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Stream: learning: questions

Topic: bicrossed pairs are monads, so distributive laws are...

fosco (Nov 08 2025 at 10:43):

I’ve recently gone back to working on this paper of mine, which appeared on arXiv not long ago. There, I prove (what I think is )a(n interesting) fact: if we look at Mealy automata and organize them as the loose cells of a double category, then a monad corresponds to an object known to algebraists as a bicrossed pair.

A bicrossed pair consists of two monoids $M$ and $N$, each acting on the other, subject to a certain compatibility condition. The interesting part is that this condition is precisely one of the coherence conditions that a cell in this double category must satisfy. So, if you’re familiar with double categories, they can help explain this algebraic construction — and vice versa, which is good.

In my particular setting, the bicrossed pairs are semifree, meaning that one of the two monoids is free on a set $A$. Conceptually, this doesn’t change the definition, but as a consequence it’s equally natural to present the action of a monoid $N$ on a set, and then extend it inductively.

Bicrossed pairs give rise to bicrossed products, which generalize semidirect products. When, in a bicrossed pair, one monoid acts trivially on the other, the nontrivial monoid acts by endomorphisms. In that case, the bicrossed product with one trivial action coincides with the semidirect product (and the same reasoning applies to groups, of course). A very natural conjecture is then that the algebras for a monad given in terms of bicrossed pairs are sets with an action of the bicrossed product: this turns out to be true.

Now, my question for you: monads themselves form a double category — for suitable choices of tight and loose morphisms — and monads in this latter double category of “modules” correspond to distributive laws between monads.

@Nathanael Arkor recently pointed out to me that there are two possible definitions for a loose morphism, and I’m now working out both to see what they give in my case. Interestingly, the “less natural” choice of loose morphism still turns out to be meaningful: it captures another construction in categories of Mealy automata, which in turn clarifies why that old definition “has to be like this for formal reasons.”

The notion of distributive law that I use in the paper at the moment follows Monads in Double Categories, page 11.

In my instance of the double category of monads à la Fiore–Gambino–Kock, a distributive law understood as "certain data and axioms ensuring that the composite loose cell forms a monad" translates into the data required for two bicrossed pairs $(A, X)$ and $(B, Y)$ to combine into a bicrossed pair $(A, X \times Y)$, allowing the bicrossed product $A \bowtie (X \times Y)$ to be constructed.

I’d be very interested to know whether this kind of construction —that is, the conditions under which two bicrossed pairs $(A, X)$ and $(A, Y)$ “compose” into a third— has been studied by algebraists, surely under a different name than “distributive law”, or if it appears in other corners of category theory that I don't know yet.