Universal Causal Inference in a Topos
Authors: Sridhar Mahadevan
Venue: NeurIPS 2025 spotlight ()
Keywords: Causal inference; Category Theory; Topos; Structural Causal Model
Primary area: probabilistic_methods
TL;DR: We characterize the universal properties of causal inference on a topos category.

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.

Universal Causal Inference in a Topos
Sridhar Mahadevan' Adobe Research San Jose, CA smahhadevaddobe.com com
Abstract
In this paper, we explore the universal properties underlying causal infer- ence by formulating it in terms of a topos. More concretely, we introduce topos causal models (TCMs), strict generalization of the popular structural causal models (SCMs). A topos category has several properties that make it attractive: general theory for how to combine local functions that define "independent causal mechanisms" into a consistent global function building the theory of sheaves in topos; generic way to define causal interventions using a subobject classifier in a topos category; and finally, internal logical language for causal and counterfactual reasoning that emerges from the topos itself. A striking characteristic of subobject classi- fiers 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-Bénabol language, a typed local set theory, associated with causal models, and its associated Kripke-Joyal intuitionistic semantics. We prove universal property of TCM, namely that any causal functor mapping decomposable structure probabilistic semantics factors uniquely through TCM representation.
1 Introduction
In recent years, there has been significant interest in categorical models of causality, based on symmetric monoidal categories [ong 2012, Fritz and Klingler 2023 Cho and Jacobs 2019
Jacobs et al. 2018), as well as simplicial sets and higher-order categories [Mahadevan] 2023
Markov categories [Fritz] 2020) define broad unifying framework for probabilistic interence and statistics using symmetric monoidal categories, where each object is additionally equipped with a comonoidal "copy-delete" operation. It enables carrying out rigorous proofs using an elegant string diagrammatic language [Selinger 2010 Any causal model based on graphs [Pear 2009 Forré and Mooij 2017 Spirtes al. 2000 or other algebraic formalisms, such as integer-valuee multisets [Studeny 2010 can be translated into string diagram over a symmetric monoidal category, or a simplicial set. Operations on causal models, such as interventions, can be modeled as functors on the objects of the associated symmetric monoidal category or simplicial set. Categorical approaches causality also extend to the potential outcomes counterfactual framework [Ibens and Rubin 2015
Categorical approaches fundamentally differ from past work in causality in their focus on the elucidation of universal properties. In our previous work [Mahadevan] 2023 2025c we introduced the framework of universal causality based the notion universal properties
'Academic affiliation: Research Professor, University of Massachusetts, Amherst; See webpage at https://people.cs umass. h.m.s...h///maaaauu/////////bbb
39th Conference Neural Information Processing Systems (NeurIPS 2025).

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Causal Model
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Lung irliios
Preciies
Lodkown
Endogenout varables
Exogenous VERMOIO
Figure 1: Topos causal models (TCMs) are defined as a category CrCи whose objects c E Creи are causal models, and whose arrows Crom(c, c') are commutative diagrams between models e and c'. A specific object e defining a model can be conceptualized as DAG (left, where information flows from top to bottom), or string diagram in Markov category (middle, where information flows from bottom to top), or in terms of its induced unique "blackbox" function mapping exogenous variables to endogenous variables (right).
in category theory [Riehl 2017: a causal property is universal if it can be defined in terms of an initial or final object in a category of causal diagrams, or in terms of a causal representable functor using the Yoneda Lemma. For example, structural causal model (SCM) [Pearl] [2099] is defined as a (deterministic) mapping from collection of exogenous variables into collection of endogenous variables, derived by "collating" local functions that serve as independent causal mechanisms [Galles and Pearl 1988 Parascandolo et al. 2017]. However, SCMs can be further analyzed in terms of their universal properties, such as categorical product, coproduct, limits and colimits, equalizers and coequalizers etc. These latter properties can be shown formally to be initial or final objects category of diagrams Riehl 2017], or as representable functors through the Yoneda Lemma [MacLane] 1971].
Our main contribution in this paper is to present topos- thoos-tteertic view of causality, and in particular, introduce topos causal models (TCMs) that strictly generalize structural causal models (SCMs) [Pearl 2009]. A topos is type of category [MacLane [971], which is particularly well-suited to modeling operations that are "set-like" |Mac Lane and leke Moerdijk 1994]. It also features an internal logical language topos provides three universal properties that make it natural as category to do causal inference in: it provides a general theory for how combine local functions, which can be viewed as "independent causal mechanisms' [Parascandolo et al.) 2017], into a consistent global function building on the theory of sheaves topos [Mac Lane and Moerdijk 1992]. It enables generic way to define causal interventions using a subobject classifier in a topos category [Johnstone] 2014). Finally, gives an internal logical language for causal and counterfactual reasoning Bell 1988
languaage [Goldblatt] 2006]. We claim that
As Figure fillustatsssssssssss.. the objects in TCM category can be conceptualized in multiple ways. First, each object can be a causal graphical model [Pearl] 1989 Spirtes et al. 22000] Each object can also be a functor: for example, directed graphs form a topos functor category [Vignal 2003]. TCMs can also be defined in terms of string diagrams in a symmetric monoidal Markou cntegory Fritz 2 where we restrict ourselves the Markov subcategory defined through deterministic morphisms. For example, the arrow h Traffic& Afriicuturl Fires - Pollution defines a deterministic mapping specifying the two potential causes of Pollution. For exogenous variables, the arrow / I + Overpopulation defines the marginal distribution on Overpopulation, where is the terminal object in the Markov category. Finally, we can view TCM object as "blackbox" function that maps some collection of exogenous variables (e-g., "Overpopulation", or "Farming Practices" into some set of endogenous variables, e-g.. "Asthma" or "Pollution").

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2 Principles of Universal Causality
We give a brief overview of the fundamentals of universal causality (UC) Mahadevan 2023
2025c] before delving into the specific details of the TCM framework. As with other work in categorical causality [Fong| 2012, J00o/Jacobb Jacobs et al. 2018 Fritz and Klingler 2023 UC uses category theory [MacLane| 1971 to define causality. A category C' a collection of abstract objects c E C. Anything technically can count as an object, from a variable in a causal model to an entire model itself. Each category C is additionally specified by set of arrows or morphisms C(c, between each pair of objects c and d. There is an identity arrow 1. E C(c, Arrows compose in the obvious way, inducing a function C(c, d) x C(d,e) - C(c,e). An initial object c in category C defined as one inducing a unique arrow from c to every object in category C. A terminal object, usually denoted by 1, is one that defines a unique arrow from every object c in category C into 1. An object is isomorphic to another object d, denoted d and g
d, if two arrows f A functor F C - D between two categories C and D is specified by an object function mapping each c C to Fc D, and an arrows function mapping each arrow f e C(c, to Ff E D(Fc, Functors come in two varieties- covariant and contravariant - the latter acts on the domain category by reversing the arrows. Given any two functors F c- D and
exist, such that g o f = 1c, and f o la-
G:C-) D between the same pair of categories, we can define mapping between F and G that is referred to a natural transformation. These are defined through a collection of mappings, one for each object c of C, thereby defining morphism in D for each object in C.
Fo
a
Ge
Ff
Gf
Fc
Gc'
de
2.1 Yoneda Lemma and the Causal Reproducing Property
UC rests on the Yoneda Lemma - any object in a category can be defined by the interactions it makes with other objects (upto isomorphism). In the setting of causal inference, it means that objects in a TCM category can be ascribed "meaning" through studying the arrows of the category, without having to "look inside" the object. The Yoneda Lemma states that the set of all morphisms into an object d in a category C, sometimes denoted as Homc(-, d), or as C(,,d), denoted as the presheaf, is sufficient to define d up to isomorphism The category of all presheaves forms a category of functors, and denoted Ć = Set This category forms a topos, and will be fundamental to the TCM framework.
For any functor F:C- Set, whose domain category C "locally small" (meaning that the collection of morphisms between each pair of objects forms set), and any object c in C, there is a bijection Hom(C(-,c), = Fc that associates natural transformation C(-,c) = F to the element (1(1.) Fc. This correspondence is natural in both c and F.
Lemma 1. [Macaane] 1971 Riehl 2017 Yoneda lemma:
Definition 1. [Rihll] 2017 A universal property of an object c E C in a category C is expressed by a representable functor F together with a universal element E Fc that defines natural isomorphism c(-,c) = F. The collection of morphisms C(-,c) into an object c is called the presheaf, and from the o
Yoneda Lemma, forms a universal representation of the object.
2.2 Diagrams and Universal Constructions
A key distinguishing feature of category theory is the use of diagrammatic reasoning. However, diagrams are also viewed more abstractly as functors mapping from some indexing category to the actual category. Diagrams are useful in understanding universal constructions, such as limits and colimits. Briefly, a diagram J C is a functor F from some finite category I into a category of interest, C. For example, I = example of "pullback" diagram. Here the refer to abstract objects that are mapped into concrete objects in C by the functor F. What we want to know whether particular diagram
- is an
3

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F or an entire class of diagrams is "solvable". What this means is whether its limit or colimit exists, that is, is the category complete or co-complete? For any object c E C and any category J, the constant funclor : J - C maps every object j of to c and every morphism f in to the identity morphisms 1,- We can define a constant functor embedding as the collection of constant functors A - Cl that send each object in C the constant functor at c and each morphism f C - c' to the constant natural transformation, that is, the natural transformation whose every component defined to be the morphism f.
Definition 2. [Riehl 2017 A cone over a diagram F | | C with the summit or apex c E C a natural transformation A c = F whose domain is the constant functor at c. The components (A; jer of the natural transformation can be viewed as its legs. Dually, cone under F with nadir is a natural trausformation À C whose legs are the components (Az: : Fj jej-
If
Fj
Fk
Aj
A
Ff
Fj
Fk
C
Cones under diagram are referred to usually as cocones. Using the concept of cones and cocones, we can now formally define the concept of limits and colimits more precisely.
Definition 3. [Richl 2017| For any diagram F J - C, there is functor Cone(-, CV - Set, which sends c é to the set of cones over F with apex c. Using the Yoneda Lemmaa a limit of F is defined as an object lim F E C together with natural transformation A lim F - F, which can be called the universal cone defining the natural isomorphism C(-,lim F) = Cone(-, F). C - Set that maps object E C to
Dually, for colimits, we can define a functor Cone(F,-) the set of cones under F with nadir c. A colimit of F is a representation for Cone(F, Once again, using the Yoneda Lemma, colimit is defined by an object ColimF E C together with natural transformation À : F colimF, which defines the colimit cone as the natural isomrrphism C(colimE,-) Cone(F,-)
Limit and colimits of diagrams over arbitrary categories can often be reduced the case of their corresponding diagram properties over sets.
2.3 The Universality of Diagrams and the Causal Reproducing Property
We state two key results that underly UC [Mahadevan] 2023]. While both these results follow directly from basic theorems in category theory, their significance for causal inference is what makes them particularly noteworthy. The first result pertains to the notion of diagrams as functors, and shows that for the functor category of presheaves, which is a universal representation of causal inference, every presheaf object can be represented as a colimit of representables through the Yoneda Lemma. This result can be seen as generalization of the very simple result in set theory that each set is union of one element sets. The second result is the causal reproducing property, which shows that the set of all causal effects between two objects is computable from the presheaf functor objects defined by them. Both these results are abstract, and apply to any category representation of a causal model.
Theorem 1. [MacLane and leke Moerdijk] 1994 [Mahadevan] 2023 Universality of Diagrams in UC: In the functor category of presheaves Set"". eeeery functor object is the colimit of a dingram of representable objects, in canonical way.
To explain the significance of this result for causal inference, note that UC represents causal diagrams as functors from an indexing category of diagrams to an actual causal model. The theorem above tells us that every presheaf object can be represented as a colimit of (simple) representable objects, namely functor objects of the form Home(-,c).
Theorem 2. [MacLane] 1971, Mahadevan 2023] Causal Reproducing Property: All causal influences between any Two objects c and d can be derived from ils funchor objects, namely
Home(c, d) Nat(Home(-,c)) Homc(-,d))

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Any causal influence of an object upon any other object d can be represented as natural transformation (a morphism) between two functor objects in the presheaf category C.
3 Topos Causal Models
A topos [Johnstone 2014 is a "set-like" category which generalizes all common operations on sets. Thus, the generalization of subset is subobject classifier in a topos. To help build some intuition, consider how to define subsets without "looking inside" a set. Essentially, a subset S of some larger set can be viewed as a "monic arrow" (an injective function). Monic arrows generalize injective functions.
Definition& 4. [Goldblatf 2006 An arrow f: a - in a category C is called monic if given a, the equality f o g = f implies that g h, namely f is
any parallel arrows 8, "left-cancellbble..
Our approach builds on this abstraction to define a category CTCM whose objects are causal models, such as SCMs or Markov categories, and submodel M, of an SCM M is simply monic arrow fx: M2 < M.
Definition A calegory C has binary products for every of objects, cadd d, there exists third object, e = с x d, along with two projection arrows,, P1 ee c and P2 e- d, such that for any other object a and arrows f a C and g - d, there exists a unique morphism u : a satisfying p1 1I 11==1 f and p2
=.
Definition& 6. |MacLane and leke Moerdijk 1994 A category C with binary products has exponen-
tial objects if for each pair of objects c, d in C, there exists an object e that defines the following bijection:
Cle d, Cle,
Definition 7. [MacLane and leke Moerdijk] 1994 A category C is Cartesian closed if has binary products, a termunal object 1,, and exponential tal objects.
Definition anecLane and leke Moerdijk 7994| In a category with finite limits, a subobject classifier is C-object D and C-arrow true T Q such that to every monic arrow S = X in C, there a unique arrow & that forms the following pullbuck square:
S
Y
m
true
X
Q
This commutative diagram enforces the constraint that every monic arrow m (i.e., every 1 - function) that maps a subobject S to an object must be characterizable in terms of "pullback", particular type of universal property that is a special type of limit. In the special case of the category of sets, subobject classifiers are defined through the characteristic (Boolean-valued) function o that defines subsets. In general, we show below, the subobject classifier Ω for causal models is not Boolean-valued, and requires using intuitionistic logic through Heyting algebra. This definition can be rephrased as saying that the subobject functor is representable. In other words, a subobject of a causal model X in category CrCM is an equivalence class of monic arrows S < X.
Definition 9. MacLane and leke Moerdijk [1994] An elementary topos is calegory C that is Cartesian closed and has a subobject classifier.
For example, the category of sets forms a topos. Limits exist because one can define Cartesian products of sets, and colimits correspond to forming set unions. Exponential objects correspond to the set of all functions between two sets. Finally, the subobject classifier is simply the subset function, which induces a boolean-valued characteristic function.
Definition 10.
The calegory CrCM of topos causal models is defined as collection of objects E CTCM each of which is triple (U, V,F) where V (Vi...., Vnl is set of endogenous
5

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variables, U is a set of exogenous variables, and F is function from U to V. The arrows CTeMiC, d) are defined through commutative diagrams as illustrated below, where f and f are the global functions induced by the TCM objects and d, respectively.
u
U
h
8
>
V
A submodel c' (U'VVV V F') of c is any subobject of c. The effect of an intervention by some submodel c'. Finally, let Y be a variable in V, and let be a subset of V. The potential outcome in response to an intervention on modeled by submodel c' - c is the solution of Y in the submodel c'.
is given
A commutative diagram, as the term suggests, is a structure showing the equivalence of two paths. Here, the diagram asserts that g 8of f = ft ht In the context our category CSCM, the arrow f u V is simply an SCM M, and f is its induced mapping from exogenous to endogenous variables. Similarly, f also the induced function mapping exogenous to endogenous variables for another SCM M'. The morphisms /t and are functions on SCMs, which transform one causal model into another. In the specific case we are interested in, these functions define causal interventions, but in general, they may be arbitrary functions.
For completeness, we define category Csси whose objects are indeed SCMs.
Definition 11. The category CSCM of structural causal models is defined as a collection of objects, each of which is triple (U, V, where V = IV,,..., Vnl is set of endogenous variables, U is set of exogenous variables, F a set (fi,..., fu) of "local functions" fi: (UUVV \ Vi) - V, whose composition induces unique function F from U to V. Let X be subset of variables in V, and x be parlicular realization of X. A submodel M, = (U, V,F&) of M is the causal model M, {U, V,,) where Fx = lff Vi X] U (X- x). The effect of an action do(X = x) on M is given by the submodel My. Finally, let Y be a variable in V, and let X a subset of V. The potential outcome of in response to an action do(X = x), denoted Y, (,),), is the solution of Y for the set of equations F..
The set of arrows morphisms between two objects c and d in the category CsCM, denoted CscM(e, d), represent ways transitioning from SCM object to d. For example, if is a submodel of c, then the arrow defines do calculus causal intervention.
4 Causal Inference in Topos Category
We show in this section that a TCM category whose objects are defined as SCMs, and whose arrows correspond to commutative diagrams defining operations on causal models does define a topos. In the next section, we generalize from SCMs to consider more complex causal models over functor categories. Now, can state the first key result of this paper. Theorem The category CscM topos.
Since we have previously defined the objects and arrows of the CsCM category, to show it forms topos, we need to construct its subobject classifier. First, we need to define what "subobject" is in the category CSCM. Since SCMs can abstractly be defined as functions, let us assume that the SCM с that defines f is a submodel of the SCM c' that induces g. We can denote that by defining commutative diagram as shown below. Let us stress the difference between the commutative diagram shown below Definition 10 for arbitrary functions g and vs. the one below, where and j are monic arrows.
Proof:
U
u
f

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m
Xv
(0)
[0, 1/2, 1)
(0)
true
Figure Left: diagram showing that CSCM has pullbacks. Right: The subobject classifier Q for the topos category CSCM is displayed on the bottom face of this cube.
An element x E U', which is a particular realization of the exogenous variables in U', can be classified in three ways by defining a characteristic function 4r:
1. x E here we set (x) = 1.
2. x & but g(x) here we set (x() 2'
3. x $ and g(x) & V - we denote this by yx(x) = 0.
The subobject classifier is illustrated as the bottom face of the cube shown on the right in Figure
true(0) (0) =
t: (0, 1, - (0, 1), where t(0) t(1) = t(t) 1.
Xv is the characteristic function of the exogenous variable set V.
The base of the cube in Figure 2 displays the subobject classifier T: 1:1 -> Q2 where
T = (t', true) that maps = id to = t: (0, 1,, 1| - 10, 1/.
This proves that the subobject classifier for the category CSCM does not have Boolean semantics, but intuitionistic semantics as its subobject classifier Q has multiple degrees of "truth", corresponding to the three types of classifications of monic arrows (in regular set theory, there are only two classifications). Moving on to show the other properties of a topos are satisfied, note that the terminal object is simply the identity function id (0) (0). Now, it remains to show that CSCM has pullbacks and exponential objects.
Pullbacks in CsCM: Consider the cube shown on the left in Figure 2 Here, f. and h can be interpreted as three SCMs, each mapping some exogenous variables to some endogenous variables. The arrows i, ensure that the bottom face of the cube is commutative diagram, and the arrows p, ensures the right face of the cube is a commutative diagram. The arrow from P to Q exists because looking at the front face of the cube, Q is the pullback of i and 9, which must exist because we are in the category Similarly, the back face of the cube is a pullback of j and p, which is again pullback in Sets. Summarizing, (u, v) and (m, n) are the pullbacks of fi, and (p,4).
Sets, which has all pullbacks.
Exponential objects in CSCM: Now it only remains to check that the category has exponential objects. Let f and N. Then, we need to define the meaning of gf in CsCM, which we can define as gf: + Y, where = VVV which must exist since Sets is a Cartesian closed category that has exponential objects (i.e., Y is simply the set of all functions from V/ to V'). Also, X is the set of all arrows in CsCM from SCM M to SCM N, which the pair of functions (h, in the commutative diagram shown below. This finally proves that CSCM is a topos.
V and g : L' V' be two functions induced by SCM models M

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Causal Models Over a Topos of Sheaves
We now describe a more general categorical framework for defining causal models as topos by using the property that Yoneda embeddings of presheaves forms topos [MacLane and leke Moerdijk [994]. To ensure consistent extension into unique global function, we build on the theory of sheaves |Mac Lane and Moerdijk, 1992], which ensures local functions can "collated" together yield a unique global function. In our setting, we will construct sheaves from categories over causal models through the Yoneda embedding
(x) C
Sets CV and impose Grothendieck topology.
5.1 Grothendieck Topology Sites
Definition 12. A sieve for any object x in any (small) category C is a subobject of its Yoneda embedding J (x) = C(-,x). If 5 is sieve on x, and h + x is any arrow in category C, then
h (())) [g cod(g) hg S)
Definition 13. MMac Lane and Moerdijk 7992 A Grothendieck topology on a calegory C is a function J which assigns each object x of collection ((x) of sieves on x such that
1. the maximum sieve tx = -fflcod(f) = x) is in J(x).
2. If S E J(x) then (S) E (y) for any arrow h : - x.
3. If S E [(x) and R is any sieve on x, such that h²R E KV) for all h : - x, then R E J(C).
We can now define categories with given Grothendieck topology as sites.
Definition 14. A site is defined as pair (C, J) consisting of a small category C and a Grothendieck topology J on C.
Definition 15. The subobject classifier Q is defined on any topos Sets C as subobjects of the representable functors:
(xx)= = [S]S is subobject of C(-,x))
and the morphism true 1 Q true(x) = x for any representable x.
5.2 Universal Property of TCM over Functor Categories
Causal models, like SCMs, must represent both decomposable structure and (probabilistic) semantics. To capture this richer structure, we define TCM over funclor calegories, where every object is functor that maps structure to semantics. For example, the category of Bayesian networks can be modeled as a functor category [Jacobs et al., 2018 Fritz and Klingler 2023 from Markov category to the category FinStoch of tinite stochastic processes.
Theorem 4. Given a causal functor A -C& &, such as the Bayesian network functor FcDu, from small category C (e.g., a symmetric monoidal category such as a Markov category) to cocomplete category (e-g., the category Prob of probability spaces (see Theorem 6), the functor R from & to presheaves, given by (where C C and E
R(E) + Home(A((c,E)
has a left adjoint I. Se + E defined for each presheaf P in C as the colimit
L(P) Colim p C
where [c PPPtt is the category of elements, whose objects are pairs (c, (), where c is an object C and is an element of P(C) (recall P is presheaf, i.e., a sel-valued funcior that maps each element into set), and its arrows are (c', p') (c.p) any morphism ::?'
such that pc = p'.
Proof: Essentially, Theorem 4 is stating that there is a pair of adjoint functors L + R, defined as:
L: Sets CV 28:R
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As defined earlier, a natural transformation between two functors T : - R(E) is family (Td) of maps indexed by the objects E C, where each map T, is defined as the mapping:
T P(C) + Home(A(C),
which is natural in can also be defined set of arrows of & as (τ.(p) : A(c) - El(p) that is indexed by the objects pc, of the category fo of elements of P. This fact implies that there is a bijection
(Pt(P, R(E)) Homg(LP,
This bijection being natural in P and in E proves that L is left adjoint functor to R.
Now, let us define general causal functor mapping from decomposable symmetric monoidal category (e.g., a Markov category) to the symmetric monoidal category of probability spaces.
Definition 16. A causal functor - Prob maps from a general symmetric monoidal category C with comonoidal "copy-delete" structure (e.g., a CDU category [acobs et al. 2018] or Markow category [Fritz] 2020)) to the calegory of probability spaces Prob, where each object (0,9, is probabulity space, and the arrows are preserig maps, namely Prob(c, d), where Fe, P.) and d = (Qa, Fa. P Pa), where Prob(c, d) is such that P.(f- Pa(A) for all A
Theorem 5. For each causal funclor A -
- & from small category C deftning the structure of causal model to a cocomplete calegory & defining its (probabilisticc) semantics, there exists a colimit preserving functor L Sets - & such that A = L k, where & is the Yoneda embedding.
Proof: The proof is just a special case of Corollary 4 on page 43 in [MacLane and leke Moerdijk 1994). To emphasize the importance of the co-completeness condition on &, we use the following result from [van Belle, 2024] that the category of probability spaces is co-complete.
Theorem 6.
(aan Belle 2024 The symmetric monoidal category Prob has all colimits of non-emply
diagrams.
The proof, given in van Belle 2024 shows that Prob has coproducts and coequalizers. Thus, we can choose Prob as our cocomplete category & in Theorem
Proof:
We can finally state the two central results of our paper, the first (Theorem 7 establishes the universal property underlying TCMs, and the second (Theorem ( shows that causal interventions define Heyting algebra whose logic is intuitionistic.
Theorem 7.
Any causal functor F: from structural causal category C (such as Markov category) to a semantic cocomplete category & (such as Prob) factors uniquely through a TCM structure defined by the Yoneda embedding, as given in Theorem
Proof: The proof follows directly from Theorem 4
Theorem
Theorem and Definition 16
D
Definition 17. A Heyting algebra is a poset with all finite products and coproducts, which is Carfesian closed. That is, Heyting algebra a lattice, including bottom and top elements, denoted by 0 and 1, respectively, which associates to each pair of elements x and y an exponential y". The exponential written = y, and defined as an adjoint functor:
z S (x = y) if and only if z ZAxSy
In other words, x = y is a least upper bound for all those elements z with z A S As concrete example, for a topological space the set of open sets O(X) is a Heyting algebra. The "law of the excluded middle", meaning V true, does not always hold in a Heyting algebra.
Theorem 8. For any TCM category defined Č S Td by the Yoneda embedding & (c) of small causal calegory C, the partially ordered set (ubx(P) of subobjects generated by causal interventions on any causal functor defined by the presheaf is a Heyting algebra.
This result follows directly from the corresponding result for any category of presheaves (see [Ma and leke Moerdijk, 1994)), and is based on constructing the complete lattice Sub(P) of all subtunctions P Fuusing a pointwise operation for each object e C, which can be shown to satisfy an infinite distributive law.
Proof:

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6 Causal Mitchell-Bénabou Language and its Kripke-Joyal Semantics
The Causal Mitchell-Bénabon language (CMBL) is typed local set theory Bell 1988 whose syntax and semantics is defined using the arrows of the CrCM topos. The types of CMBL as causal model objects M of CTCM- For each type M, we assume the existence of variables XM, .M where each such variable has as its interpretation the identity arrow 1 M - M. We can construct product objects, such as A x B C, where terms like that define arrows are given the interpretation o X B X C D.
Each variable XM of type M is a term of type M, and its interpretation is the identity XM = 1: M + M, Here, M may represent an entire SCM, an individual variable such as Overpopulation in Figure [ or a causal functor mapping a Markov category to the cocomplete Prob category.
Terms and т of types C and D that are interpreted as a : A - Cnd т B - D can be combined to yield a term (o, 7) of type C D, whose joint interpretation is given as (op, Tq) X - C x D, where X has the required projections p: X - A and q B. causal intervention modeled as an arrow f X Y in CrCM can be
composed with term o: - X to yield a term of type Y as f foo: u x Y.
Terms of type Q are defined as formulae of CMBL and can be combined with the usual logical connectives A, V, =, and quantifiers V, 3 to obtain further terms again of type Q. An expression such as Vx (xxx, y) is interpreted by an arrow Y - Q (since x is not free variable). A formula 4xxx, y) in the topos Creи is defined to be universally valid in the topos if the corresponding arrow (xx y) : X x Y - Q factors through true: 1 Q. A formula y without free variables is interpreted as arrow 4 1 Q and is valid it coincides with the arrow true: - Ω
Indirect proofs (i.e., reductio ad absurdum) cannot be used CMBL because the rule of the excluded middle & V -u is not in general valid, nor is the axiom of choice generally true. Instead, the rules of intuitionistic predicate calculus need to be used, illustrated by de Araujo Fernandes and Haeusler [2009].
The Sripke-Joyal semantics for CMBL is specified using generalized elements. We - M. Thus, generalized element a: N - M represents the "stage of definition" of M by N. We specify the semantics of how an ICM model N supports any formula ((a), denoted by N Φ(a) by N + Φ(a) if and only if Im a S (x]((x)). Stated in the form of commutative diagram, this "forcing" relationship holds if and only if a factors through (xlΦ(x)). See MacLane and leke Moerdijk 1994 Bell 1988) for additional details.
define an element of a causal model by the morphism
(xx(((x)
true
x(x
N
>
M
>
Ω
7 Limitations and Future Work
There are Tigittioa limitations of our TCM framework, which we are currently investigating. We have recently developed an intuitionistic generalization of Pearl's do-calculus termed j-stable causal inference, which uses the nawvere-Tierree topology on a topos defined by a modal operator on the subobject classifier Q |Mahadevan] 2025a]. In this paper, we define an intuitionistic logic called j-do-calculus, where we replace global truth with local truth defined by Kripke-Joyal semantics. We are currently working on another paper Mahadevan, 2025b that implements j-do-calculus with well-known causal discovery procedures (e.g., score-based and constraint-based methods) [Zanga and Stella 2023 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.
10

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Acknowledgments
This research has been funded by Adobe Corporation.
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Appendix: Category Theory Background
We give an introduction to categories and functors in Section Section B gives brief overview of the theory of sheaves over toposes. Section C defines local set theories, another way to characterize the internal language of topos. Section R contains overview of affine CDU and Markov categories, which have been previously studied as categorical models of causality and probability [aacobs et al. 2018 Fritz and Klingler 2023
A Categories and Functors
Functor
Category
Category
Figure 3: Categories are collections of objects, with a collection of arrows defined between each pair. functor must map both objects and arrows from domain category into co-domain category.
Category theory perhaps the most transformative rethinking of mathematics since antiquitt. Rather than focus on the internals of an object, like the elements of a set, it focuses on the arrows or morphisms that define the interactions between objects. Applied to causal models, it suggests that we can construct languages where individual causal models can be defined as variables in a logical language with intuitionistic semantics. The Yoneda Lemma, one of the most celebrated results in pure mathematics of the 20th century, states that objects are completely defined up to isomorphism purely in terms of their interactions, Applied nonetheless startling. SCMs in effect can be defined not just
to SCMs, its implications in terms of their internal structure, but in terms how they interact with other SCMs in category of such objects. Being conditioned to think in terms of internal structure, this statement may seem counterintuitive. But, as category theory has shown in numerous cases, such as in algebraic topology May| [1992], we can often understand deep properties of complex objects such as topological spaces by modeling them as combinatorial objects and analyzing their interactions. Our framework builds the theory of categories, functors, and natural transformations [MacLane] 1971 Riehl 2017 (see Figure and Figure Many common mathematical structures groups, rings, modules, measurable spaces, topological spaces etc. form categories. More interesting categories for Al are those associated with compositional machine learning models, such as deep learning [Fong et al. 2019], where symmetric monoidal category Learn is defined that combines an 1mple ement routine that maps an input A into some output B parameterized by some parameter P, and an Update routine that given an input-output pair A, B and a parameter P, returns new parameter P. Causal models, such as DAGs, structural equation models etc. can be straightforwardly mapped into categories.
More formally, category C is collection of abstract objects e C. Anything technically can count as an object, from a variable in a causal model to an entire model itself. Each category is additionally specified by a set of arrows or morphisms C(c, d) between each pair of objects and d. There is an identity arrow 1, E C(c, c). Arrows compose in the obvious way, C(c, e). An initial object c in category C defined as one
inducing function C(c, d) x e(d,e) inducing a unique arrow from to every object in category C. A terminal object, usually denoted by 1, is one that defines unique arrow from every object in category into 1. An object e is isomorphic to another object d, denoted c = d, if two arrows ff - d and g d - c exist, such that g 1,, and fo gog- = 1 A functor F C - between two categories C and D is specified by an object function mapping each e C to Fc E D, and an arrows function mapping each arrow f T C(c, d) to Ff D(Fc, Fd). Functors come in two varieties covariant and contravariant the latter acts on the domain category by reversing the arrows. Given
20

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Set theory
Topos theory
set
object
subset
subobject
truth values {0, 1]
subobject classifier Q power object P(A) = Q24
power set P(A) = 2
bijection
isomorphims
injection
monic arrow
surjection
epic arrow terminal object initial object morphism : X
singleton set [+]
empty set elements of set X
functors, natural transformations limits, colimits, adjunctions
Figure 4: A topos is a category that generalizes set theory: subsets become subobjects and the characteristic function for subset, which is Boolean, turns into subobject classifier that may have multiple "degrees of truth".
any two functors F: C D and G - D between the same pair categories, we can define a mapping between F and G that is referred to as natural transformation. These are defined through a collection of mappings, one for each object of C, thereby defining morphism in D for each object in C.
Fe
a:
Gc
Ef
Gf
Fd
Ge'
0
1971 states that any set-valued functor F C Sets can be modeled in terms natural transformations between F and the Yoneda embedding C(x,-) (or C(-,x)) of a representable functor. In essence the action of set valued functor on an object x completely determined by natural transformations with C(-,x). What are natural transformations? These specify how functors interact, much like how objects interact through arrows. remarkable property of category theory is that the set of all sets is not set, but the category of all categories is indeed category, where the arrows are functors, and the objects are categories.
The celebrated Yoneda Lemma MMcLane
Definition 18. Given categories C and D, and functors F.G: C + D, natural transformation = G is defined by the following data:
arrow a, Fc - Gc in D for each object c E C, which together define the components of the natural transformation.
For each morphism - c', the following commutativee diagram holds true:
Fc
Gc
Ff
Gf
Fc'
> Gd'
natural isomorphism a natural transformation a in which every component a, an isomorphism.
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A.1 Yoneda Lemma
The Yoneda Lemma states that the set of all morphisms into an object d in category C, denoted as Homc(-,d) and called the contravariant functor (or presheaf), is sufficient to define d up to isomorphism. The category of all presheaves forms a calegory of functors, and is denoted Č = Set The Yoneda lemma plays a crucial role in this paper because it defines the concept of universal representation in category theory. We first show that associated with universal arrows is the corresponding induced isomorphisms between Hom sets of morphisms in categories. This universal property then leads to the Yoneda lemma.
D(r,)) $
Src,
pwr) d(r,d)
Lassp)
4u
Sd)
As the two paths shown here must be equal in a commutative diagram, we get the property that bijection between the Hom sets holds precisely when (r, - Sr) is a universal arrow from c to S. Note that for the case when the categories and D are small, meaning their Hom collection of arrows forms set, the induced functor Home(c, S-) to Set is isomorphic to the functor (rmp(r, -) This type of isomorphism defines a universal representation, and is at the heart of the causal reproducing property (CRP) defined below.
Lemma 2. Yoneda lemma: For any functor F: C + Set, whose domain category C is "locally small" (meaning that the collection of morphisms between each pair of objects forms a set), any object c in C, there is bijection
Hom(C(c,-)) = Fc
that defines natural transformation C(cc,) F to the element a,(1.) E Fc. This correspondence is natural in both c and F.
There is of course dual form of the Yoneda Lemma in terms of the contravariant functor C(-,c) as well using the natural transformation C(-,c) = F. A very useful way to interpret the Yoneda Lemma is through the notion of universal representability through covariant or contravariant functor.
Definition 19. A universal representation of an object E in a category C is defined contravariant functor F together with a functorial representation C(-,c) = F or by a covariant functor F together with a representation C(c,-) The collection of morphisms C(-,c) into an object is called the presheaf, and from the Yoneda Lemma, forma, a universal representation of the object.
A.2 Universal Properties in Categories
A fundamental principle of category theory is to characterize objects by universal properties. Take the Cartesian product of two sets. The conventional way define this product of two sets is as the set of ordered pairs, one drawn from each set. But this definition does not specify its universal property. We explain how define universal properties below, which will be essential to understanding a topos. A key distinguishing feature of category theory is the use of diagrammatic reasoning. However, diagrams are also viewed more abstractly as functors mapping from some indexing category to the actual category. Diagrams are useful in understanding universal constructions, such as limits and colimits of diagrams. To make this somewhat abstract definition concrete, let us look at some simpler examples of universal properties, including co-products and quotients (which in set theory correspond to disjoint unions). Coproducts refer to the universal property of abstracting a group of elements into a larger one.
Before we formally the concept of limit and colimits, we consider some examples. These notions generalize the more familiar notions of Cartesian products and disjoint unions in the category of Sets, the notion of meets and joins in the category Preord of preorders, as well
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as the least upper bounds and greatest lower bounds in lattices, and many other concrete examples from mathematics.
Example 1. If we consider small "discrete" calegory D whose only morphisms are identity arrows, then the colimit of a functor F : - category D, is denoted
- C is the categorical coproduct of (D) D, an object of
Colimit F(D)
In the special case when the category C is the category Sets, then the colimit of this functor is simply the disjoint union of all the sets F(D) that are mapped from objects D E D.
Dual to the notion of colimit of a funcior the nolion of limit. Once again, if we consider small "discrete" category D whose only morphisms are identity arrows, then the limit of functor FF D - C is the categorical product of F (D) for D, an object of category D, is denoted as
Example 2.
limit F= = II F())
In the special case when the category C. is the category Sets, then the limit of this functor is simply the Cartesian product of all the sets F(D) that are mapped from objects D E D.
Category theory relies extensively on universal constructions, which satisfy universal property. One of the central building blocks is the identification of universal properties through formal diagrams. Before introducing these definitions in their most abstract form, greatly helps to see some simple examples. We can illustrate the limits and colimits in diagrams using pullback and pushforward mappings.
P
XUY
R
An example of universal construction is given by the above commutative diagram, where the coproduct object X L Y uniquely factorizes any two mappings h : - R and i: Y R, such that any mapping i: Y Y-RR R, so that h = ro fo and furthermore i r g. Co-products are themselves special cases of the more general notion of co-limits. Figure illlustatts the fundamental property of a pullback, which along with pushforward, is one of the core ideas in category theory. The pullback square with the objects u Y and Z implies that the composite mappings o must equal g' o f In this example, the morphisms f and represent a pullbuck pair, as they share common co-domain Z. The pair of morphisms f", emanating from U define a cone, because the pullback square "commutes" appropriately. Thus, the pullback of the pair of morphisms f.g with the common co-domain Z the pair of morphisms f',g' with common domain U. Furthermore, to satisfy the universal property, given another pair morphisms x, with common domain T, there must exist another morphism k T + U that "factorizes" x, y appropriately. so that the composite morphisms f = y and g' k x. Here, and U are referred as cones, where is the limit of the set of all cones "above" Z. If we reverse arrow directions appropriately, we get the corresponding notion of pushforward. So, in this example, the pair of morphisms f',g' that share common domain represent pushforward pair. For any set-valued functor 8: S Sets, the Grothendieck category of elements f can be shown to be pullback in the diagram of categories. Here, Set. is the category of pointed sets, and is projection that sends pointed set (X, x E X) to its underlying set X.
We can now proceed to define limits and colimits more generally. We define diagram F of shape in category formally as functor F: J C. We want to define the somewhat abstract concepts of limits and colimits, which will play central role in this paper in defining
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T
8
8 Set.
R5
8
8 Set
Figure 5: (Left) Universal Property of pullback mappings. (Right) The Grothendieck category of elements f of any set-valued functor 5 : - Set can be described as a pullback in the diagram of categories. Here, Set. is the category of pointed sets (X, x E X), and n is the "forgetful" functor that sends pointed set (X, X) into the underlying set X.
topos. A convenient way to introduce these concepts is through the of universal cones that are over and under a diagram.
For any object E C and any category J, the constant functor c: C maps every object of J to c and every morphism f in to the identity morphisms 1.. We can define constant functor embedding as the collection of constant functors C C/ that send each object in C to the constant functor at and each morphism f - c' to the constant natural transformation, that is, the natural transformation whose every component is defined to be the morphism f.
Definition 20. A cone over diagram F - C. with the summit or apex E C is a natural transformation A c = F whose domain is the constant functor at c. The components (A :c - Fj)ja of the natural transformation can be viewed as its legs. Dually, a cone under F with nadir is natural transformation À = c whose legs are the components (A : F, ()
Ff
Fj
Fk
Aj
Aj
Ax
Ff
Fj
Fk
C
Cones under diagram referred to usually cocones. Using the concept of cones and cocones, we can now formally define the concept of limits and colimits more precisely. Definition 21. For any diagram F: - C, there is a functor
Cone(-,F) C - Set
which sends c E C to the set of cones over F with apex Using the Yoneda Lemma, limit of is defined as an object lim F E C together with a natural transformation lim F which can be called the universal cone defining the natural isomorphism
C(-limm Cone(-,F)
Dually, for colimits, we can define a functor
Cone(F,-): Set
that maps object c E C to the set of cones under with nadir A colimit of is a representation for Cone(F,-). Once again, using the Yoneda Lemmna, a colimit is defined by an object ColimF C together with a natural transformation A. colimF, which defines the colimit cone as the natural isomorphism
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C(colimE; Cone(F,-)
Limit and colimits of diagrams over arbitrary categories can often be reduced to the case of their corresponding diagram properties over sets. One important stepping stone is to understand how functors interact with limits and colimits.
Definition 22. For any class of diagrams K - C, a functor C
preserves limits if for any diagram K J C. and limit cone over K, the image of the cone defines limit cone over the composite diagram -
reflects limits for any cone over diagram K J C whose image upon applying is limit cone for the diagram FK : D is a limit cone over K
creates limits if whenever FK : J D has a limit in D, there is some limit cone over FK that can be lifted to limit cone over K and moreoever F reflects the limits in the class of diagrams.
To interpret these abstract definitions, it helps to concretize them in terms of specific universal construction, like the pullback defined above c' - c c" in C. Specifically, for pullbacks:
A functor preserves pullbacks if whenever p the pullback of c c" C,
it follows that Fp is the pullback of Fc' - Fc - Fc" in D.
A functor F reflects pullbacks if is the pullback of d' - - d" in whenever Fp is the pullback of Fc' - Fc - Fc" in D.
A functor creates pullbacks if there exists some that the pullback of + C c c" in C whenever there exists such that d is the pullback of Fc' - Fc Fe" in F.
A.3 Symmetric Monoidal Categories
Categorical models of causality Fong 2012 Fritz and Klingler 2023 Jacobs et al. 2018 Mahadevan 2023 are usually defined over symmetric monoidal categories, which we briefly review now [MacLane] 1971, Richter 2020].
Definition 23. A monoidal category is a category C together with a functor & : CCx x C - C, an identity object of C and natural isomorphisms a, A, defined as:
ac,,c &
(C10C))CC.
Ac:e8C = C,
C:e C,
The natural isomorphisms must satisfy coherence conditions called the "pentagon" and "triangle" diagrams [MacLane| 1971]. An important result shown in [MacLane] 1971] is that these coherence conditions guarantee that all well-formed diagrams must commute. There are many natural examples of monoidal categories, the simplest one being the category of finite sets, termed FinSet in [Fritz, 2020], where each object C. is a set, and the tensor product is the Cartesian product of sets, with functions acting arrows. Deterministic causal models can be formulated in the category FinSet. Other examples include the category of sets with relations as morphisms, and the category of Hilbert spaces [Heunen and Vicary) 2019]. Markov categories Fritz and Klingler 2023) are monoidal categories, where the identity element is also terminal object, meaning there is a unique "delete" morphism de X + associated with each object X. [Fritz 2020 shows they form unifying foundation for probabilistic and statistical reasoning.
Definition 24. A symmetric monoidal category is a monoidal category (C, (,,,,,,,, a,,,a,,, p, together with natural isomorphism
TC,C, C, @ C2 = C2 C1, for all objects C1, C2
where 7 salisfies the additional conditions: for all objects C₁, C2 TC2. TC, 10,8,, and for all objects C, pC Ac TC,: CéezC
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An additional hexagon axiom is required to ensure that the ? natural isomorphism is compatible with a. The t operator is called "swap" in Markov categories [Fritz [020]. In most cases of interest in AI, the symmetric monoidal categories are enriched over some convenient base category V including vector spaces, or preorders such as the unit interval [0, 1I, where the unique morphism from - b exists if and only if a - b. 5
Definition 25. A V-enriched category consists of a regular calegory C, such that for each pair of objects x and y in C, the morpkisms C(x, y) E V, often referred to V-hoo object. For the case when (calV, S, 8, 1) is a commutative monoidal preorder, we have the following conditions
1 1 x(x,x)
C(y,z) @ C(x,y) 2(x,z)
A.4 Topos Category
To help build intuition for topos, let's understand why sets are special as category. They have all limits and colimits, meaning that one can always construct the categorical product of two sets as the Cartesian product, and the disjoint union as the coproduct. They have all pullbacks, which are commutative diagrams that define in essence a categorical definition of products and coproducts. In terms of the Yoneda Lemma, a universal property is either an initial or final property in a category of diagrams (functors). The product is the final object in category of diagrams, and the coproduct is the initial object. Every concept in category theory essentially reduces down to these simple notions. Sets also have exponential objects: the set of all functions between two sets is once again a set! They also have a subobject classifier: each set has subsets, defining its parts, which correspond to a characteristic function that evaluates to true for elements in the subset. Topos theory generalizes all these properties to categories. defines subobject of an arbitrary object. A causal intervention subobject of that SCM object.
important to causal inference because a causal intervention SCM produces
To define a topos, we need to go through a few more definitions.
Definition 26. An object x in a category C is called initial if there is a unique morphism f:x to every other object y in the category. Dually, an object is called final if there is unique morphism f: x into x.
In the category Sets, the null or empty set 0 is the initial and the single element set (+) is the final object. From an empty set, there is only one function possible to any other set, namely the empty function. From any set there is exactly one function into the single element set. A topos generalizes the property of subobject classifiers in Sets. Given any subset S c X, we can define as the monic arrow S + X defined by the inclusion of S in X, or as the characteristic function Øs that is equal to 1 for all elements x E X that belong to S, and takes the value 0 otherwise. We can define the set (0, 1/ and treat true as the inclusion [1] in The characteristic function Φs can then be defined as the pullback of true along Øs.
S
1
m
true
X
We can now define subobject classifiers in category C as follows.
Definition 27. In a calegory C with finite limits, a subobject classifier is a monic arrow true 1 Q such that to every other monic arrow S - X in C, there is a unique arrow o that forms the following pullback square:
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S
X
This definition can be rephrased saying that the subobject functor is representable. In other words, a subobject of an object x in a category C an equivalence class of monic arrows -
Definition 28. A topos is category & with
1. A pullback for every diagram X - B
2. A terminal object 1.
3. An object Q and a monic arrow true - Q such that any monic m S = B, there is unique arrow Φ B in & for which the following square is pullback:
s
true
X
4. To each object x object Px and an arrow x Px Q such that for every arrow
f ::x x - Q there a unique arrow g : Px for which the following diagrams commute:
f
x xxy
>
Q
1xg
Px
X Px
Q
Let us understand these definitions in the category of Sets. Clearly, the single point set (+) is terminal object for Sets, because there is unique function from any set Š to single element set ,, and the categorical product of two sets
is just the Cartesian product. Furthermore, given two sets A and B, we can define B as the exponential object representing the set of all functions f f B. We can define exponential objects in any category more generally as follows.
Definition 29, Given any category with products, for a fixed object x in C, we can define the functor
XX-:-C
this functor has a right adjoint, which can be denoted as
c-
then we say x is an exponentiable object of
Definition 30. category Cartesian closed it has finite products (which equinalent to saying it has terminal object and binary products) and if all objects in C are exponentiable.
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Graphs as Topos
H
E
MM 10.1)
8
Figure 6: Causal models over arbitrary directed graphs define topos. classifier is illustrated on the right.
The subobject
A.5 The Topos of Causal Graphs
To begin with a relatively elementary construction involving sheaves, T how directed graphs, widely used in causal modeling, can be modeled topos that only has two objects: a generic "vertex" object V, representing an abstract causal variable, and generic "edge" object E, denoting an abstract causal path. Any actual graph, such the two shown in the figure, are covariant functors from Cr, the graph topos, to the actual graphs (we can treat graphs equivalently to E). Sample object mappings of V for the two graphs are shown. For the topos category Cr, the "representable functor" is defined as the presheaf Cr(-,c) for each object in the category r, which means the set of all morphisms going into object c. Let us calculate the representable functors for the topos of graphs Cr. Since Cr has only two objects, V and E, the representable functors are given as the sets:
contravariant functors by reversing the edges from V
Cr(-,V), Cr(-,E)
Since Cr has no arrows from object E to object V, we can easily check that the representable functor Cr(-, is given as:
Cr(V [1vl, Cr(E, =)=
where lv is the self-loop arrow that maps object V to itself. On the other hand, the representable functor E) is defined as:
Cr(V, f), Cr(E, = (1g)
Additional details, like constructing exponential objects H for the Cr category, are given in |Vigna| 2003]. For the subobject classifier, the idea as follows. For any causal model N represented as submodel of a more complex causal model M, defined by monic arrow m N M, the generalization of the usual set-theoretic characteristic function is the
classifying map Xm N - Q. As in Theorem 3, Q is not Boolean, but has multiple "degrees of truth" (see Figure
1. Causal variables not in N are mapped to Ov.
2. Causal variables in N are mapped to V.
3. If an edge is in N, four cases emerge:
(a) An whose source and target are not in N is mapped to Og-
(b) An whose source is in N, but target is not, is mapped to edge
(c) An are whose target is in N, but source is not, mapped to an edge f.
(d) An arc having both source and target in N is mapped to ()
B Sheaves and Toposes: Categories of Functors
general way to construct a topos category is through covariant Yoneda embeddings
J C + SetC, or contravariant Yoneda embeddings k C - e In simpler terms,
²() V+V E is map defined by the universal property of coproducts.
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each object x in the category C is either mapped to the functor C(x, C - Sets or C(-,x) C Sets. These structures are called presheaves or copresheaves. To construct proper sheaf, we need to include an additional condition that is illustrated in Figure] 7 The sheaf condition plays an important role in many applications of machine learning, from dimensionality reduction [Mcinnes et al. 2018] to causal inference [Mahadevan] 2023
Mac Lane and Moerdijk 1992 provides an excellent overview of sheaves and topoi, and how remarkably they unity much of mathematics, from geometry to logic and topology. For causal inference, the structure of causality shapes the arrows in a causal model, such as Markov category, and that imposes Grothendieck topology with associated internal logic. Logic and causality are interwoven in ths sense.
Figure Two applications of sheaf theory in Al: (top) minimizing travel costs in weighted graphs satisfies the sheaf principle, one example of which is the Bellman optimality principle in reinforcement learning [Bertsekas] [219 (bottom): Approximating a function over topological space must satisty the sheat condition.
Chicago
New York
San Francisco
F(C)
A
F(A)
F(B)
7 two concrete examples of sheaves (in both cases, these are enriched sheaves). In a minimum cost transportation problem, say reinforcement learning [Bertsekas] 2019)) any optimal solution has the property that any restriction of the solution must also optimal. In RL, this sheaf principle is codified by the Bellman equation, and leads to the fundamental principle of dynamic programming [Bertsekas] 2019]. Consider routing candy bars from San Francisco to New York city. If the cheapest way to route candy bars is through Chicago, then the restriction of the overall route the (sub) route from Chicago to New York City must also be optimal, otherwise it is possible to find shortest overall route by switching to lower cost route. Similarly, in function approximation with real-valued functions F:C R, where C is the category of topological spaces, the (sub)functions F(A), F(B) and F(C) restricted to the open sets A, B and must agree on the values they map the elements in the intersections A n B, A C, A n B n C and
Sheaves can be defined over arbitrary categories, and we introduce the main idea by focusing on the category of sheaves over Sets.
Definition 31. [Mac Lane and Moerdijk 1992 A sheaf of sets F on topological space X is functor F 8P Sets such that each open covering U U, U, I of an open set of X yields an equalizer diagram
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Figure 8: Sieves are subobjects of of J (x) Yoneda embeddings of category C, which generalizes the concept of sheaves over sets in Figure
Sieves are subobjects of Yoneda embeddings
FU S
I1 Fu:
P
Il
F(U: n Uj)
The above definition succinctly generalizes the idea in SCMs of combining local functions to get unique global function.
Definition 32. The category Sh(X) of sheaves over a space X is a full subcategory of the functor category OCX
B.1 Grothendieck Topologies
We can generalize the notion of sheaves to arbitrary categories using the Yoneda embedding } (x) = x)-,x). We explain this generalization in the context of a more abstract topology categories called the Grothendieck topology defined by sieves. A sieve can be viewed subobject S S (x) in the presheaf Se but we can define it more elegantly as a family of morphisms in C, all with codomain x such that
f
Figure 8 illustrates the idea of sieves. A simple way to think of a sieve is as right ideal. We can defne that more formally as follows:
Definition 33. If S a sieve on x, and h: D x is any arrow in category C, then
I (S) 1g cod(g) D, e S)
Definition 34. (Mac Lane and Moerdijk 1992 A Grothendieck topology on category C is function J which assigns each object x of C collection J(xx of sieves on x such that
1. the maximum sieve tx = -fflcod(f) x) is in J(x).
2. If S J(x) then ht( E [(D) for any arrow h D
3. If S J(x) and R is any sieve x, such that Ir(()) (D))for all h D x, then R J(C).
We can now define categories with given Grothendieck topology as sites.
Definition 35. A site is defined as pair (C, consisting of a small category C and a Grothendieck topology J on C.
An intuitive way to interpret a site is as generalization of the notion of topology on space X, which defined set X together with collection of open sets O(X). The sieves on category play the role of "open sets".
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B.2 Mitchell-Bénabou Language
We define the Mitchell-Bénabou language (MBL), typed local set theory (see Section associated with causal topos. Given the topos category Co, we define the types of MBL as causal model objects M of Co. For each type M (e.g., an SCM), we assume the existence of variables XM, VM,..., where each such variable has as its interpretation the identity arrow x C, where terms like a that define arrows are given the interpretation A B X C - D. We can inductively define the terms and their interpretations in a topos category as follows (see MMc Lane and Moerdijk
1 M M. We can construct product objects, such as
1992 for additional details):
of type M, and its interpretation the identity XM = 11 M - M (e.g,, M may be an SCM or causal model on Markov category).
Each variable XM of type M is
Terms a and t of types and that are interpreted as a - C and t B can be combined to yield term (a, 7) of type Cx D, whose joint interpretation is given as
(op, Tq) X C
where X has the required projections p A and x
Terms A B and T C - B of the same type B yield term 0 7 of type Q interpreted as
(o= W (apzu)
Bx B
Ω
where öy is the characteristic map of the diagonal functor AB - B x B. These diagonal maps correspond to the "copy" procedure in Markov categories Fritz 2020].
Arrows f:A A- and a term a: C A of type A can be combined to yield a term foo of type B, whose interpretation is naturally a composite arrow:
foo: Co::c444
For exponential objects, terms 0 A B and D C of types B and C, respectively, combine to give an "evaluation" map type B, defined as
0()) W - B
where is the evaluation map, and W defines map (0p, op),oq), where once again
p: W - A and q: W - D are projection maps.
Terms a: A - B and r: D - Ω combine to yield term E r of type Q, with the following interpretation:
0 O
(dp.zg)
B BXQBS Ω
Finally, can define local functions A objects, such as
Axco : - BC
where xe is variable of type Cnd a: Cx B.
We combine terms a etc. of type Q using logical connectives A, ,,,, as well as quantifiers, to get composite terms, where each of the logical connectives is now defined over the subobject classifier Q
A 1xQ0 - Q is interpreted as the meel operation in the partially ordered set of subobjects (given by the Heyting algebra).
V Q - Q is interpreted as the join operation in the partially ordered set of subobjects (given by the Heyting algebra). This operation gives the definition of disjunction of two properties.
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=: Q - Q is interpreted as an adjoint functor, as defined previously for Heyting algebra. Thus, the property of implication over SCMs is modeled an adjoint functor.
We can combine these logical connectives with the term interpretation as arrows, relegating some details to Mac Lane and Moerdijk 1992]. We now turn to the Kripke-Joyal semantics of this language.
B.3 Kripke-Joyal Semantics for Causal Topos
We now define the Kripke-Joyal semantics for the Mitchell-Bénabou language of causal topos. Any free variable x must have some causal model X of Ca as its type. For any causal model M in Co, define generalized element definition, note that we can define an element of a causal model by the morphism x - M. Thus, a generalized element a N M represents the "stage of definition" of M by N. We specify the semantics of how an SCM N supports any formula ((a), denoted by IF <(a), as follows:
morphism
N - M. To understand this
N + ((a) if and only if Im a S (xlσ(x))
Stated in the form of a commutative diagram, this "forcing" relationship holds and only factors through (xlσ(x)), where x is variable of type M (recall that objects M of topos form its types), as shown in the following commutative diagram.
(x)(())
True
((x)
N
M
Ω
This diagram provides an interesting way to define causal interventions causal topos, because it defines submodels of M. Building this definition, if a, N - M are parallel arrows, we can give semantics to the formula ß by the following statement:
$ N
AM
M
M
Q
following the definitions in the previous section for the composite (a, B) and ōx in the Mitchell-Bénabou language. We can extend the previous commutative diagram to show that U I a = B holds if and only if (a, B) factors through the diagonal map A:
M
1
True
dy
N
M
M
Ω
(n,9)
Monotonicity: If U F (xx), then pullback the interpretation through any arrow f Li
u in a topos to obtain U' If ςaa o f).
[x(((x)
True
600
U
U
X
Ω
Local character: Analogously, if L' U is an epic arrow, then from L' ó(a f), we can conclude U I- (xx).
'The concept of "forcing" is generalized from set theory Mac Lane and Moerdijk 1992
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Theorem 9. If - M is a generalized element of causal model M, and ((xx) and (xxx) are formulas with a free variable x of type M, we can conclude that
1. N ((a) (b)) holds if I <(a) and N + ((a).
2. N + ((x) V (xxx holds if there are morphisms p O - N and q - N such that p p+9 9- N + O - M is an epic arrow, and N IF ((ap) and O It p(aq).
3. N It ((a) = ( if it holds that for any morphism p : N - M, where N IF <(ap), the assertion N + ((app) also holds.
4. N (a(a) holds if whenever the morphism P:M- - N satisfies the property N I- o(ap), then N 0.
5. M + 36(x, y) holds if there exists an epic arrow : N - and generalized elements B: V - Y such that M lt ((ap,B).
6. M + Vym(x, y) holds for every structural causal model N, and every arrow p N - M, and every generalized element N - O, it holds that V ((ap,p).
Proof: The proof follows readily from the general result on Kripke-Joyal semantics for the languages of any topos [Mac Lane and Moerdijk| 1992] The Kripke-Joyal semantics takes on a simpler form when using a Grothendieck topology on a topos, and we postpone the details to the Supplementary Materials.
B.4 Kripke-Joyal Semantics for Sheaves
In the main paper, we introduced Kripke-Joyal semantics can be specialized to a topos equipped with a Grothendieck topology, that is a site. This specialized structure captures how causal inference is woven in the fabric of the internal logic of causal topos. Define Sh(C,J) be a topos of sheaves with specified Grothendieck topology J, defined by the following diagram:
semaattsccss for any topos. These semantics
C - S Sh(C, C
where we know that the Yoneda embedding : creates a full and faithful copy of the original category C. Let define the semantics for sheaf element E X(C), where X(C) = Sh ()((,,)), X)). Since we know that (x|((x)) is subsheaf, and given an arrow f: of and X(C), then if is one of the elements that satisfies the property that (x]o(x)), the monotonicity property stated above implies that a f f (xl((x))( S X(D). Also, the local character condition stated above implies that if (fi : C C} is a cover in the Grothendieck topology I such that CJI I δl o fi) for all i, then C It ((aa).
With these assumptions, sheaves as follows:
restate the Kripke-Joyal semantics for the topos category of
1. C If p(a)) A ((a) if it holds that C It ((a) and C IH (a)).
2. C IH ((a) v (aaa if there a covering (fi C - C) such that for each i, either C; It ((at)
C; (a))
3. C If ((a) ~ (a) if for all f: - C, and D I Φ(a o f), it holds that D It vaa o f).
4. C -o(a) holds if for all arrows f D C in C, if D It gla o f) holds, then the
empty family is a cover of D.
5. C By Φ(x, y) holds if there is covering (fi : C C) and elements B, e Y(C.) such that C, + ạ(a fi,f.) holds for each i.
6. Finally, for universal quantification, C Vy 6(x, y) holds if for all arrows f: D C in the category C, and all f e Y(D), it holds that D t Φ(a f.
C Local Set Theory
The Mitchell-Bénabon language is an example of "local set theory" Bell 1988 Formally, the Mittchll--Bénbbo language for the Generalized Do-Calculus is a local set theory, defined
33

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by set of types that correspond to each structural causal model M object in Cn- A local set theory [Bell 1988 is defined as a language L specified by the following classes of symbols:
1. Symbols 1 and Q representing the unity type and truth-oulue type symbols.
2. A collection of symbols A, B, C, called ground type symbols.
3. A collection of symbols f, g, h,... called function symbols.
To instantiate this definition for our paper, the ground types will be SCMs, each of which will be interpreted as primitive type in Section [32 We will use the apposttheoretical constructions to construct composite types. We can use an inductive procedure to recursively construct type symbols of L as follows:
1. Symbols and Ω are type symbols (the terminal object and the subobject classifier in causal topos).
2. Any ground type symbol is type symbol. For a causal topos, each SCM is ground type symbol.
3. If Ai...., A. type symbols, so is their product A1 A.. where for 0, the type of I[ A; is 1. The product A x A, has the product type symbol. These constructs allow defining an algebra of causal models.
4. If A is a type symbol, so is PA. The type PA called the power type. We thus can give meaning to concept of "powerset" of causal model, where we interpret the subobject classifier as defining the abstract semantics powerset for each SCM.
Thus, a product of SCMs will define product types. Given an SCM M, we can define its power type as well, which is an abstract notion of the "power set" of a causal model (if you interpret this in the context of subobject classifiers, it means that we are defining a family of submodels). For each type symbol A, the language L contains set of variables XA, ZA,...... In addition, L contains the distinguished symbol. Each function symbol in L is assigned signature of the form A B. We can define the terms of the local set theory language L recursively as follows:
is a term of type 1.
for each type symbol A, variables XA, VA.... are terms of type A.
if is function symbol with signature A and is term of type A, then f(T)) is a term of type B.
If Ti....., Tn are terms of types A,...., Ag, then (Tı,... Tn) is a term type A x... Agr where if πt = 0, then (TI.... Tn) is of type
If t is a term of type A₁ x Ag, then for 1 sis n, (T), a term of type A;
if a term of type Q and XA is variable of type A, then (xx : a) is term of type PA.
if a, т are terms of the same type, o = T is a term of type Q.
if a, T are terms of the types A, PA, respectively, then a t is term of type Q.
A term of type Q is called formula. The language L does not yet have defined any logical operations, because in a typed language, logical operations can be defined in terms of the types, as illustrated below.
ß is interpreted as =
true is interpreted as * =
a B is interpreted as (a, (3) = (true, false).
a β interpreted (a () =
NNoe that topos, these will be interpreted as power objects, generalizing the concept of power sets.
*In topos, these will correspond to arrows of the category.
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Vx a is interpreted as (x a) = (x true]
false is interpreted as Yw w.
a is interpreted as a = false.
a ß is interpreted as Yw [(a = a B p= =)=aa
Ex is interpreted Va|Vx(a = @) = al
Finally, we have to specify the inference rules, which are given in the form of sequents. We will just sketch out few, and the rest can be seen in [Bell 1988]. A sequent is a formula I: where a formula, and I' is a possibly empty finite set of formulae. The basic axioms include a (tautology), equality, and another for comprehension. Finally, the inference rules are given the form:
(unity), a rule for forming projections of products, rule for
Thinning:
::a a
;,T:a
Cut:
F: a, a,T';
T:
Equitvllenc::
a, [ : 8,T a + B
A full list of inference rules with examples of proofs is given in Bell 1988 Now that we have the elements of local theory defined shown above, need connect its definitions with causal topos. That is the topic of the next section.
C.1 Counterfactuals using Topos Categories
de Araujo Fernandes and Haeusler 2009 describe an intuitionistic logic for Lewis' [Lewis] 1973] theory of countertactuals, where the neighborhood system of possible worlds is governed by the graph topos Cr illustrated in Figure 6 However, as Galles and Pearl 1988 argued, many counterfactuals have causal meaning, For example, the well-known counterfactual "If Kangaroos had no tails, they would fall over" attains causal meaning as the logic of counterfactual, proposed by Lewis evaluates its truth in the nearest possible worlds, where the laws physics are the same as our world, except for "kangaroo surgery", Lewis introduces two counterfactual connectives, and we illustrate how to model one of them in our framework:
D B is true in world according to neighborhood system Co, if either for world w in Cn(u), For a, or there is some neighborhood N Ca(u) that has a world such that Fiv & and F,
β in every world of N.
Crucially, unlike the case for de Araujo Fernandes and Haeusler 2009 our TCM framework imposes causal structure on the neighborhood system.
D Intuitionistic j-Do-Calculus on Sites
In a recent paper, we have developed an intuitionistic generalization of Pearl's do-calculus on sites Mahadevan 2025a]. / A site (C,J) is a small category C whose objects index regimes (e.g., labs, contexts) and a Grothendieck topology T that specifies which families (u: - u) "cover" stage u. The topos toposSy(C)) of J-sheaves behaves like universe of sets varying over regimes. A formula @p is Issablt at iff holds on cover of (Kripke-Joyal semantics). Intuitively: truth is verified locally and then glued. Random variables and probabilities live
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internally (as objects/morphiisms of SC)( When J is trivial, these reduce to the classical notions.
Pearl's do-calculus [Pearl 2009 is a complete axiom system for interventional identification in acyclic causal models under classical (Boolean) logic. In [Mahadevan 2025a we generalize it to j-stable causal inference inside topos of sheaves Shy(C), where regimes are sheaves on that site. We define j-
form site (C,J) and abservatioonsiiteerenttt stability for conditional independences and interventional claims as local truth in the internal logic of (C). We give three inference rules that mirror Pearl's insertion/deletion and vetton/observation exchange, and we characterize their soundness using the Kripke-Joyal semantics. companion paper [Mahadevan] 2025b], which currently in preparation, we provide concrete algorithmss for j-stable causal discovery on sites, and compare a number of popular causal discovery procedures (score-based and constraint-based) [Zanga and Stella 2023 with their j-stable variants.
E
Affine CDU and Markov Categories
In this section, we review previous work on affine CDU ("copy-delet--uniform"" categories [Cho and Jacobs 2019] and Markov categories Frittz 2020 which have been proposed as unitying categorical model for causal inference, probability and statistics. They are symmetric monoidal categories, which we reviewed above, combined with comonoidal structure on each object. Importantly, Markov categories are semi-Cartesian because they do not use uniform copying, but contain Cartesian subcategory defined by deterministic morphisms (see below). We give a brief review of Markov categories, and significant additional details that are omitted can be found in [Cho and Jacobs 2019 Fritz (2020] Fritz and Klingler [023]. The equations are written in the diagrammatic language of string diagrams, which can be shown to represent formal language that is equivalent to writing down algebraic equations [Selinger 2010].
Definition 36. Markov category [Fritz 2020 is symmetric monoidal category in which every object X E C is equipped with a commutative comonoid structure given by comultiplication copyx: X X @ X and counit delx X I, depicted in string diagrams
copy X
delx
(1)
and satisfying the commutative comonoid equations,
(2)
(3)
as well compatibility with the monoidal structure,
XOYXOY
X
Y
x
X Y X Y
X
X
and naturality of del, which means that
(5)
for every morphism f.
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Let us briefly explain these definitions. The delx: X - I is essentially like integrating over probability distribution, which always yields 1. Hence, I, the unit of the tensor product, is the terminal object in affine CDU and Markov categories. Bayes rule turns into disinlegration rule [Cho and Jacobs 2019,, which is only available Markov categories with conditionals (i.e., where one can categorically refine P(ylx) conditional distributions). Note that in the continuous case of random variables defined as measurable functions on real numbers, one has to take considerable care in defining conditioning [Halmos] 1974]. The copyx x procedure is uniform, and deterministic, meaning you take the tensor product of two variables X @ and then copy the resulting object, that's exactly the same as first copying X into X @ X and Y into Y @ Y, and then taking the tensor product, along with a swap operation (see Equation T. Only dely acts "uniformly"" meaning that if you process variable X using some function f and then delete f(X) (meaning marginalize it), that's equivalent to simply deleting X. However, copyx is not defined this way, and we discuss that subtlety below, as it will be important in understanding why Markov categories are semi-Cartesian. To convert them into a topos, we need the result to be Cartesian closed, which is why we need to use the Yoneda Lemma to construct the category of presheaves to guarantee obtaining a topos.
E.1 Cartesian Structure in Markov Categories
We now discuss subcategory of Cartesian categories within Markov that involves uniform copyx and delx morphisms. One fundamental property of Markov categories is that they are semi-Cartesian, the unit object is also terminal object. But, subtlety arises in how these copy and delete operators are modeled, as we discuss below.
Definition 37. A symmetric monoidal category C is Cartesian if the tensor product @ is the categorical product.
C and D are symmetric monoidal categories, then functor C D is monoidal if the tensor product is preserved up to coherent natural isomorphisms. is strictly monoidal if all the monoidal structures are preserved exactly, including &, unit object 1,, symmetry, associative and unit natural isomorphisms. Denote the category of symmetric monoidal categories with strict functors as arrows as MON. Let us review the basic definitions given by Heunen and Vicary [2019), which will give some further clarity on the Cartesian structure in affine CDU and Markov categories.
Definition 38. The subcategory of comonoids coMON in the ambient category MON of all symmetrri monoidal categories is defined for any specific category C as a collection of "coalgebraic" objects (X, x. delx), where X is in C, and arrows defined as comonoid homomrrphisms from (X, x delx) to copy,, dely) that act uniformly, in the sense that if : ->> Y any morphism in C, then:
(f 2 f) copyx = copy, f dely o f = delx
Heunen and Vicary [0199] define the process of "uniform copying and deleting" in the category CoMON, whuch we now relate to Markov categories. A subtle difference worth emphasizing with Definition i6 that in Markov categories, only dely is "uniform", but not copyx in the sense defined by Heunen and Vicary| [2019] This distinction can be modeled in cPROP category that is Cart-Cartnsian like Markov categories by suitably modifying the definition of the associated PROP map for copying.
Definition 39. [Heunen and Vicary 2019] A symmetric monoidal category C admits uniform deleting if there is a natural transformation ex X 2,, 1 for all objects in the subcategory CooMON of comonoidal objects, where ei = id, as shown in Equation
This condition was referred to Cho and Jacobs 2019 as a causality condition on the arrow ex. Essentially, it states that if you process some object and then discard it, it's equivalent to discarding it without processing.
Theorem 10. (Heunen and Vicary 2019 symmettrc monoidal category C has uniform deleting if
and only if 1 is terminal.
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This property holds for Markov categories, as noted in Fritz 2020 and simple diagram chasing proof is given in [Heunen and Vicary 2019).
Definition 40. [Heunen and Vicary 2019] A symmetrrc monoidal calegory Chas uniform copying
there is natural transformation copyx X X @ X such that del; PT satisfying Equation
and Equation
We can now state an important result proved in [Heunen and Vicary 2019
(Theorem 4.28),
which relates to the more general results shown earlier by Fox (1966]
Fox 1976 Heunen and Vicary 2019 The following conditions are equivalent for symmetric monodal category C.
Theorem 11.
The category C is Cartesian with tensor products @ given by the categorical product and the tensor unit is given by the terminal object.
The symmetrie monoidal category C has uniform copying and deleting, and Equation holds.
As noted by Fritz 2020 not all Markov categories are Cartesian, because their copyx not uniform, but only delx is. For example, consider the category FinStoch, where joint distribution is specified by the morphism distributions can be formed as the composite morphisms
X @ Y. In this case, the marginal
- X del,
X
1 - X
dely Y
But to require that in this case @ is the categorical product implies that the marginal distributions defined as the above composites must be in bijection with the joint distribution.
Social Impact
Causal inference addresses real-world problems with significant social impact, including healthcare, education, climate change, and economics. Discovering causal models has enormous potential for improving human lives in all these areas. However, our paper is largely a theoretical study of the universal properties that underlie causality. It does not ress concrete algorithmic questions, and is complementary the empirical literature in the field.
38


Reviews:
[{"review_id": "Z6dBA7hRqx", "reviewer": "Reviewer_giku", "summary": "This 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.", "questions": "- 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?", "limitations": "yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths\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.", "was_revised": false, "final_justification": "", "rebuttal": ""}, {"review_id": "Z0P9wQulA3", "reviewer": "Reviewer_H13Y", "summary": "The 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.", "questions": "Q1: 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.", "limitations": "yes", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 4, "significance": 4, "originality": 4, "strengths_and_weaknesses": "Overall: 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.", "was_revised": true, "final_justification": "As already written in the comment section:\nIn the causality community we only just begin to understand what causality is or what notion of causality works best. Pearl's notion is very influential, but by far not the only contender. In my view this paper could be a key insight to move this whole research line further. I agree that the paper can be improved, but in my view most papers can and this paper really has a novel contribution with -in my expectation- big impact down the road. I think that this paper deserves the visibility NeurIPS provides and I would expect it to have some strong impact down the road.", "rebuttal": ""}, {"review_id": "valWSAwAsW", "reviewer": "Reviewer_EdR2", "summary": "This 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.", "questions": "What 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.", "limitations": "In 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.", "rating": 3, "confidence": 1, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 1, "originality": 3, "strengths_and_weaknesses": "Strength:\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.", "was_revised": false, "final_justification": "", "rebuttal": ""}, {"review_id": "3oac6kuOFQ", "reviewer": "Reviewer_HARx", "summary": "Recall 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\".", "questions": "- - 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\"?", "limitations": "yes", "rating": 4, "confidence": 1, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 1, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Quality\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.", "was_revised": true, "final_justification": "Thank you for the multiple answers. Please for non-specialists, please improve the writing. I have updated my score.", "rebuttal": "Dear Reviewer HARx: \n\nThank you for your careful reading of the paper and your detailed comments. They will be invaluable in improving the paper. We appreciate the opportunity to address your concerns below. \n\nQUESTIONS regarding Quality: \n\nQUESTION 1: The results are interesting but difficult to grasp.\n\nANSWER: We agree that category theory is a novel mathematical area for causal inference researchers in AI/ML, but as we mentioned in the paper, there is a recent surge of published journal papers that are showing the usefulness of categorical thinking in causality. It will take some time for ideas like topos theory to percolate into the field of causal inference in AI/ML, but we believe this will happen in due course as more researchers become familiar with the concepts. Recognizing that causal models such as SCMs form a topos category gives new insights. For example, it shows that causal interventions can be meaningfully thought about as constructing sub-objects in a category, and there is a wealth of known properties regarding sub-object classifiers in topos theory. Similarly, it introduces for the first time to our knowledge the idea of constructing \"exponential objects\" in causal models, so we can now precisely define what it. means to exponentiate one causal model M1 over M2, or M1^M2. The topos theoretic nature of SCM categories mandates that exponential objects must exist. For example, as we described in the paper, directed graphs form a topos, and hence they have exponential objects as well (Vigna's paper on Arxiv on the topos of graphs constructs such exponential objects, and shows their relevance to  automata models).  Similarly, we believe our work will show new ways to reason about causal models using the Mitchell-Benabou language of the SCM topos category, and this will be a topic that we aim to explore in future papers. \n\nQUESTION 2: The definitions are not crystal-clean.\n\nANSWER: We apologize in advance for any lack of clarity, but we attempted within the confines of a conference paper to be as thorough as we could. We will of course address all your concerns in the revised paper regarding lack of clarity. We believe that each of our definitions can be rigorously stated and all our theorems can be proved rigorously, subject of course to limitations of space. Where possible, we intend to provide references to the published literature and textbooks on classic results. \n\nQUESTION 3: 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\nANSWER: We apologize for the deficiencies in organization and will strive to improve the organization in the revised paper. We attempted to give a concise overview of the results in the main paper, and left the more detailed definitions to the background supplementary material. In the revised paper, we will of course ensure that all key definitions are given  in the main paper. \n\nQUESTION:  The contribution is Theorem 1-4 but Section 4 for instance is very obscure.\n\nANSWER: Section 4 describes how every topos can be characterized by an internal logical language called the Mitchell-Benabou language. It is a unique aspect of a topos as a category. For causal inference, it brings new insights because it shows that we can now use the MB language to reason about causal models. This topic of course deserves a new paper in itself, and in fact, it is an entirely new line of research that has yet to be explored in causal inference. For the revised paper, we will attempt to expand this section and give some simple examples to illustrate what the MB language could reason about in causal inference. \n\n\nQUESTIONS regarding Significance\n\nQUESTION 1: The work could be of interest.\nBut the paper fails showing the significance. 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\nANSWER:  To answer your valid concern here, we can clearly state what do we better understand now by mentioning the major points made in the paper. First, showing that SCMs and other causal models form a topos category means that we can now model causal interventions in terms of sub-object classifiers. We now have a new way to understand and analyze a causal intervention, something that was not known before. Second, we can give new meaning to terms that have never been explored previously, such as \"exponential objects\" in causal inference. We can rigorously define what it means to exponentiate one causal model M1 with respect to another, M2, so we can now talk about M1^M2, building on the property that exponential objects must exist in every topos. Third, we know now that because SCMs form a topos category, they can be studied with respect to the Mitchell-Benabou logical language and given the Kripke-Joyal semantics. This gives us a new logic to study causal inference. Finally, knowing that casual models form a topos category, we can bring in the vast power of category theory in terms of functors that map a topos category in the category of topological spaces. A recent example of this type of analysis is given in the Entropy journal paper by Mahadevan (vol 7, no. 5, 2025, \"Higher Algebraic K-theory of Causal Inference), which shows that causal equivalence classes of the type studied by Chickering and Meek can be given topological meaning and we can use powerful tools from abstract homotopy theory to study the structure of causal equivalence classes. We can apply similarly many techniques from topos theory to study causal inference. \n\nQUESTIONS regarding Originality: \n\nQUESTION 1: Section 2 and theorems 1-4 are new. Section 4 seems to only recall existing notions and the impact on the work is not explicitly stated.\n\nANSWER: As we mentioned above, Section 4 shows that each topos has an internal logic that is described by the Mitchell-Benabou language. In the revised paper, we will use a simple example to illustrate the use of the MB language to reason about causal models. \n\nQUESTION 2: Theorem 1: the category in Theorem 1 is the same as in Definition 1: objects are SCMs, and arrows are operations on SCMs, such as interventions. We begin with this theorem since its proof is simpler than the following theorems (3 and 4). \n\nANSWER: Theorems 3 and 4 generalize Theorem 1, since they apply to a larger class of causal models (Markov categories and simplicial sets), and use a more powerful set of theoretical tools (Yoneda embeddings, sheaves over a topos), whereas the proof of Theorem 1 is much simpler in comparison. \n\nWe are indebted to you for your careful reading of the paper, and your detailed comments regarding typos. We will of course make the necessary revisions to address all your concerns.  A few highlighted comments: \n\nQUESTION: 49 I do not understand \"our main contribution in this paper is to argue for...\". The contribution cannot be about arguing. => What about saying clearly the result? \"Our contribution is 1. to define a category CAUSAL of causal models, 2. to prove that CAUSAL is a topos\". (also give a name to the category you introduce)\n\nANSWER: We absolutely agree with you and will revise the paper accordingly. \n\nQUESTION: between 72-73: please insert an outline of the paper because we are left lost at the end of the introduction.\n\nANSWER: Yes, thank you, we will include the outline as requested. \n\n74-94: seems to be kind of motivation. Should be in the intro as well?\n\nANSWER: Yes, we will include it, thank you. \n\nQUESTION regarding Figure 1. The left and middle parts should be drawn in the same direction. The right part is not understandable.\n\nANSWER: The right part of  Figure 1 describes the construction of the sub-object classifier for the SCM category. We will revise the figure to make it clearer. \n\nQUESTION regarding 102 Very difficult to know whether Definition 1 is from the litterature or whether it is new\n\nANSWER: We will ensure that all standard definitions are cited. \n\nQUESTION regarding 134 add phi after \"pullback\" (I am not sure, if it is not phi, refer to the appropriate object or morphism)\n\nANSWER: Pullback refers to the whole diagram. We will clarify in the revised paper. \n\nQUESTION about Theorem 1. \"The category where objects blablabla...\" => define the category before with a name CAUSAL (in the tradition of category theory like SET, GRP, etc.). By the way, to be sure, is it the category defined in Definition 1?\n\nANSWER: Yes, Theorem 1 is for the category defined in Definition 1. \n\nQUESTION about 145 \"The proof essentially involves...\". Then do it. The proof have the shape. \"Proof of i), proof of ii)....\". The proof has not this shape. And it is not what it is done. It seems like you only tackle point (iv), but it is not said explicitely so I am not sure.\n\nANSWER: We only wrote the proof for the sub-object classifier, and left the rest to the Supplementary Materials. We will revise and condense the proof in the revised paper. \n\nQUESTION about 822 \"Our framework\": what do you mean? topos already exist, SCM too.\n\nANSWER: By \"Our framework\", we mean treating the SCM category as a topos, which is our novel contribution, to the best of our knowledge. \n\nThank you again for your careful reading of the paper, and for your many excellent comments regarding the presentation and its clarity. We will of course strive to incorporate all your suggestions in the revised paper."}]

Author rebuttal:

Meta-review: This paper explores the use of topos theory, a sophisticated concept from category theory, as a foundational framework for interpreting causal models. It contributes to a growing trend in machine learning that seeks to recast probabilistic and statistical concepts in abstract mathematical terms. The central claim is that toposes naturally accommodate the structure needed to represent various causal models, and the paper identifies three key benefits of doing so. It also succeeds in making these abstract ideas accessible, offering intuitive explanations that connect categorical concepts to causal inference, which could help broaden understanding among researchers unfamiliar with category theory.
However, the paper’s impact is limited by several factors. While it effectively outlines the properties gained by interpreting causal models within toposes, it does not convincingly explain why these properties matter from a causal inference perspective. The practical implications of the universal properties it highlights remain vague, and the paper does not clearly articulate what a causal inference researcher gains by adopting this framework. This lack of motivation weakens the case for its relevance, especially for readers seeking concrete benefits or applications.
The mathematical rigor is also uneven. Although the proofs are generally correct and serve to validate the interpretations, they are less formal than expected in this domain. Some reviewers find this acceptable given the paper’s conceptual nature, but others note that the lack of depth in the theorems and the informal presentation may detract from its credibility in more mathematically rigorous circles.
The organization of the paper is another point of critique. Key concepts are scattered between the main text and appendices, making it difficult to follow. Some sections, particularly Section 4, are described as obscure, and the definitions are not always presented with the clarity needed for such a complex topic. Moreover, while the paper emphasizes its theoretical contribution, it does not sufficiently connect its ideas to existing philosophical literature on causality, which could have strengthened its conceptual grounding.
Despite these weaknesses, the paper is recognized for its originality. The use of topos theory in causal inference is novel, and the foundational perspective it offers could pave the way for deeper insights into the nature of causality. However, to fully realize its potential, the paper needs to better articulate the significance of its results, clarify its methodology, and connect more explicitly to both practical scenarios and philosophical discussions.
The discussion was intense and with several posts from authors and reviewers. This helped some reviewers to re-evaluate her/his first rating of the manuscript and in particular to increase the final rating.
Decision: Accept (spotlight)