Jason (jcreed) wrote,
Jason
jcreed

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 μS and μT 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 tricategorical 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.
Tags: categories, math
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