### (no subject)

I'm pretty certain Iterated Distributive Laws is a stunningly beautiful piece of mathematics, even though I have not fully understood it. One of the things it is about is composing monads. You can compose two of them S and T and get a monad TS if you have a natural transformation λ : ST → TS showing how S distributes over T (so that the multiplication of TS arises from λ and μ

(This is also called the "third Reidemeister move" in knot theory)

The punchline is that this is all you need for four or more monads composed in a row too! Just individual swaps for each pair and Yang-Baxter equations for each triplet. Crazy.

---

ETA:

I think I got the idea now. It's quite beautiful. In order to show associativity of the composite monad, you just need to fudge the lines around in a string diagram. The pentagon axioms for distributivity itself let you push a string across a multiplication 2-cell, and Yang-Baxter lets you push strings across crossings of other pairs of monads.

All in all it has a very

_{S}and μ_{T}as suggested by TSTS → TTSS → TS) as long as λ satisfies a few equations. You can compose three of them S, T, and U if you have such several "distributive laws" that swap each of S, T, and U pairwise as long as they further satisfy a Yang-Baxter equation that essentially means the following two string diagrams are the same:S T U S T U | | | | | | | | | | | / \ / | | \ \ | | \ \ | | \ \ | | / \ / \ | \ / | | \ | \ | | \ = \ | | \ \ | | / \ / \ / \ / | | \ \ | | \ \ | | \ \ | | / \ / \ | | / \ U T S U T S

(This is also called the "third Reidemeister move" in knot theory)

The punchline is that this is all you need for four or more monads composed in a row too! Just individual swaps for each pair and Yang-Baxter equations for each triplet. Crazy.

---

ETA:

I think I got the idea now. It's quite beautiful. In order to show associativity of the composite monad, you just need to fudge the lines around in a string diagram. The pentagon axioms for distributivity itself let you push a string across a multiplication 2-cell, and Yang-Baxter lets you push strings across crossings of other pairs of monads.

All in all it has a very

*tri*categorical feel to it, since the distributivity laws give you "just enough" 3-dimensionality to be able to compose monads. Strangely, if you were talking monads that arise as monoid objects in a braided monoidal category, (a twice-degenerate tricategory, remember) you'd have so much wiggle room as to be able to compose monads in a*commutative*way, since you have a clockwise and counterclockwise twist.