A Higher Algebraic K-Theory of Causality

Causal discovery involves searching intractably large spaces. Decomposing the search space into classes of observationally equivalent causal models is a well-studied avenue to making discovery tractable. In this paper, we study the topological structure underlying causal equivalence to develop a categorical formulation of Chickering’s transformational characterization of Bayesian networks. We describe a homotopic generalization of the Meek-Chickering theorem on the connectivity structure within causal equivalence classes, and a topological representation of Greedy Equivalence Search (GES) that moves from one equivalence class of models to the next. Specifically, we define causal models as propable symmetric monoidal categories (cPROPs), a functor category CP from a coalgebraic PROP P to a symmetric monoidal category C. Such functor categories were first studied by [], who showed that they define the right adjoint of the inclusion of Cartesian categories in the larger category of all symmetric monoidal categories. cPROPs are an algebraic theory in the sense of Lawvere. CPROPs are related to previous categorical causal models, such as Markov categories and affine CDU categories, which can be viewed as defined by cPROP maps specifying the semantics of comonoidal structures corresponding to the “copy-delete" mechanisms. We develop a higher algebraic K-theory of causality by studying the classifying spaces of observationally equivalent causal cPROP models by constructing their simplicial realization through the nerve functor. We show that Meek-Chickering causal DAG equivalence generalizes to induce a homotopic equivalence across observationally equivalent cPROP functors. We present a homotopic generalization of the Meek-Chickering theorem, where covered edge reversals connecting equivalent DAGs induce natural transformations between homotopically equivalent cPROP functors and correspond to an equivalence structure on the corresponding string diagrams. We give a topological characerization of the Meek-Chickering Greedy Equivalence Search (GES) procedure. Finally, we present the Grothendieck group completion of cPROP causal models corresponding to causal DAGs using the Grayson-Quillen construction and relate the classifying space of Meek-Chickering equivalence classes to classifying spaces of an induced groupoid.