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Stream: theory: categorical probability

Topic: Coproducts in BorelStoch

Nathaniel Virgo (Dec 10 2023 at 13:39):

This was briefly discussed in a previous thread but I don't think this particular question got answered (and the more I try to follow up the points raised there the less confident I am in my understanding). So I'm wondering if anyone knows the following:

• Does $\mathbf{BorelStoch}$ (or its deterministic subcategory) have finite coproducts?

• Does it have countable coproducts? (I'm guessing it doesn't have uncountable ones)

• Are the coproducts "what they should be", i.e. can we think of a map into $A+B$ as first flipping a biased coin to choose $A$ or $B$ and then choosing an element of the appropriate space?

• Does the tensor distribute over the coproducts?

(I think that if the category of standard Borel spaces and measurable maps has coproducts then these will also be coproducts in $\mathbf{BorelStoch}$, although if I'm wrong I'm happy to be corrrected.)

Paolo Perrone (Dec 10 2023 at 13:49):

At least countable coproducts are there, and they are the disjoint union of (standard Borel) measurable spaces.
(Since left-adjoints preserve coproducts.)

Todd Trimble (Dec 10 2023 at 14:02):

Sorry, but what's the definition of $\mathbf{BorelStoch}$?

Paolo Perrone (Dec 10 2023 at 14:04):

Equivalently, either the category of standard Borel spaces and Markov kernels, or the Kleisli category of the Giry monad on standard Borel spaces.

Nathaniel Virgo (Dec 10 2023 at 14:06):

Paolo Perrone said:

At least countable coproducts are there, and they are the disjoint union of (standard Borel) measurable spaces.
(Since left-adjoints preserve coproducts.)

what's the left adjoint you're referring to here?

Paolo Perrone (Dec 10 2023 at 14:10):

Nathaniel Virgo said:

Paolo Perrone said:

At least countable coproducts are there, and they are the disjoint union of (standard Borel) measurable spaces.
(Since left-adjoints preserve coproducts.)

what's the left adjoint you're referring to here?

The inclusion functor from measurable maps to Markov kernels.

Todd Trimble (Dec 10 2023 at 14:32):

I was just going to say, based on the Kleisli description: for any Kleisli category on a monad $M: C \to C$, the forgetful functor $\mathrm{Kleisli}(M) \to C$ has a left adjoint.

Sam Staton (Dec 10 2023 at 17:26):

Also, yes the tensor does distribute over the coproducts (i.e. distributive monoidal category). This is because it does in standard Borel spaces (e.g. this is part of Ruiyuan Chen's characterization of standard Borel spaces).

Not sure that "distributive Markov categories" have been used before, but we used them in the paper I mentioned recently (see page 8).