Necessity of atomicity and its relationship to embeddings, supports, and universal dilations

Ascertain the necessity of atomicity within the framework of Markov categories considered; specifically, determine whether an embedding of a Markov category into a traced monoidal category implies atomicity, and characterize the precise relationships between atomicity, supports, and universal dilations.

Background

Atomicity ensures the validity of contraction identities and the existence of causal traces for non-signalling morphisms. The paper shows that important categories like BorelStoch are not atomic, which blocks certain contraction identities and embeddings into traced categories, while many other categories (e.g., FinStoch) are atomic.

Clarifying whether atomicity is a necessary assumption for the results developed, and whether embedding into traced monoidal categories forces atomicity, would delineate the boundaries of the theory. Since atomicity is linked to supports and universal dilations, a precise characterization of these relationships is a natural next step.