Proposition [efr-001D]

The set of indexed stochastic charts is just the set of commutative squares in \mathsf {Kl}(\Delta ) of this form:

(where the vertical maps are, by assumption, deterministic, but the horizontal ones aren't necessarily)

Proof

Note that an indexed stochastic chart X \to Y is equivalently a family of stochastic charts \binom {\bar {X}_{x}}{\{x\}} \to \binom {\bar {Y}}{Y} for each x \in X. The analogous statement is true for commutative diagrams of this form (since coproducts in diagram categories are pointwise, and coproducts of sets are again coproducts in Kleisli categories). Hence it suffices to check the case X = *. In this case, a commutative square amounts to a family of distributions on \bar {Y}, each of which induces the same distribution on Y. On the other hand, we are looking at the convex set
\sum _{y \in Y}[\Delta (\bar {X}),\Delta (\bar {Y}_y)] = \sum _y \Delta (\bar {Y}_y)^{\bar {X}},

where this exponential denotes the iterated Cartesian product in convex spaces. An element of this set clearly gives a commutative square, because for each y \in Y, there is a square which deterministically chooses that y, and uses the given family of distributions on \bar {Y}_y depending on \bar {X}. This mapping is injective, since each map \bar {X} \to \Delta (\bar {Y}_y) can be recovered as a conditional distribution, and the convex combination of ys is merely the underlying distribution on Y. On the other hand, by choosing conditional distributions, it is also seen to be surjective, finishing the proof.

Context

Stochastic Categorical Systems Theory (The Discrete Case) [efr-001B]

It will be instructive to work out the theory in the discrete case, which is to say, for the Kleisli category of the discrete (i.e finite-support) distribution monad on \mathsf {Set}. Note that even for discrete-time discrete-space applications, this setup is not really adequate, as we lack the structure to construct terminal systems.

Definition [efr-001C]

Let \Delta : \mathsf {Set} \to \mathsf {Set} carry a set X to the set of finite-support probability distributions on X. Recall that \Delta is a monad.

Given two indexed sets \bar {X} \to X, \bar {Y} \to Y, an indexed stochastic lens \binom {\bar {X}}{X} \to \binom {\bar {Y}}{Y} is an element of the convex space \prod _{x \in X} \sum _{y \in Y} [\Delta (\bar {Y}_y),\Delta (\bar {X}_x)], where the internal hom, product and coproduct are taken in the category of convex spaces. Note that if each \bar {Y}_y is identical, say B, this is isomorphism to \prod _{x \in X} \Delta (Y) \otimes [\Delta (B),\Delta (\bar {X}_x)]. Recall also that an element of [\Delta (X),\Delta (Y)] is equivalently a function X \to \Delta (Y), i.e. a Kleisli map X \to Y.

An indexed stochastic chart is an element of the convex space \prod _{x \in X} \sum _{y \in Y} [\Delta (\bar {X}_x),\Delta (\bar {Y}_y)]

Proposition [efr-001D]

The set of indexed stochastic charts is just the set of commutative squares in \mathsf {Kl}(\Delta ) of this form:

(where the vertical maps are, by assumption, deterministic, but the horizontal ones aren't necessarily)

Proof

Note that an indexed stochastic chart X \to Y is equivalently a family of stochastic charts \binom {\bar {X}_{x}}{\{x\}} \to \binom {\bar {Y}}{Y} for each x \in X. The analogous statement is true for commutative diagrams of this form (since coproducts in diagram categories are pointwise, and coproducts of sets are again coproducts in Kleisli categories). Hence it suffices to check the case X = *. In this case, a commutative square amounts to a family of distributions on \bar {Y}, each of which induces the same distribution on Y. On the other hand, we are looking at the convex set
\sum _{y \in Y}[\Delta (\bar {X}),\Delta (\bar {Y}_y)] = \sum _y \Delta (\bar {Y}_y)^{\bar {X}},

where this exponential denotes the iterated Cartesian product in convex spaces. An element of this set clearly gives a commutative square, because for each y \in Y, there is a square which deterministically chooses that y, and uses the given family of distributions on \bar {Y}_y depending on \bar {X}. This mapping is injective, since each map \bar {X} \to \Delta (\bar {Y}_y) can be recovered as a conditional distribution, and the convex combination of ys is merely the underlying distribution on Y. On the other hand, by choosing conditional distributions, it is also seen to be surjective, finishing the proof.