# Intuitionistic j-Do-Calculus in Topos Causal Models

arXiv: https://arxiv.org/abs/2510.17944 · HTML: https://mlsanity.com/paper/2510.17944/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 28 references that are in the map.
- First posted on arXiv 20 Oct 2025.

## Builds on (most cited first, 7 of 7)

- [Learning Independent Causal Mechanisms](https://mlsanity.com/paper/1712.00961/) (2017)
- [A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics](https://mlsanity.com/paper/1908.07021/) (2019)
- [Causal de Finetti: On the Identification of Invariant Causal Structure in Exchangeable Data](https://mlsanity.com/paper/2203.15756/) (2022)
- [Disintegration and Bayesian Inversion via String Diagrams](https://mlsanity.com/paper/1709.00322/) (2017)
- [Markov Properties for Graphical Models with Cycles and Latent Variables](https://mlsanity.com/paper/1710.08775/) (2017)
- [Causal Inference by String Diagram Surgery](https://mlsanity.com/paper/1811.08338/) (2018)
- [The d-separation criterion in Categorical Probability](https://mlsanity.com/paper/2207.05740/) (2022)

## Built on by (most cited first, 5 of 5)

- [Large Causal Models from Large Language Models](https://mlsanity.com/paper/2512.07796/) (2025)
- [Decentralized Causal Discovery using Judo Calculus](https://mlsanity.com/paper/2510.23942/) (2025)
- [Causal Density Functions](https://mlsanity.com/paper/2606.00754/) (2026)
- [CSQL: Mapping Documents into Causal Databases](https://mlsanity.com/paper/2601.08109/) (2026)
- [A cubical formalisation of topos causal models: intervention, forcing, and a contextuality obstruction](https://mlsanity.com/paper/2607.15629/) (2026)

## Abstract (arXiv)

In this paper, we generalize Pearl's do-calculus to an Intuitionistic setting called j-stable causal inference inside a topos of sheaves. Our framework is an elaboration of the recently proposed framework of Topos Causal Models (TCMs), where causal interventions are defined as subobjects. We generalize the original setting of TCM using the Lawvere-Tierney topology on a topos, defined by a modal operator j on the subobject classifier Omega. We introduce j-do-calculus, where we replace global truth with local truth defined by Kripke-Joyal semantics, and formalize causal reasoning as structure-preserving morphisms that are stable along j-covers. j-do-calculus is a sound rule system whose premises and conclusions are formulas of the internal Intuitionistic logic of the causal topos. We define j-stability for conditional independences and interventional claims as local truth in the internal logic of the causal topos. We give three inference rules that mirror Pearl's insertion/deletion and action/observation exchange, and we prove soundness in the Kripke-Joyal semantics. A companion paper in preparation will describe how to estimate the required entities from data and instantiate j-do with standard discovery procedures (e.g., score-based and constraint-based methods), and will include experimental results on how to (i) form data-driven j-covers (via regime/section constructions), (ii) compute chartwise conditional independences after graph surgeries, and (iii) glue them to certify the premises of the j-do rules in practice