Title: Universal Causal Inference in a Topos
Venue: NeurIPS 2025
Accepted: yes
Keywords: Causal inference; Category Theory; Topos; Structural Causal Model
Abstract: In this paper, we explore the universal properties underlying causal inference by  formulating it in terms of a topos. More concretely, we introduce topos causal models (TCMs), a strict generalization of the popular structural causal models (SCMs).  A topos category has several properties that make it attractive: a general theory for how to combine local functions that define ``independent causal mechanisms" into a consistent global function building on the theory of sheaves in a topos; a generic way to define causal interventions using a subobject classifier in a topos  category; and finally, an internal logical language for causal and counterfactual reasoning that emerges from the topos itself.  A striking characteristic of subobject classifiers is that they induce an intuitionistic logic, whose semantics is based on the partially ordered lattice of subobjects. We show that the underlying subobject classifier for causal inference is not Boolean in general, but forms a Heyting algebra.  We define the internal Mitchell-B\'enabou language, a typed local set theory, associated with causal models, and  its associated Kripke-Joyal intuitionistic semantics.  We prove a universal property of TCM, namely that any causal functor mapping decomposable structure to probabilistic semantics  factors uniquely through a TCM representation.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nRecall that a category is a \"multigraph\" where vertices are objects (here typically causal models) while edges/arrows are \"morphisms\" (here transformations of causal models, interventions). The paper starts with the definition of a \"causal topos\" which is this category made up of causal models. They recall the notion of topos which is a category that behaves like the category \"SET\" of sets. In particular, there is the notion of subobject classifier, which is the equivalent of the Boolean set {false, true}. Its existence morally says that there is a correspondence between subobjects (subsets) and characteristic functions.\nThe authors prove that the causal topos (or maybe a variation, not sure) is a topos (Theorem 1).\nSection 3 shows that other categories based on causal models are topos (Theorem 2, 3, 4).\nSection 4 recalls the notion of Heyting algebra for replacing the notion of \"Boolean values\".\n\nSTRENGTHS AND WEAKNESSES:\nQuality\n+ The results are interesting but difficult to grasp.\n- The definitions are not crystal-clean.\n\n\nClarity\n- The paper is not well-organized. Some notions of category theory are in the text (subobject classifier) while others are in the appendix. It makes difficult to navigate in the document.\n- The contribution is Theorem 1-4 but Section 4 for instance is very obscure.\n\n\nSignificance\n+ The work could be of interest.\n- But the paper fails showing the signifiance. We perfectly understand that there is no algorithmic contribution. That is recalled several times in the document. But the question is then: - what do we better understand now? \n\n\nOriginality\n+ Section 2 and theorems 1-4 are new.\n- Section 4 seems to only recall existing notions and the impact on the work is not explicitely stated.\n\nQUESTIONS:\n- - Is the category in Theorem 1 the same than the one in Definition 1? It seems that it is not the case because the definition is recalled in Theorem 1.\n- What is the purpose of Section 4 (in simple words)\n- 203 I am lost. objects are not SCM? Why are they \"vertex\" or \"edge\"?\n- You say \"general framework\" that seems your own. But topos is not a new notion, same for many other existing notions presented in the document. What do you mean \"your general framework\"?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 1, "CLARITY": 1, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-06-22"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes a unified framework for causal inference based on topos theory, aiming to reveal the universal properties of causal inference from a category theory perspective. The authors argue that topos, as a category type with \"set-like\" properties, provides three key universal properties for causal inference: first, a general theory based on sheaves theory in topos that can combine local functions defining \"independent causal mechanisms\" into consistent global functions; second, using subobject classifiers in topos categories to provide a universal method for defining causal interventions; finally, generating an internal logical language for causal and counterfactual reasoning from the topos itself.\n\nSTRENGTHS AND WEAKNESSES:\nStrength:\n1. This paper reduces the gaps in past causal inference work by providing a unified mathematical model for different causal representations from a topos perspective;\n2. The proof process is rigorous, definitions are clear, forming a relatively complete theoretical system;\n3. The concept is novel, and related work indicates that there have been no similar attempts in the past.\n\nWeakness\n1. The theory is quite complex, lacks useful scenarios, and has a high barrier for causal inference researchers;\n2.  Insufficient motivation, fails to adequately explain the necessity of using a unified framework.\n\nQUESTIONS:\nWhat real-world causal inference problems are better suited for this framework compared to existing frameworks? The paper does not seem to mention how this framework improves upon existing causal inference methods.\n\nLIMITATIONS:\nIn my personal view, causal inference exists primarily as a tool for scientific discovery. This paper conducts excessive mathematical formalization of existing causal frameworks, which may not reflect the demands of real-world scenarios.\n\nTo AC: I have insufficient knowledge of the topos field, perhaps the AC needs to seek an emergency reviewer.", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 1, "CLARITY": 3, "IMPACT": 1, "ORIGINALITY": 3, "DATE": "2025-06-28"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors take a novel look at causal inference models, most notably Structural Causal Models (SCMs), and reformulate them as a topos. With this they aim to uncover a deeper understanding of causality. \nI find this deeply interesting, because different Definitions/representations of casualty do vary, but seem connected on some level. It looks to me like topoi might be a good lense through which to view these connections. I love this paper, because the idea seems obvious once you read it, but very hard to spot without. As such I am curious to see what this discovery will yield in the future.\n\nSTRENGTHS AND WEAKNESSES:\nOverall: I love the core idea and think it will have wide ranging impact on our understanding of causality - especially different notions of it. I think this paper is a great foundational frame to find similarities and analogies between definitions and come closer to a 'universal' understanding of causality. \n\nStrength: \n- Great idea.\n- The text is very well structured and all relevant background is explained in the appendices. \n- The math - as far as I can fully understand it in my available time- seems to check out fine.\n\nWeakness: \n- Like any deep paper it would benefit from more available space... \n- I would find it interesting to connect this work with Sander Beckers' work on causal sufficiency and different definitions of casualty:\nBeckers, S. (2021). Causal sufficiency and actual causation. Journal of Philosophical Logic, 50(6), 1341-1374.\n- In general, this paper would benefit from closer ties to the philosophy part of the causality literature - but I totally understand that space here is too limited for that.\n\nQUESTIONS:\nQ1: Would it be possible (within page limits...) to briefly connect to the philosophy of causality literature, e.g. Sander Beckers work: \nBeckers, S. (2021). Causal sufficiency and actual causation. Journal of Philosophical Logic, 50(6), 1341-1374.\n(A 'No' is a perfectly fine answer!)\n\nQ2:  I believe this work to have great impact on the future understanding of causality. Can you perhaps add a brief outlook on what viewing causality as a topos might allow us to see? Maybe some space can be gained by omitting the pointer that this paper is not for practioners and/or causal discovery.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 4, "CLARITY": 4, "IMPACT": 4, "ORIGINALITY": 4, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper approaches causal inference using the language of category\ntheory, and shows that several causal models (e.g. structural causal\nmodels and bayesian networks) can be interpreted as toposes, a\nspecific kind of category. The paper claims that the benefit of the\ninterpretation is to have access to three properties: 1) the\npossibility to reason about global functions starting from local\nfunctions using sheaves in a topos, 2) the possibility of reasoning\nabout causal interventions using an intuitionistic logic which emerges\nfrom their definition using subobject classifiers, 3) the possibility\nof using the internal language of a topos for causal and\ncounterfactual reasoning.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths\n\n- the paper shows that topos have the natural mathematical structure\n  needed to interpret several causal model\n\n- the paper identifies three benefits of interpreting causal models in\n  toposes\n\n- the paper does a good job of explaining abstract categorical\n  concepts and their connection to causality\n\nWeaknesses\n\n- the paper explains well the properties that one can get from using\n  toposes to interpret causal models but does not explain what one gains\n  in doing this\n\n- Some of the proofs are less rigorous than what one could expect \n\n\nThis is a rather unorthodox neurips submission aiming at using\nabstract mathematical concepts from category theory to explain causal\ninference models. This is part of a trend that has emerged in recent\nyears and that has brought to interpretations of several probabilistic\nand statistical machine learning concepts in terms of categorical\nconcepts.\n\nThe main observation that the paper makes, i.e. that causal models can\nbe interpreted in toposes, is interesting since toposes are a\nwell-studied kind of category which emerges naturally in several\napplications. The paper shows several benefits, from the modeling\nperspective, of doing this. However, I find that the paper stops too\nearly and does not clarify how these modeling benefits can be really\ntranslated to the understanding of causal models. In other terms, I\nfind that the paper does not explain enough what one gains from having\nthe three universal properties that the paper focuses on, and why a\ncausal inference scholar should care about them. I think this is\nimportant for a paper of this kind, to be really appreciated.\n\nThe presentation of the paper is good. The paper introduces the\ndifferent abstract concepts in an intuitive way and connects them well\nwith causal inference. As a result, I think this paper could result\naccessible also for reader without a deep background in category\ntheory. However, a downside of this is that the proofs in the paper\nare less rigorous than the usual in this area. The theorem themselves\nare not very deep, they are just showing the correctness of the\ndifferent interpretations, so I think it is fine to have more informal\nproofs but I was a bit surprised at first.\n\nOverall I think this is a good paper but it would be even better if\nthe authors could provide more explanation of what one gains from\nthe causal inference perspective by understanding the paper results.\n\nQUESTIONS:\n- Can you elaborate on the concrete benefit you see for a causal inference scholar to using the framework you propose?\n\n- I didn't get why in causal models the subobject classifier is not boolean. I think I see this intuitively, but now where it comes from technically. Can you elaborate on it?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}