### What Is This Category Enriched In?

#### Posted by David Corfield

I’ve never felt completely happy with enriched category theory, so perhaps people could help me think through this example.

This question got me wondering again what can be said about a category of conditional probabilities. Let’s take as our objects finite sets, and morphisms to be of the form $f: A \to B$, a conditional probability distribution $M_{i j} = P(b_j|a_i)$, that is, a row stochastic matrix. This is an arrow in the Kleisli category for the Giry monad. An equivalent category with column stochastic matrices is described by Tobias Fritz in A presentation of the category of stochastic matrices.

Now $Hom(A, B)$ has quite some structure. If $|A|= m$ and $|B| = n$, $Hom(A, B)$ is the $m$-fold product of $(n - 1)$-simplices. On each of these simplices, representing the space of probability distributions over $B$, there is defined a very natural metric – the Fisher information metric. On $Hom(A, B)$ we can then choose the *product* Fisher metric, as described in Lebanon’s Axiomatic Geometry of Conditional Models.

So now, how do I go about working out what this category may be said to be enriched in? Given $M$ in $Hom(A, B)$ and $N$ in $Hom(B, C)$, there is a composite $M N$ in $Hom(A, C)$, corresponding to $P(c_k|a_i) = \sum_j P(b_j|a_i) \cdot P(c_k|b_j)$. So I think I need to come up with a category which has objects including these Hom spaces, and a monoidal product to cater for the pairs of composable morphisms, such that composition is an arrow in that category.

I take it that the category of convex spaces and convex mappings won’t do, because the composition from pairs of composable morphisms is not convex. How about Riemannian manifolds with corners and whatever the right morphisms for these are?

## Re: What Is This Category Enriched In?

This category (actually a small variation described below) is enriched over partially additive monoids. In fact the category can be defined more generally. Start with the category Mes of measurable spaces (sets equipped with a sigma-algebra) with morphisms measurable functions. Instead of Giry’s monad use the small variation where there are all *sub*-probability distributions. The Kleisli category of this is something that is called SRel (stochastic relations). The homsets are partially additive (partial because you cannot add a family of morphisms if the total measure adds up to more than 1) monoids under pointwise sum. Arbib and Manes describe the theory of partially additive categories at some length and this is an example. I have used this for semantics of probabilistic programming languages. [Sorry, I am MathML illiterate, otherwise I would have posted more details.]

Prakash