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Stream: learning: questions

Topic: lifting a subset-valued operation

Naso (May 18 2023 at 02:23):

Let $P$ be the covariant powerset functor, and suppose we have an operation $* : X \times X \to P(X)$.
We can lift this to an operation $*' : P(X) \times P(X) \to P(X)$ by defining

$a *' b = \cup \{ x * y \mid x \in a \land y \in b \}$

How can we express what's going on here in a categorical way?

Ralph Sarkis (May 18 2023 at 02:41):

You can factorize $*'$ as

$$PX \times PX \xrightarrow{\times} P(X\times X) \xrightarrow{P*} PPX \xrightarrow{\mu^P_X} PX$$

Where $\mu^P$ is the multiplication of the monad $P$. Also, the composition of the last two arrows is the extension of $*$ to the free $P$-algebra on $X\times X$.

I am not sure if there is a nice way to see the first morphism that takes $(a,b)$ to $a \times b$ in a categorical way. It is a right inverse to the pairing $\langle P\pi_1,P\pi_2\rangle : P(X\times X) \to PX \times PX$ given by the universal property of the product.

Mike Shulman (May 18 2023 at 02:50):

I think this is a two-variable Kleisli extension for the monad $P$. That is, given $X \to P(Y)$ there is a unique way to extend it to a $P$-algebra morphism $P(X) \to P(Y)$, and for a [[strong monad]] we can do this in two variables one after the other -- and when the monad is commutative the order doesn't matter. In functional programming syntax this would be

let x <- a in
let y <- b in
return (x * y)