Universal Causal Inference in a Topos
Sridhar Mahadevan
In this paper, we aim to shed light on the universal properties underlying causal inference by  formulating it in terms of a topos, a type of category that has set-like properties.  We argue that a topos provides three universal properties that make it attractive in formalizing causal inference: a general theory for how to combine local functions that define independent causal mechanisms into a consistent global function building on the theory of sheaves in a topos; a generic way to define causal interventions using a subobject classifier in a topos  category; and finally, an internal logical language for causal and counterfactual reasoning that emerges from the topos itself. A striking characteristic of subobject classifiers is that they induce an intuitionistic logic, whose semantics is based on the partially ordered lattice of subobjects. We show that the underlying subobject classifier for causal inference is not Boolean in general, but forms a Heyting algebra.  We study the internal language of the topos induced by causal models by defining its corresponding Mitchell-Benabou language, a typed local set theory where causal models define types.  We describe the Kripke-Joyal intuitionistic semantics of a topos that can support causal inference.  We illustrate the application of our framework to a range of representations, from  structural causal models to causal models over Markov categories and simplicial sets, as well as Lewis' possible-worlds logic of counterfactuals.