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: reading & references

Topic: Reference for a certain class of monads

fosco (Jan 01 2025 at 11:53):

Regard the monoid of natural numbers as a bicategory with a single object, and only identity 2-cells.

Then, for every monoidal category $V$, you can form the bicategory of pseudofunctors $\mathbb N \to V$, where an object is an object $A$ of $V$, a 1-cell $(X,\phi)$ is a morphism $\phi : A\otimes X \to X\otimes B$ in $V$, and a 2-cell $\alpha : (X,\phi) \to (Y,\psi)$ is a certain morphism in $V$, $\alpha : X\to Y$, plus the obvious compatibility with $\phi,\psi$.

What is a monad in this bicategory?

It's an "intertwiner" $(X,m : A\otimes X\to X\otimes A)$, plus monad axioms to the effect that

• $X$ is a monoid, internal to $V$, call $\mu$ its multiplication;
• $m$ and $\mu$ interact as follows:
  image.png
  

Where else is this notion studied, and what keywords might help in understanding questions like

• what's a more hands-on description of the monad structure (I look for a characterization in terms of $V$ alone: for example, I suspect one can prove that $(E,\mu)$ is acting on $A$ in some sense);
• what's a description of the category of algebras for a monad? (ditto as above: I can, and to some extent know how, to spell it out, but I don't know of a characterization in terms of $V$ alone)