# Free gs-monoidal categories and free Markov categories

arXiv: https://arxiv.org/abs/2204.02284 · HTML: https://mlsanity.com/paper/2204.02284/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Apr 2022.

## Built on by (most cited first, 6 of 6)

- [Causal models in string diagrams](https://mlsanity.com/paper/2304.07638/) (2023)
- [The d-separation criterion in Categorical Probability](https://mlsanity.com/paper/2207.05740/) (2022)
- [An Algebraic Approach to Moralisation and Triangulation of Probabilistic Graphical Models](https://mlsanity.com/paper/2503.11820/) (2025)
- [String Diagrams with Factorized Densities](https://mlsanity.com/paper/2305.02506/) (2023)
- [Causal Abstractions, Categorically Unified](https://mlsanity.com/paper/2510.05033/) (2025)
- [Bayesian Networks, Markov Networks, Moralisation, Triangulation: a Categorical Perspective](https://mlsanity.com/paper/2512.09908/) (2025)

## Abstract (arXiv)

Categorical probability has recently seen significant advances through the formalism of Markov categories, within which several classical theorems have been proven in entirely abstract categorical terms. Closely related to Markov categories are gs-monoidal categories, also known as CD categories. These omit a condition that implements the normalization of probability. Extending work of Corradini and Gadducci, we construct free gs-monoidal and free Markov categories generated by a collection of morphisms of arbitrary arity and coarity. For free gs-monoidal categories, this comes in the form of an explicit combinatorial description of their morphisms as structured cospans of labeled hypergraphs. These can be thought of as a formalization of gs-monoidal string diagrams (=term graphs) as a combinatorial data structure. We formulate the appropriate 2-categorical universal property based on ideas of Walters and prove that our categories satisfy it. We expect our free categories to be relevant for computer implementations and we also argue that they can be used as statistical causal models generalizing Bayesian networks.