Post #2729845

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.