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Stream: theory: category theory

Topic: Relative monads

Bruno Gavranović (Nov 30 2022 at 12:04):

If a monad can be viewed as a lax functor $1 \to \mathbf{Cat}$, what kind of analogous construction can a relative monad be viewed as?

Nathanael Arkor (Nov 30 2022 at 14:40):

I can at least explain why the answer is not obvious. Giving a lax functor $1 \to \mathcal K$ is equivalent to giving an object $A \in \mathcal K$ together with a lax monoidal functor $1 \to \mathcal K[A, A]$, hence a monoid in $\mathcal K[A, A]$, which is precisely a monad on $A$ in $\mathcal K$.

We can try to go in the opposite direction for relative monads. Assuming the existence of enough pointwise left extensions, $(j \colon A \to E)$-relative monads can be presented as monoids in skew-monoidal categories $[A, E]_j$ (I've made the unit $j$ explicit to reduce ambiguity). These are equivalent to lax skew-monoidal functors $1 \to [A, E]_j$. Notice to generalise from lax monoidal functors to lax functors, in the case of monads, we needed to specify the data of an object in a 2-category. To generalise from a lax skew-monoidal functor to some functor-like structure $F \colon 1 \to \mathcal K$, we therefore need $F$ to specify not an object $A$, but rather a 1-cell $j \colon A \to E$. However, it is not clear what such a thing should look like in general, particularly if $A$ is entirely unrelated to $E$ (in an informal sense). Notice that figuring out what $F$ should look like requires us first to work out what kind of structure $\mathcal K$ should be in this situation, which is also not obvious.

If we have a coherent choice of $j_A \colon A \to E_A$ for each $A$, then there is one potential candidate: we may take $\mathcal K$ to be a skew-bicategory in the sense of Lack and Street and $F$ to be a "lax skew-functor". In particular, this generalises the "monads as lax functors" perspective, taking $j_A = 1_A$. However, it does not permit us to capture arbitrary relative monads (e.g. two relative monads with domain $A$ but different codomains).

Nathanael Arkor (Nov 30 2022 at 14:44):

(The story also becomes more complicated when we don't have enough pointwise left extensions, for which skew-monoidal categories do not suffice. For this, one needs to move to some kind of multicategorical structure: cf. my Octoberfest slides.)

Bruno Gavranović (Dec 02 2022 at 19:21):

Thanks @Nathanael Arkor , this is really useful.
Indeed, something funny is going on here. I don't have much to say other than the way you broke down the problem resonates with me.

I don't understand the details of the lax-skew functor yet, but it is strange that it doesn't capture arbitrary relative monads :thinking: