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: Monads whose algebras are / give rise to monads

Patrick Nicodemus (Nov 21 2022 at 00:46):

There is a free group monad, and an algebra of the free group monad is a group. A group G defines a monad $A \mapsto G \times A$. An algebra of this monad is a G-set, or a group equipped with a G-action.

Similarly we have that rings are algebras of a certain monad on Abelian groups ( the free tensor algebra monad) and a ring R itself also defines a monad on Abelian groups whose algebras are the R-modules.

I am interested in the general theory of these structures where one has an algebraic class of structures which can have their own modules/algebras. What is this situation called? Where can I read about it?

Mike Shulman (Nov 21 2022 at 05:00):

Note that there's a "universal" case: there is a free monad monad, whose algebras are (finitary) monads, which of course have their own algebras.

Patrick Nicodemus (Nov 21 2022 at 15:28):

Thank you for the paper reference.

John Baez (Nov 21 2022 at 20:05):

In general, monoid objects in monoidal categories are the kind of thing that that have 'modules' or 'algebras', also called 'actions'.

A monad on a category C is a monoid in the monoidal category of endofunctors on C. A ring is a monoid in the monoidal category of abelian groups. An operad is a monoid in a the monoidal category of species. Etcetera.

Mike Shulman (Nov 21 2022 at 21:28):

So I guess one general context would be when you have a monad $T$ on a monoidal category $C$ and a map of monads $\mathrm{fm} \to T$, where $\mathrm{fm}$ is the free monoid monad on $C$. Then any $T$-algebra has an underlying monoid, which therefore acts on $C$ and induces a new monad.

Mike Shulman (Nov 21 2022 at 21:29):

So for instance $T$ might be the free group monad on $\rm Set$, for your first example.

Tobias Fritz (Nov 21 2022 at 21:37):

There's also a general notion of [[module over an algebra over an operad]]. I don't know how this works in detail and there isn't much on the nLab page either, but presumably the paper linked there contains more details.

James Deikun (Feb 08 2023 at 23:15):

Example 4.3.14 in Tom Leinster's "Higher Operads, Higher Categories" references that slices of the category of algebras of a monad $T$ are monadic over the corresponding slices of the underlying category, so if you allow the monad to be on a different-but-related category this happens for any monad.

As far as the [[module over an algebra over an operad]] idea goes, it seems like you could generalize it to algebras of any [[polynomial monad]] where the base category allows a good notion of total differential of polynomial functors.