nLab FinStoch

Idea

The category whose objects are finite sets and whose morphisms are stochastic maps (or stochastic matrices) is often denotes FinStoch, or similar; a full subcategory of Stoch.

This is an elementary but nontrivial example of a Markov category, and together with BorelStoch, one of the most important categories of category-theoretic probability.

Definition

FinStoch is the category whose

• Objects are finite sets;
• Morphisms are stochastic matrices.

See at stochastic matrix for more details.

Properties

• FinStoch is closely related to the Kleisli category of the distribution monad: it is its full subcategory whose objects are finite sets.

• FinStoch is a Cauchy-complete category.

• As a Markov category, it has conditionals, and is hence positive and causal.

Characterizations

• FinStoch is the opposite category of the Lawvere theory for convex spaces. One can then dualize the presentation of convex spaces to obtain a characterization of FinStoch as a category with coproducts. It being symmetric monoidal corresponds to the fact that the algebraic theory is commutative. A presentation would have generators $c_p:1\to 1+1$ (corresponding to the binary operations $c_p$) together with equations amounting to the equations presenting convex spaces.

• For a simple specific case, let DyFinStoch be the wide subcategory of FinStoch where the probabilities are dyadic rational numbers. This is the opposite category of the Lawvere theory for midpoint algebras. It is also therefore the free distributive monoidal category that is semicartesian and with a map $c:1\to 1+1$ such that $s\circ c=c$ where $s:1+1\to 1+1$ is the symmetry of the coproduct. For the third axiom at midpoint algebras says that the theory is commutative, the first says that it is semicartesian, and the law $s\circ c=c$ is the second axiom.

• Let BinStoch be the full subcategory of FinStoch where the objects are powers of two. Then this category as a symmetric monoidal category is characterized in Piedeleu et al.

Related concepts

• Markov category
• Markov kernel
• category of couplings
• probability monad
• Giry monad
• Kleisli category
• categorical probability

References:

• Marshall H Stone, Postulates for the barycentric calculus, Ann. Mat. Pura. Appl. (4), 29:25–30, 1949.

• Tobias Fritz, Tomáš Gonda, Antonio Lorenzin, Paolo Perrone, Dario Stein, Absolute continuity, supports and idempotent splitting in categorical probability [arXiv:2308.00651]

• Robin Piedeleu, Mateo Torres-Ruiz, Alexandra Silva, Fabio Zanasi, ‘’A Complete Axiomatisation of Equivalence for Discrete Probabilistic Programming’’, arXiv:2408.14701, 2024.

category: category, probability