Combs, Causality and Contractions in Atomic Markov Categories

Abstract

We present a counterexample showing that Markov categories with conditionals (such as BorelStoch) need not validate a natural scheme of axioms which we call contraction identities. These identities hold in every traced monoidal category, so in particular this shows that BorelStoch cannot be embedded in any traced monoidal category. We remedy this under the additional assumption of atomicity: Atomic Markov categories validate all contraction identities, and furthermore admit a notion of trace defined for non-signalling morphisms. We conclude that atomic Markov categories admit an intrinsic calculus of combs without having to assume an embedding into a compact-closed category.