  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.
