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Stream: community: general

Topic: Multiset and Distribution monads and physics

Ben Sprott (Sep 12 2020 at 16:23):

From his papers, Bart Jacobs seems fairly convinced that data monads, especially the multiset monad and it's probability monad, the distribution monad, should play a role in physics. I would like to know if anyone here is also looking into this or has seen other similar work.

Jules Hedges (Sep 12 2020 at 16:44):

Not an answer (no mention of physics), but there is a small conversation about these 2 monads at #practice: applied ct > histogram monad

Ben Sprott (Sep 13 2020 at 00:41):

Nice, thanks

John Baez (Sep 14 2020 at 15:06):

How does Bart Jacobs think these monads should play a role in physics?

The multiset monad is closely connected to the "Fock space" endofunctor on Hilb, which we use to describe collections of identical particles (bosons).

John Baez (Sep 14 2020 at 15:07):

I talk about that here (keeping monads deep in the shadows):

• nth Quantization.

John Baez (Sep 14 2020 at 15:08):

The Fock space functor K is like a lift of the multiset monad from Set to Hilb along the functor I call F: Set $\to$ Hilb.

Cole Comfort (Sep 14 2020 at 15:13):

The kleisli category of the finite (I think) multiset monad over a semiring is equivalent to matrices over that semiring, so what Bart cares about is the kleisli category of the $\mathbb C$-multiset monad which is pretty much just $\sf FHillb$.

John Baez (Sep 14 2020 at 15:20):

What's a "$\mathbb{C}$-multiset"? The set of finite complex linear combinations of elements of some set? That's usually called the free vector space on a set. It naturally gets completed to form a Hilbert space.

I was talking about actual multisets in my comment about second quantization.

Cole Comfort (Sep 14 2020 at 15:22):

Yeah, that's what his paper is about, which isn't as exciting as you might hope.

Ben Sprott (Sep 15 2020 at 01:57):

I believe he sees multisets as the middle part of this view of science:

Apparatus-> data monads-> probabilities

He calls the natural transformation from data monads to probabilities "learning".

Martti Karvonen (Sep 18 2020 at 15:00):

John Baez said:

The multiset monad is closely connected to the "Fock space" endofunctor on Hilb, which we use to describe collections of identical particles (bosons).

I might be misremembering, but was there a sublety here in that "Hilb" should mean "Hilbert spaces and contractive/non-expansive maps" rather than "Hilbert spaces and bounded maps" for the Fock space construction to result in a well-defined endofunctor?

Martti Karvonen (Sep 18 2020 at 18:00):

And I guess the functor $F:Set\to Hilb$ should have sets and partial injections rather than Set as its domain - for instance, the unique map $\mathbb{N}\to 1$ doesn't seem to induce a bounded map $\ell^2(\mathbb{N})\to\mathbb{C}$.

Oscar Cunningham (Sep 18 2020 at 18:25):

That's what Chris Heunen does in 'On The Functor $\mathcal{l}^2$'.

Martti Karvonen (Sep 18 2020 at 18:31):

And Barr earlier in "algebraically compact functors", except that the codomain is Hilbert spaces and contractive maps.

Martti Karvonen (Sep 18 2020 at 18:44):

Not sure whether the multiset monad survives the move to PInj though.