  Conditional independence has been widely used in AI, causal inference,
machine learning, and statistics. We introduce categoroids, an algebraic
structure for characterizing universal properties of conditional independence.
Categoroids are defined as a hybrid of two categories: one encoding a
preordered lattice structure defined by objects and arrows between them; the
second dual parameterization involves trigonoidal objects and morphisms
defining a conditional independence structure, with bridge morphisms providing
the interface between the binary and ternary structures. We illustrate
categoroids using three well-known examples of axiom sets: graphoids,
integer-valued multisets, and separoids. Functoroids map one categoroid to
another, preserving the relationships defined by all three types of arrows in
the co-domain categoroid. We describe a natural transformation across
functoroids, which is natural across regular objects and trigonoidal objects,
to construct universal representations of conditional independence.. We use
adjunctions and monads between categoroids to abstractly characterize
faithfulness of graphical and non-graphical representations of conditional
independence.
