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We propose Universal Causality, an overarching framework based on categorytheory that defines the universal property that underlies causal inferenceindependent of the underlying representational formalism used. More formally,universal causal models are defined as categories consisting of objects andmorphisms between them representing causal influences, as well as structuresfor carrying out interventions (experiments) and evaluating their outcomes(observations). Functors map between categories, and natural transformationsmap between a pair of functors across the same two categories. Abstract causaldiagrams in our framework are built using universal constructions from categorytheory, including the limit or co-limit of an abstract causal diagram, or moregenerally, the Kan extension. We present two foundational results in universalcausal inference. The first result, called the Universal Causality Theorem(UCT), pertains to the universality of diagrams, which are viewed as functorsmapping both objects and relationships from an indexing category of abstractcausal diagrams to an actual causal model whose nodes are labeled by randomvariables, and edges represent functional or probabilistic relationships. UCTstates that any causal inference can be represented in a canonical way as theco-limit of an abstract causal diagram of representable objects. UCT followsfrom a basic result in the theory of sheaves. The second result, the CausalReproducing Property (CRP), states that any causal influence of a object X onanother object Y is representable as a natural transformation between twoabstract causal diagrams. CRP follows from the Yoneda Lemma, one of the deepestresults in category theory. The CRP property is analogous to the reproducingproperty in Reproducing Kernel Hilbert Spaces that served as the foundation forkernel methods in machine learning.

On The Universality of Diagrams for Causal Inference and The Causal Reproducing Property