Embedding Markov categories into traced monoidal categories

Determine necessary and sufficient conditions under which a Markov category admits an embedding into a traced monoidal category, and clarify how the weaker causal trace axioms compare to partial trace axioms in this embedding context, potentially via constructions analogous to Houghton-Larsen’s causal channels.

Background

The paper develops a notion of causal trace for non-signalling morphisms in atomic Markov categories and shows that contraction identities hold under atomicity and conditionals. Classical results establish that every partially traced category embeds in a traced monoidal category, but the causal trace axioms introduced here are strictly weaker than partial trace axioms. Understanding embeddings of Markov categories (which model stochastic processes abstractly) into traced monoidal categories is important for equational reasoning and for connecting categorical probability to established traced frameworks.

The authors speculate that differences between partial trace axioms and their causal trace axioms might be bridged by constructions similar to Houghton-Larsen’s causal channels, motivating a precise characterization of when such embeddings exist.