  The d-separation criterion detects the compatibility of a joint probability
distribution with a directed acyclic graph through certain conditional
independences. In this work, we study this problem in the context of
categorical probability theory by introducing a categorical definition of
causal models, a categorical notion of d-separation, and proving an abstract
version of the d-separation criterion. This approach has two main benefits.
First, categorical d-separation is a very intuitive criterion based on
topological connectedness. Second, our results apply both to measure-theoretic
probability (with standard Borel spaces) and beyond probability theory,
including to deterministic and possibilistic networks. It therefore provides a
clean proof of the equivalence of local and global Markov properties with
causal compatibility for continuous and mixed random variables as well as
deterministic and possibilistic variables.
