Markov Processes for Category Theorists

Definitions

Recall the definition of an affine space over a ring A: an affine space is a space X equipped with the ability to take “affine combinations” of points in X, i.e. a_1 x_1 + \ldots a_n x_n where a_1 + \ldots + a_n = 1.

We can similarly define an “affine space” over a rig (ring without negation). An “affine space” over the rig \mathbb{R}_{\geq 0} is precisely a convex space. Using this definition makes it obvious how to define “convex morphisms”: a convex morphism is one that preserves affine combinations over \mathbb{R}_{\geq 0}. Note that this is different from the typical definition of a “convex function”, which has

f(\lambda x + (1-\lambda)y) \leq \lambda f(x) + (1-\lambda)f(y)

However, this definition of “convex morphism” has

f(\lambda x + (1-\lambda)y) = \lambda f(x) + (1-\lambda)f(y)

Soon we will pick a different name for “convex morphism” that fixes this difficulty.

The most important convex spaces are the n-simplices:

\Delta^n = \{x \in \mathbb{R}_{\geq 0}^{n+1} \mid x_0 + \ldots + x_n = 1 \}

These are important for many reasons, one of which is that an element of \Delta^n represents a probability mass function on n+1 elements. More generally, the set of (probability) measures on any space X with sigma-algebra \Omega of What has previously been called a stochastic map is precisely a convex map from a probability simplex to itself.

I propose that we should generalize the concept of stochastic map to cases where to domain and codomain are not a probability simplex. Namely, I propose that we define a stochastic map to be what we previously called a “convex morphism”.

We then can consider the category of convex spaces and stochastic maps. Call this category \mathbf{Cvx}.