Post #2729844

2026-04-30 16:11 UTC

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.

Replies (1)

@eigil@mathstodon.xyz 2026-04-30 16:23

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.

Open ##2729845