# Categoroids: Universal Conditional Independence

arXiv: https://arxiv.org/abs/2208.11077 · HTML: https://mlsanity.com/paper/2208.11077/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 44 references that are in the map.
- First posted on arXiv 23 Aug 2022.

## Builds on (most cited first, 9 of 9)

- [Markov Properties for Graphical Models with Cycles and Latent Variables](https://mlsanity.com/paper/1710.08775/) (2017)
- [Margins of discrete Bayesian networks](https://mlsanity.com/paper/1501.02103/) (2015)
- [An enriched category theory of language: from syntax to semantics](https://mlsanity.com/paper/2106.07890/) (2021)
- [A Mathematical Theory of Co-Design](https://mlsanity.com/paper/1512.08055/) (2015)
- [The operad of wiring diagrams: formalizing a graphical language for databases, recursion, and plug-and-play circuits](https://mlsanity.com/paper/1305.0297/) (2013)
- [Causal Homotopy](https://mlsanity.com/paper/2112.01847/) (2021)
- [Learning general conditional independence structures via the neighbourhood lattice](https://mlsanity.com/paper/2206.05829/) (2022)
- [An Alternative Markov Property for Chain Graphs](https://mlsanity.com/paper/1302.3553/) (2013)
- [On The Universality of Diagrams for Causal Inference and The Causal Reproducing Property](https://mlsanity.com/paper/2207.02917/) (2022)

## Built on by (most cited first, 3 of 3)

- [On the Power of Foundation Models](https://mlsanity.com/paper/2211.16327/) (2022)
- [Going Beyond Neural Network Feature Similarity: The Network Feature Complexity and Its Interpretation Using Category Theory](https://mlsanity.com/paper/2310.06756/) (2023)
- [From Intent to Evidence: A Categorical Approach for Structural Evaluation of Deep Research Agents](https://mlsanity.com/paper/2603.25342/) (2026)

## Abstract (arXiv)

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.