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Stream: theory: category theory

Topic: Enriching a monad in a monad

Noah Chrein (Apr 18 2024 at 19:23):

I am looking for a yoneda lemma in an odd situation. I will abstract some details away so they aren't too distracting.

Let $(\mathbb C,M)$ and $(\mathbb D,N)$ be monads, that is, 2-functors $M,N:[\Delta_+, \mathbb C\text{at}]$ .
$[\Delta_+,\mathbb C\text{at}]$, as a category, seems to be cartesian closed: there is a monad
$(\;[\Delta_+,\mathbb C\text{at}](M,N)\;,\;+1\;)$ given by $N\circ F \cong +1(F)\cong F\circ M$.
I'm looking for a yoneda structure like $M \overset{y}\to [\Delta_+,\mathbb C \text{at}](M^\text{op}, N)$
so that $[\Delta_+,\mathbb C \text{at}](M^\text{op},N)(\;y(m)\;,\;F\;)\cong F(m)$

this seems to imply M would need to be "enriched" in N, is there a standard notion of a monad enriched in another monad?

As a riff:
I've been playing around with some toy ways of doing this, like enriching $(\mathbb C, M)$ in $(\mathbb D, N)$ by enriching $\mathbb C$ in $\mathbb D$ and having M and N respect a number of things, but this favors the categories over the monads and it feels like there should be a way to enrich M in N without even mentioning the underlying categories. Perhaps this has less to do with monads and more to do with 2-functors in general.

Morgan Rogers (he/him) (Apr 19 2024 at 09:10):

I would like to understand the details of how this construction is adjoint to the cartesian product (out of curiosity).
The product of monads is what you would expect (product of underlying categories with respective monads applied component-wise). Suppose we are given a morphism of monads $F:(\mathbb{C},M) \times (\mathbb{D},N) \to (\mathbb{E},P)$, so we have a family of transformations $\lambda_{c,d}: P(F(c,d)) \to F(M(c),N(d))$ natural in $c$ and $d$ and compatible with the unit and multiplications. If we naively curry the underlying functor, we get $F': c \mapsto F(c,-)$. To complete this to the data of a morphism of monads $N \to P$, we need transformations $P(F(c,d)) \to F(c,N(d))$ for each fixed $c$, natural in $d$... If I have the variance right on the natural transformations then it's not obvious what that should be.
I could fix that by taking a skewed version, namely $F': c \mapsto F(M(c),-)$, but then there's more I need to check vis-a-vis compatibility.
And if that works we have to check that this gives a monad morphism $M \to [\Delta_+,\mathbb{C}\mathrm{at}](N,P)$ !! Is there a slick way to see that this all works out?

Morgan Rogers (he/him) (Apr 19 2024 at 09:17):

I also notice that $M^{\mathrm{op}}$ doesn't make sense to me because the composition monoidal structure on a category is not symmetric: we have no canonical "switching" transformation $M \circ M \Rightarrow M \circ M$. If you meant the structure on $\mathbb{C}^\mathrm{op}$, that ends up being a comonad!

Noah Chrein (Apr 19 2024 at 10:34):

I have not fully checked the cartesian closure, I used the word "seems", but I should be more careful. Are you using lambda as in the lax $P\circ F \overset \lambda \to F\circ (M\times N)$? I was thinking more strictly, i.e. that $\lambda$ was an iso, which may yield the $P\circ F \to F\circ (1\times N)$ by just composing with the unit $1 \to M$. I have to teach today so I will return to this later, but thank you for the analysis.

You're right about $M^\text{op}$ being a comonad, that was silly of me. But I still think there is a way to enrich a monad in another. Is there a notion of profunctor between monads $M \rlap{+}{\to} N$ if there is no "opposite" monad? Perhaps instead of enriched (profs) I should consider internal (spans) and ask for "a monad internal to a monad". I'll work on this a bit

Morgan Rogers (he/him) (Apr 20 2024 at 14:12):

The strict version of morphism either yields a lax version on the transpose by composing with the unit, or a strict morphism with the skew version of the transpose (although again I haven't checked the other compatibility conditions)