Atomicity and supports in free Markov categories, and absence of conditionals

Prove that free Markov categories over a monoidal signature are atomic and admit supports, while not admitting conditionals.

Background

Free Markov categories provide a combinatorial hypergraph-based setting for Markov string diagrams. The paper constructs causal traces in free Markov categories via a contraction operation and shows preservation under interpretations.

The authors conjecture structural properties of free Markov categories—atomicity, existence of supports, and lack of conditionals—motivated by their information-flow characteristics and the constructions presented. Establishing these properties would clarify the internal probabilistic structure of free Markov categories and their role within categorical probability.