Free gs-Monoidal Categories and Free Markov Categories

Abstract Categorical probability has recently seen significant advances through the formalism of Markov categories, within which several classical theorems have been proven in entirely abstract categorical terms. Closely related to categories are gs-monoidal also known as CD categories. These omit a condition that implements normalization probability. Extending work Corradini and Gadducci, we construct free generated by collection morphisms arbitrary arity coarity. For this comes form an explicit combinatorial description their structured cospans labeled hypergraphs. can be thought formalization string diagrams ( $$=$$ = term graphs) data structure. We formulate appropriate 2-categorical universal property based on ideas Walters prove our satisfy it. expect relevant for computer implementations argue they used statistical causal models generalizing Bayesian networks.