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: learning: questions

Topic: ✔ The category of monads?

Anand Rao (May 12 2023 at 05:34):

For a given category $\mathcal{C}$, I was wondering if it is possible to construct a corresponding "category of monads" on $\mathcal{C}$. The objects of the category are monads on $\mathcal{C}$; given that a monad is a (endo-)functor with additional structure, it makes sense for the morphisms of the category to be natural transformations between the underlying functors with some additional properties. However, a naive approach along these lines presents some problems.

Let $T$ and $S$ be monads on $\mathcal{C}$, and let $\eta$ be a natural transformation from $T$ to $S$. Then the "unit" of the monad $T$, which is a natural transformation from $\mathbb{1}_{\mathcal{C}}$ to $T$, can be post-composed with $\eta$ to give a natural transformation from $\mathbb{1}_{\mathcal{C}}$ to $S$; one can impose the condition that this natural transformation must be equal to the "unit" of the monad $S$. This kind of approach seems to fail for the "multiplication" of the monad. The difficulty lies in changing $T^2$ to $S^2$ using $\eta$ -- one round of application of $\eta$ transforms $T^2$ to $ST$, and it does not seem to be possible to simplify this further to $S^2$.

Is it possible to construct a sensible definition of the category of monads on $\mathcal{C}$ along these lines?

Nathanael Arkor (May 12 2023 at 06:11):

The category of monads on a category $\mathcal C$ is the category of monoids in the strict monoidal category $[\mathcal C, \mathcal C]$. If you expand the definitions, you should get the description you are looking for.

Dylan Braithwaite (May 12 2023 at 07:17):

Namely the extra step you need is that using the functoriality of $S$ you get a natural transformation $S(\eta) : ST \Rightarrow SS$

Anand Rao (May 13 2023 at 03:07):

Ah, I see. Thank you.

Notification Bot (May 13 2023 at 03:08):

Anand Rao has marked this topic as resolved.