Eigil Rischel

@skewray@mathstodon.xyz I'm sorry to say I think most of the available material on Markov categories is written for category theorists. @paolop@mathstodon.xyz might know of a good source here. (Tobias' paper https://arxiv.org/abs/1908.07021 is a comprehensive introduction which is not *too* heavy on category theory, but still very much written for categorists)

@johncarlosbaez@mathstodon.xyz @julesh@mathstodon.xyz @pigworker@types.pl Mathematicians tend to call containers "polynomial functors". David Spivak has written a lot about them under this name.

@julesh@mathstodon.xyz X can be recovered from PX as the set of elements with exactly one other element below them (in the inclusion order, which can be recovered from the order on 2 and observing that Hom(X,-) preserves limits, so carries posets to posets).

Lemma: in any category as above, let \(\chi: 2^\mathbb{N} \to 2\) be the (deterministic) indicator function for the subobject of sequences that contain infinitely many ones (note that this is definable internally). Let \(u: 1 \to 2^\mathbb{N}\) be the infinite independent pairing of the coinflip with itself. Then the map \(\chi \circ u : 1 \to 2\) is equal to the constant \(1\) map. It follows that \(u\) factors over the inclusion of the subobject \(\chi^{-1}(1)\). In particular it factors over the inclusion of the subobject of sequences with at least one \(1\). In other words, given an infinite sequence of independent fair coinflips, there will be infinitely many ones. This is of course true in classical probability theory (but false in some Markov categories, for example of sets and total relations). Proof(sketch): it is apparent that \(\chi\) is independent of any finite prefix of its argument. It follows from the abstract version of Kolmogorov's 0-1 law (proved in https://arxiv.org/abs/1912.02769) that \(\chi u : 1 \to 2\) is deterministic. If we let \(\chi'\) be the indicator of sequences with infinitely many zeroes, then \((\chi,\chi') u : 1 \to 2 \times 2\) is also deterministic. It follows that it is equal to \((\chi u, \chi' u)\). But this latter is clearly equal to \((\chi u, \chi u)\) by the symmetry of the coinflip. By postcomposing with \(\vee : 2 \times 2 \to 2\), and observing that each sequence must have infinitely many ones or zeroes, we obtain the desired identity.

It is not so hard to see why you get every kernel this way. Every kernel \(A \to B\) can be written as a measurable function \(f: A \times [0,1] \to B\) composed with the Lebesgue measure \(1 \to [0,1]\), which in turn is measure-isomorphic to the "infinite independent coinflip" measure \(1 \to 2^\mathbb{N}\). This can be constructed using the coinflip and the property that \(2^\mathbb{N}\) is a Kolmogorov product. It is much more subtle to see that every identity between such kernels is a consequence of the axioms. The full proof is complicated (and very ugly currently, although I have ideas for how to make it more conceptual), but I'll sketch a key lemma which contains the general idea.

I actually find this lemma quite surprising. A priori, I would expect this map \(1 \to 2\) to correspond to some sort of infinitesimal probability. But apparently the axioms here rule this out. Currently I'm trying to work out how much these axioms can be weakened. The proof relies heavily on the assumption that \(\mathcal{C}_\mathrm{det}\) is Boolean, but this is very strong. It would be great to apply this idea to toposes or similar to construct canonical probability monads. If \(\mathcal{C}_\mathrm{det}\) has enough structure to construct a Dedekind real numbers object and a subobject classifier, we can define a kernel \(b: [0,1] \to \Omega\), given by sampling a stream from \(u: 1 \to 2^\mathbb{N}\), viewing it as the binary expansion of a real number in the interval, and comparing it with the input. One can try to ask under what assumptions this family of distributions satisfies the expected equations (\(b(x) \wedge b(y) = b(xy), \neg b(x) = b(1-x)\) , etc). I think I have some ideas here but this is still work in progress.

An interesting question here is whether there exists such a category where \(\mathcal{C}_\mathrm{det} \cong \mathsf{Set}\), or more generally a commutative monad on \(\mathsf{Set}\) so that the Kleisli category has Kolmogorov products. The Vitali sets prove that it can't be given by the ordinary distribution monad on countable sets, but doesn't rule out some more exotic construction. It is interesting to ask which, say, toposes, admit a probability monad with Kolmogorov products which behaves "as expected", maybe formalized by saying the distributions on the natural numbers object should all be discrete, i.e given by a point of the countable-dimensional simplex.

@bentnib @mkerjean Are extended abstracts (or papers, fot that matter) supposed to be anonymized? I can't see any info about this on the website.

@andrejbauer@mathstodon.xyz I created a separate user for it, which only has access to the stuff I want it to work on, then run claude code from a shell logged into that account. This may be more annoying to do with the desktop app, not sure.

@olynch@mathstodon.xyz I like this way of displaying slides on the web - how did you make this?