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Stream: theory: category theory

Topic: Categories of Algebras

Joshua Meyers (Feb 10 2021 at 03:21):

If $C$ is a category and $F$ is an endofunctor, we can form the category $F\textbf{-Alg}$ of algebras of $F$.

But it turns out that the mapping $\Psi:(C,F)\mapsto F\textbf{-Coalg}$ is functorial!

The domain of $\Psi$ is the category of endofunctors. Objects are endofunctors $F:C\to C$ and a morphism from $F:C\to C$ to $G:D\to D$ consists of

• A functor $P:C\to D$
• A natural transformation $p: GP\to PF$

So this is the oplax arrow category $2\textbf{Cat}_\textbf{lax}(\mathbb{N},\textbf{Cat})$, where $\mathbb{N}$ is the monoid $(\mathbb{N}, +,0)$ considered as a one-object category. Cf. https://ncatlab.org/nlab/show/lax+natural+transformation

The codomain of $\Psi$ is $\textbf{Cat}$. The morphism $(P,p)$ maps to the functor $F\textbf{-Alg}\to F\textbf{-Alg}$ which sends the algebra $(c, x:Fc\to c)$ to the algebra $(Pc,G(Pc)\xrightarrow{p_c} P(Fc) \xrightarrow{Px} Pc)$.

Anybody heard of this before, or know what it is a special case of or something?

Joshua Meyers (Feb 10 2021 at 03:28):

Relevant earlier discussion https://categorytheory.zulipchat.com/#narrow/stream/229136-theory.3A-category.20theory/topic/functors.20between.20categories.20of.20F-algebras/near/218334677

Joshua Meyers (Feb 10 2021 at 03:38):

Also BTW if you take oplax natural transformations instead it then it works for coalgebras

Tom Hirschowitz (Feb 10 2021 at 07:53):

It is certainly well-known. E.g., IIRC, if you unfold Theorem 6 of "the formal theory of monads", it relates monads on a cat to monadic functors over it. There might be more elementary references...

Joshua Meyers (Feb 10 2021 at 12:19):

@Tom Hirschowitz Can you explain the relevance of that to this situation? Not sure where the monads or monadic functors are here...

Nathanael Arkor (Feb 10 2021 at 12:24):

The analagous construction for (co)monad (co)algebras is in The formal theory of monads, though I don't think that the construction for arbitrary endofunctor (co)algebras follows directly. If you look at the 2-category of monads that Street defines, you'll see it matches your category of endofunctors.