Lemma [efr-297V]

Let \mathcal {C} be a Markov category with good deterministic pullbacks, and let X \in \mathcal {C}

  1. If \mathcal {C} is causal, so is \mathcal {C}_{/X}.
  2. If P \to A \in \mathcal {C}_{/X} admits a support S \subseteq A in \mathcal {C}, then this is also a support in \mathcal {C}_{/X}

Proof

Since the inclusions A \times _X B \to A \times B are all monomorphisms, it is easy to verify causality. Similarly, it is easy to see that being a support in \mathcal {C} is simply stronger than being a support in \mathcal {C}_{/X}.

Context

Slice Markov Categories [efr-OKM1]

Definition Slice Markov Category [efr-EANM]

Let \mathcal {C} be a Markov category. Suppose \mathcal {C} admits deterministic pullbacks. Then for each object X \in \mathcal {C}, let \mathcal {C}_{/X} denote the category where objects are deterministic maps A \to X and morphisms are commutative triangles (whose map A \to B is not necessarily deterministic).

Note that \mathcal {C}_{/X} admits the structure of a Markov category---given A_1 \to B_1 and A_2 \to B_2 over X, the induced map A_1 \times _X A_2 \to B_1 \times B_2 necessarily lands within the subobject B_1 \times _X B_2

Lemma [efr-297V]

Let \mathcal {C} be a Markov category with good deterministic pullbacks, and let X \in \mathcal {C}

  1. If \mathcal {C} is causal, so is \mathcal {C}_{/X}.
  2. If P \to A \in \mathcal {C}_{/X} admits a support S \subseteq A in \mathcal {C}, then this is also a support in \mathcal {C}_{/X}

Proof

Since the inclusions A \times _X B \to A \times B are all monomorphisms, it is easy to verify causality. Similarly, it is easy to see that being a support in \mathcal {C} is simply stronger than being a support in \mathcal {C}_{/X}.