# On the Equivalence of Causal Models: A Category-Theoretic Approach

arXiv: https://arxiv.org/abs/2201.06981 · HTML: https://mlsanity.com/paper/2201.06981/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 9 references that are in the map.
- First posted on arXiv 18 Jan 2022.

## Builds on (most cited first, 6 of 6)

- [Causal Consistency of Structural Equation Models](https://mlsanity.com/paper/1707.00819/) (2017)
- [Abstracting Causal Models](https://mlsanity.com/paper/1812.03789/) (2018)
- [Approximate Causal Abstraction](https://mlsanity.com/paper/1906.11583/) (2019)
- [Visual Causal Feature Learning](https://mlsanity.com/paper/1412.2309/) (2014)
- [Multi-Level Cause-Effect Systems](https://mlsanity.com/paper/1512.07942/) (2015)
- [Causal Inference by String Diagram Surgery](https://mlsanity.com/paper/1811.08338/) (2018)

## Built on by (most cited first, 13 of 13)

- [Jointly Learning Consistent Causal Abstractions Over Multiple Interventional Distributions](https://mlsanity.com/paper/2301.05893/) (2023)
- [Abstraction between Structural Causal Models: A Review of Definitions and Properties](https://mlsanity.com/paper/2207.08603/) (2022)
- [Causal Abstraction Learning based on the Semantic Embedding Principle](https://mlsanity.com/paper/2502.00407/) (2025)
- [Quantifying Consistency and Information Loss for Causal Abstraction Learning](https://mlsanity.com/paper/2305.04357/) (2023)
- [What is causal about causal models and representations?](https://mlsanity.com/paper/2501.19335/) (2025)
- [Coarsening Causal DAG Models](https://mlsanity.com/paper/2601.10531/) (2026)
- [Causal Explanations for Sequential Decision Making Under Uncertainty](https://mlsanity.com/paper/2205.15462/) (2022)
- [Causal Abstraction in Model Interpretability: A Compact Survey](https://mlsanity.com/paper/2410.20161/) (2024)
- [CASR: Refining Action Segmentation via Marginalizing Frame-levle Causal Relationships](https://mlsanity.com/paper/2311.12401/) (2023)
- [On the Identifiability of Causal Abstractions](https://mlsanity.com/paper/2503.10834/) (2025)
- [Causal Abstractions, Categorically Unified](https://mlsanity.com/paper/2510.05033/) (2025)
- [Multi-Level Causal Embeddings](https://mlsanity.com/paper/2602.22287/) (2026)
- [Validating Causal Abstraction Metrics on Simulated Complex Systems](https://mlsanity.com/paper/2607.00267/) (2026)

## Abstract (arXiv)

We develop a category-theoretic criterion for determining the equivalence of causal models having different but homomorphic directed acyclic graphs over discrete variables. Following Jacobs et al. (2019), we define a causal model as a probabilistic interpretation of a causal string diagram, i.e., a functor from the ``syntactic'' category textsfSynG of graph G to the category textsfStoch of finite sets and stochastic matrices. The equivalence of causal models is then defined in terms of a natural transformation or isomorphism between two such functors, which we call a Phi-abstraction and Phi-equivalence, respectively. It is shown that when one model is a Phi-abstraction of another, the intervention calculus of the former can be consistently translated into that of the latter. We also identify the condition under which a model accommodates a Phi-abstraction, when transformations are deterministic.