  We give a category-theoretic treatment of causal models that formalizes the
syntax for causal reasoning over a directed acyclic graph (DAG) by associating
a free Markov category with the DAG in a canonical way. This framework enables
us to define and study important concepts in causal reasoning from an abstract
and "purely causal" point of view, such as causal independence/separation,
causal conditionals, and decomposition of intervention effects. Our results
regarding these concepts abstract away from the details of the commonly adopted
causal models such as (recursive) structural equation models or causal Bayesian
networks. They are therefore more widely applicable and in a way conceptually
clearer. Our results are also intimately related to Judea Pearl's celebrated
do-calculus, and yield a syntactic version of a core part of the calculus that
is inherited in all causal models. In particular, it induces a simpler and
specialized version of Pearl's do-calculus in the context of causal Bayesian
networks, which we show is as strong as the full version.
