ with the coalgebras of the opposite cooperad in the opposite category $O(n)^{op} \in C^{op}$. These are different!]

My question is whether there is a sensible notion of coalgebra for (more) general non-operadic monads $M$?

[My monad $M$ is probably monoidal, and also has algebra maps $m:M(X) \otimes M(X) \to M(X), e: I \to M(X)$. ]

[Certainly, we can look at the underlying endofunctor of $M$ and consider coalgebras for that functor. However, I don't think this recovers the right notion of coalgebra. Indeed, for the symmetric algebra monad $Sym$ on $Vect_k$ this gives coalgebras which don't neccesarily satisfy any of the coalgebra laws like coassociativity]

John Baez (Dec 20 2020 at 18:06):

Alexander Gietelink Oldenziel said:

Given an operad $O(n)$ in a sym. mon. closed category $C$ we can study their algebras, given by operad morphisms $O(n) \to End(X)(n)$ for $X\in C$. Here $End(X)(n)$ is the operad $Hom(X^n, X)$.

A fundamental theorem states that $O(n)$ has an associated monad $M$ such that the algebras coincide: $Alg(M)=Alg(O(n))$.

Don't you need some extra condition, like $C$ being cocomplete, for this to work?

Say $C = \mathsf{FinSet}$ is the category of finite sets with its $\times$ monoidal structure. Then I think there's an operad $O$ whose algebras are monoids in $\mathsf{FinSet}$ - that is, finite monoids. But I don't think there's a "free finite monoid" on a finite set, so I don't think there's a monad $M : \mathsf{FinSet} \to \mathsf{FinSet}$ whose algebras are finite monoids.

By the way, I'd really like to know a reference for this "fundamental theorem".

John Baez (Dec 20 2020 at 18:14):

I know Todd Trimble has proved something similar:

• Todd Trimble, Multisorted Lawvere theories.

See Theorem 1.3. I needed this result in my work... that's why I'm interested in references!