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Stream: theory: categorical probability

Topic: 'almost closed'?

Matteo Capucci (he/him) (Aug 29 2022 at 09:20):

I'm sure I'm late to the party, but I just noticed measurable spaces are not cartesian closed, but they almost are? You can surely form a product $\prod_{x \in X} Y$ with all the projections. The map $X \times \prod_{x \in X} Y \to Y$ is not measurable in general but all the maps $\pi_{\bar x} : \prod_{x \in X} Y \to Y$, for any $\bar x \in X$, are.
This gives a map $\sum_{x \in X} \pi_x$, which means $ev : (\prod_{x \in X} Y) \times \Delta X \to Y$ exists and it's measurable. If $X=\Delta X = \sum_{x \in X} 1$, this looks like a bona-fide evaluation map. In other words, 'discrete measurable spaces' can exponentiate.

Now, one now can ponder how to extend this to a wider class of measurable spaces. It seems that what's crucial here is being able to take 'colimits indexed by a measurable space'. What I mean is, if I had something like $\sum_{x \in (X, \Sigma_X}$ I could use it to sum up the pointwise evaluation maps to a global evaluation map without having to forget $X$'s measurable structure.

On the other hand, that looks awfully a lot like a $\Sigma$-type constructor, and we surely have one in the category of measurable spaces? So it can't be the whole story... where am I going off-road?

Dmitri Pavlov (Jan 09 2023 at 06:10):

Matteo Capucci (he/him) said:

I'm sure I'm late to the party, but I just noticed measurable spaces are not cartesian closed, but they almost are? You can surely form a product $\prod_{x \in X} Y$ with all the projections. The map $X \times \prod_{x \in X} Y \to Y$ is not measurable in general but all the maps $\pi_{\bar x} : \prod_{x \in X} Y \to Y$, for any $\bar x \in X$, are.
This gives a map $\sum_{x \in X} \pi_x$, which means $ev : (\prod_{x \in X} Y) \times \Delta X \to Y$ exists and it's measurable. If $X=\Delta X = \sum_{x \in X} 1$, this looks like a bona-fide evaluation map. In other words, 'discrete measurable spaces' can exponentiate.

If you are willing to add a σ-ideal of negligible (alias measure 0) sets to the data of a measurable space, then the resulting category of enhanced measurable spaces admits a noncartesian closed symmetric monoidal structure. The internal hom functor can be seen as an appropriate replacement for the cartesian internal hom (which provably does not exist, since taking products with a fixed space is not a cocontinuous functor).

Matteo Capucci (he/him) (Jan 09 2023 at 18:57):

Nice!!

Matteo Capucci (he/him) (Jan 09 2023 at 18:57):

I am willing to add that $\sigma$-ideal

Evan Patterson (Jan 09 2023 at 22:58):

@Dmitri Pavlov is too modest to say so, but he has a really nice paper about "enhanced measurable spaces": https://arxiv.org/abs/2005.05284 I like this work a lot.

Evan Patterson (Jan 09 2023 at 23:06):

As you might expect, people have proposed other categories to work around the fact that Meas is not cartesian closed. See quasi-Borel space on nLab.

Dmitri Pavlov (Jan 10 2023 at 01:01):

Matteo Capucci (he/him) said:

I am willing to add that $\sigma$-ideal

Yes, the arXiv paper cited above does develop a bit of theory of enhanced measurable spaces. Unfortunately, it says nothing about the closed symmetric monoidal structure, since this stuff is meant to go into the second paper (in a series of about 5 papers) on this topic, but the project is proceeding at a glacial speed since there are no collaborators and I am busy working on a lot of other stuff.

The monoidal product is the usual product of measurable spaces familiar from measure theory: if (X,M,N) and (X',M',N') are two enhanced measurable spaces, then their product has X⨯X' as its underlying set, the σ-algebra of measurable sets is generated by sets of the form m⨯m', m∈M, m'∈M', and the σ-ideal of negligible sets is generated by sets of the form m⨯n' and n⨯m', n∈N, n'∈N'. Using the adjoint functor theorem we can now construct the closed structure adjoint to this monoidal structure. The solution set condition is easy. It remains to show that monoidal product with a fixed object preserves small colimits. Preservation of small coproducts is easy. To show that coequalizers are preserved, use the Gelfand duality from the cited paper (and here we need to impose very mild restrictions on enhanced measurable spaces: they must be localizable) to reduce the problem to the following dual problem: the spatial tensor product with a fixed von Neumann algebra preserves equalizers. The equalizer of two maps can be computed as the kernel of their difference. Now use the following remarkable fact about kernels of morphisms of von Neumann algebras: they canonically split off as direct summands. This immediately establishes the desired property.

Thus, the closed monoidal structure exists. I am pretty sure it is quite relevant for stochastic processes, by the way, since the internal hom of enhanced measurable spaces is the “correct” construction of the (enhanced) measurable space of all measurable maps, which circumvents the existing nonexistence results for the nonenhanced setting.