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

We propose Universal Causality, an overarching framework based on category theory that defines the universal property that underlies causal inference independent of the underlying representational formalism used. More formally, universal causal models are defined as categories consisting of objects and morphisms between them representing causal influences, as well as structures for carrying out interventions (experiments) and evaluating their outcomes (observations). Functors map between categories, and natural transformations map between a pair of functors across the same two categories. Abstract causal diagrams in our framework are built using universal constructions from category theory, including the limit or co-limit of an abstract causal diagram, or more generally, the Kan extension. We present two foundational results in universal causal inference. The first result, called the Universal Causality Theorem (UCT), pertains to the universality of diagrams, which are viewed as functors mapping both objects and relationships from an indexing category of abstract causal diagrams to an actual causal model whose nodes are labeled by random variables, and edges represent functional or probabilistic relationships. UCT states that any causal inference can be represented in a canonical way as the co-limit of an abstract causal diagram of representable objects. UCT follows from a basic result in the theory of sheaves. The second result, the Causal Reproducing Property (CRP), states that any causal influence of a object X on another object Y is representable as a natural transformation between two abstract causal diagrams. CRP follows from the Yoneda Lemma, one of the deepest results in category theory. The CRP property is analogous to the reproducing property in Reproducing Kernel Hilbert Spaces that served as the foundation for kernel methods in machine learning.