Categorial subsystem independence as morphism co-possibility

This paper formulates a notion of independence of subobjects of an object in a general (i.e. not necessarily concrete) category. Subobject independence is the categorial generalization of what is known as subsystem independence in the context of algebraic relativistic quantum field theory. The content of subobject independence formulated in this paper is morphism co-possibility: two subobjects of an object will be defined to be independent if any two morphisms on the two subobjects of an object are jointly implementable by a single morphism on the larger object. The paper investigates features of subobject independence in general, and subobject independence in the category of C* algebras with respect to operations (completely positive unit preserving linear maps on C*-algebras) as morphisms is suggested as a natural subsystem independence axiom to express relativistic locality of the covariant functor in the categorial approach to quantum field theory.

establish a generally covariant quantum field theory on a general, non-flat spacetime that might not have any non-trivial global symmetry. Lack of global spacetime symmetry makes it impossible to postulate covariance of quantum fields in the usual way by requiring observables to transform covariantly with respect to representations of the global symmetry group of the spacetime. Instead, general covariance is implemented in Categorial Local Quantum Physics very naturally by postulating that the functor representing quantum field theory is a covariant functor. Locality also has to be implemented in categorial quantum field theory. This is done by formulating axioms for the covariant functor that express independence of subsystems. In the original paper formulating Categorial Local Quantum Physics [8] Einstein Locality (local commutativity) is taken as the expression of locality as independence. In subsequent publications [6], [7] a categorial version of the split property is added to the Einstein Locality axiom. It is then shown in [7] that (under the further assumption of weak additivity) the categorial split property is equivalent to the functor being extendable to a tensor functor between the tensor category formed by spacetimes with respect to disjoint union as tensor operation and the tensor category of C * -algebras taken with the minimal tensor products of C * -algebras.
In what sense are Einstein Locality and Einstein Locality together with the categorial split property (i.e. the tensor property) of the functor subsystem independence conditions? This is a non-trivial question, which is shown by the remark of Buchholz and Summers on Einstein Locality: "This postulate, often called the condition of locality [16], has become one of the basic ingredients in both the construction and the analysis of relativistic theories [37]. Yet, in spite of its central role in the theoretical framework, the question of whether locality can be deduced from physically meaningful properties of the physical states has been open for more than four decades." [11] In particular, Buchholz and Summers point out [11] that some of the standard notions of subsystem independence (such as C * -independence) do not entail Einstein Locality. And conversely: Einstein Locality alone does not entail C * -independence. Nor does Einstein Locality, in and by itself, entail the independence condition known as prohibition of superluminal signaling: local commutativity of C *algebras pertaining to spacelike separated spacetime regions only entails prohibition of superluminal signaling with respect to measurements of local observables representable by the projection postulate and by the operations given by local Kraus operators -but not with respect to general operations that do not have a local Kraus representation (this was shown in [33]). Einstein Locality alone also does not ensure another subsystem independence called operational C * -independence: That any two (nonselective) operations (completely positive, unit preserving linear maps on C * -algebras) performed on spacelike separated subsystems of a larger system are jointly implementable as a single operation on the larger system [32], [29]. Since the tensor product of two operations is again an operation ( [4][p. 190], see also Proposition 9. in [32]), the tensorial property of the functor does entail operational C * -independence of the components of the tensor product within the tensor product algebra; however, the tensor property entails more than this: it entails operational C * -independence in the product sense [32], which is a strictly stronger condition than simple operational C * -independence. But for the purposes of expressing locality in quantum field theory subsystem independence in the sense of operational C * -independence does not have to be implemented in the strong form of requiring existence of a joint product extension of operations on subsystems.
Thus requiring only Einstein Locality of the functor seems too weak, imposing the categorial split property (i.e. demanding the functor to be tensorial) seems to demand a bit more than needed to implement locality interpreted as subsystem independence in Categorial Local Quantum Physics. What is then the right concept of subsystem independence that expresses locality in Categorial Local Quantum Physics?
The subsystem independence hierarchy in Local Quantum Physics suggests a general concept of subsystem independence that has a natural formulation in terms of categories objects of which are sets with morphisms as maps: independence as morphism co-possibility. According to this independence concept two objects are independent in a larger object with respect to a class of morphisms if any two morphisms on the two smaller objects have a joint extension to a morphism on the larger object. This independence notion appeared in [30] and was suggested in [31] as a possible axiom to require in Categorial Local Quantum Physics. Taking specific subclasses of the operations as the class of morphisms, one can recover the standard concepts of subsystem independence in the independence hierarchy as special cases of morphism co-possibility (see [30]).
The way subsystem independence as morphism co-possibility was formulated above and in the papers [30] and [31] is not entirely satisfactory however because it is not purely categorial: in a general category objects are not necessarily sets and morphisms are not necessarily functions -not every category is a concrete category (e.g. the real numbers R regarded as a poset category) [3]. In a general category subsystem independence as morphism co-possibility should be formulated as subobject independence with respect to some class of morphisms. The aim of the present paper is to define subobject independence in this way, as morphism co-possibility in a general category, and to investigate the basic properties of such an independence notion. This notion is of interest in its own right and, after defining it in section 2, we give several examples of this sort of independence in different categories in section 3. Section 4 proves some propositions on the relation of subobject independence and tensor structure in a category. In section 5 subobject independence is specified in the context of the category of C * -algebras taken with the class of operations between C * -algebras. The resulting notion of operational independence is suggested then in section 6 as a possible axiom to express relativistic locality of the covariant functor describing a generally covariant quantum field theory.

Categorial independence of subobjects
In this section C = (Ob, Mor) denotes a general category, and Hom is a subclass of Mor such that (Ob, Hom) also is a category. It is not assumed that C is a concrete category; i.e. that it is categorically equivalent to a category objects of which are sets and monomorphisms are functions. Morphisms in Hom will be referred to as Hom-morphisms, morphisms in Mor will be called Mor-morphisms. We wish to define a notion of independence of subobjects A, B of an object C, where the concept of subobject is understood with respect to Hom-morphisms, and the independence expresses that any two Mor-morphisms on the Hom-subobjects A and B are jointly implementable by a single Mor-morphism on C. The two morphism classes Hom and Mor should be considered as variables in this categorial concept of independence: Choosing different morphism classes one obtains independence notions contents of which can vary considerably.
Recall that a Mor-morphism f : A → B is a monomorphism ("mono", for short) if for any object C ∈ Ob and any Mor-morphisms g 1 , g 2 : C → A it holds that g 1 f = g 2 f implies g 1 = g 2 . Monomorphisms are the categorial equivalents of injective functions.
The notion of Hom-subobject is formulated in terms of Hom-monomorphisms: a Hom-subobject of an object X is an equivalence class of Hom-monomorphisms Hom ∋ i A : A → X where i A is defined to be In what follows, |i A | Hom denotes the equivalence class of Hom-morphisms equivalent to i A . We are now in the position to give the definition of the central concept of this paper: Definition 2.2 (Mor-independence of Hom-subobjects). Two Hom-subobjects of object X represented by the two equivalence classes |f 1 | Hom and |f 2 | Hom are called Mor-independent if any two Hom-monomorphisms g 1 ∈ |f 1 | Hom and g 2 ∈ |f 2 | Hom are Mor-independent.
The content of Mor-independence of Hom-subobjects is that two Hom-subobjects of object C are The next proposition is useful when it comes to determine whether two subobjects are independent.
By definition, independence of Hom-subobjects implies Mor-independence of any of their representatives.
The following proposition states the converse: if one pair of representatives of two Hom-subobjects are Mor-independent, then the two Hom-subobjects are Mor-independent. Mor-independent. Proof.
are Mor-independent and consider the diagram below.
Take and arbitrary Mor-morphism α 1 : D 1 → D 1 . Let By assumption C 1 are Mor-independent, therefore there is a suitable Mor-morphism γ : X → X. Then we obtain and similarly This completes the proof.
Our next proposition formulates a very natural necessary condition for independence. The content of the necessary condition can be illustrated on the example of the category C of structures. Let A and B substructures of C. If we take two morphisms α A : A → A, α B : B → B, then a morphism γ : C → C that extends both α A and α B can exist only in the case when α A and α B act on Y = A ∩ B exactly the same way, i.e. if one has The next proposition we wish to establish expresses this condition in the case of every category. To state the proposition, first we formulate the condition (3) in general categorial terms. Since the intersection Y in the category of structures is the pullback A × C B, for the next definition it is assumed that pullbacks exist in C.

Definition 2.4 (Mor-compatibility). We say that Mor-morphisms
The next proposition states the sought-after necessary condition for independence in a general category: are Mor-independent, then they are also Mor-compatible.
Proof. Consider the diagram, where α A , α B are arbitrary Mor-morphisms and A × C B is the pullback.
We need to show that the diagram without the dashed arrow commutes. By Mor-independence, for Mor-morphisms α A , α B there exists a suitable Mor-morphism γ. Then For completeness we note that the existence of the pullback A × C B in the definition of Morcompatibility could be relaxed by replacing the pullback A × C B in the diagram by any Y that can be mapped into A and B (i.e. there are Mor-arrows Y → A and Y → B) and universally quantifying over Y .
In the next proposition let ⊕ be a coproduct in the category (Ob, Mor), i.e. X 1 ⊕ X 2 be an element such that there exist Mor-morphisms (called the coproduct injections) having the universal property.
be a coproduct with coproduct injections i 1 and i 2 . From the diagram below on the left-hand side, by composing arrows, one gets the diagram on the right-hand side which is a coproduct diagram, therefore a suitable m with the dotted arrow (which is the copair exists and completes the proof. We remark that coproduct injections in general are not necessarily monic, however, in certain categories (such as extensive or distributive categories) coproduct injections are automatically monic.

Sets
Set is the category of sets as objects with functions as Mor-morphisms.

Vector spaces
Let Vect F be the category of vector spaces over the field F with linear mappings as Mor-morphisms. Take

Pregeometries (Matroids)
Pregeometries (or matroids in the combinatorial terminology) are defined in order to capture the notion of independence in a very general framework. Formally, a pregeometry is a tuple (X, cl) where X is a set and cl : ℘(X) → ℘(X) is a closure operator having a finite character satisfying the Steinitz exchange principle. Independence and basis can be defined as in vector spaces. A morphism between If the two closure operators are topological closure, then morphisms are just the continuous functions. If (X, cl) is a pregeometry, then a sub-pregeometry

Boolean algebras
Let Bool be the category of Boolean algebras as objects with injective homomorphisms as Mor-morphisms.
As before, we set Hom = Mor. Two subalgebras A,  Conversely any permutation of atoms extend to an automorphism. Given automorphisms α A and α B of A and B, respectively, give rise to a permutation of all the atoms of C which extends to an automorphism of C. Consequently, A and B are Mor-independent.  x, y ∈ L, then write x▽y if for all z ∈ L we have (x ∨ z) ∧ y = z ∧ y. Clearly x ∧ y = 0 implies x▽y. Let S and Q be subsets of L. We say that L is the internal direct sum of S and Q (and we write L = S ⊕ Q) if

Logical and categorial subobject independence in quantum logic
(1) each x ∈ L can be written as x = s ∨ q with s ∈ S and q ∈ Q; (2) s ∈ S, q ∈ Q entails s▽q.
If S and Q are (orthomodular) lattices, then their direct product is an (orthomodular) lattice, and there is a natural (ortho)-isomorphism between their direct product and their internal direct sum given by (s, q) → s ∨ q (see [23]). It follows that any homomorphisms given on the direct summands S and Q extends to a homomorphism on their internal direct sum. We have then the following characterization of the relation of logical independence and subobject independence in the category of orthomodular lattices: The question whether γ can be further extended to an Mor-morphism C → C is non-trivial and is related to the injectivity of C. We are not aware of any useful characterization of injective objects in OML.

Categorial subobject independence and tensor product structure
Components of tensor products are typically regarded "independent" within the tensor product. The paradigm example is the standard product of probability measure spaces with the product measure on the product of the component measurable spaces. In this section we investigate the relation of categorial subobject independence and the tensor product structure in a category. We will see that categorial subobject independence of the components of the tensor product is not automatic. We will however isolate conditions on the tensor category that entail subobject independence of the components in the tensor product (Proposition 4.4).
Recall first the definition of a tensor product in a category (cf. Section 7.8 in [3]) Definition 4.1. A bifunctor ⊗ : C × C → C is a tensor product if it is associative up to a natural isomorphism and there is an element I that acts as a left and right identity (up to isomorphism).
A category with a tensor product (C, ⊗) is a tensorial category (monoidal category) if ⊗ satisfies the pentagon and triangle axioms. If a category has products or coproducts for all finite sets of objects, then the category can be turned into a tensor category by adding the product or coproduct as a bifunctor (due to the universal property of products and coproducts).
For the next definition suppose that (C, ⊗) is such that for any two objects A, B there are Mor- If ⊗ is the coproduct, then the universal property of coproducts implies the existence of such a h in the definition.
Mor-independence of components of tensor products is not automatic. As a counterexample consider the category of sets with the tensor product being the union operation. We call these Hom-monomorphisms canonical injections.
(ii) For any pairs of Mor-morphisms m A : A → A ′ and m B : B → B ′ the tensor product arrow m A ⊗m B makes the following diagram commute.
We then have as an immediate consequence of regularity:   are Hom-monomorphisms representing two Hom-subobjects which are (Hom, ⊗)-independent. Let Q be Hom-injective and consider the diagram below.  Hom-injectivity of Q does this favour to us: Hom-injectivity guarantees that any Mor-morphism defined on a Hom-subobject can be extended as a Mor-morphism acting on Q.
⊗-independence of Mor-morphisms (Definition 4.2) and the notion of a regular category (Definition 4.3) was introduced and studied in [12] under different names. In [12] the notion of a tensor product with projections or with inclusions has been defined (essentially, this is our Definition 4.3). It was shown in [12] that the definition of stochastic independence relies on such a structure and that independence can be defined in an arbitrary category with a tensor product with inclusions or projections in a manner similar to Definition 4.2. It turns out that the standard notion of stochastic independence of classical random variables is equivalent to ⊗-independence of objects in the category of random variables (for more detail see [12]). Moreover, the classifications of quantum stochastic independence by Muraki, Ben Ghorbal, and Schürmann has been shown to be classifications of the tensor products with inclusions for the categories of algebraic probability spaces and non-unital algebraic probability spaces. Thus ⊗-independence of Examples of operations include states, conditional expectations (in particular the projection postulate), operations that are given by Kraus operators, and more (see [22] for the elementary theory and physical interpretation of operations, [2] for some basic properties, and [24] for a systematic treatment of operations from the perspective of operator spaces.) Specifically, C * -algebra homomorphisms are completely positive; hence the class hom Alg of injective C * -algebra homomorphism is a subclass of Op Alg . Thus it is meaningful to talk about Op Alg -independence in the sense of the following definition: Op Alg -independent as hom Alg -subobjects of object C in the category (Alg, Op Alg ) of C * -algebras in the sense of Definition 2.2.
The notion of Op Alg -independence of C * -subalgebras was first formulated in categorial terms in [30] but its content, expressed in a non-categorial terminology and called "operational C * -independence" appeared already in [32]. The content of Op Alg -independence of C * -subalgebras A, B of C * -algebra C is that operations on the C * -subalgebras A, B have a joint extension to the C * -algebra C. This kind of independence has a direct physical interpretation: The physical content of Op Alg -independence is that any two physical operations (for instance measurement interaction) performed on the two subsystems observables of which are represented by A and B, respectively, can be performed as a single physical operation on the larger system observables of which are represented by C.
Note that Op Alg -independence of A, B in C has two components: (i) that operations on A and B can be extended to C; and (ii) that there exists a joint extension. Already (i) is a non-trivial demand because operations on C * -subalgebras are not always extendable to the larger algebra [2]. As a corollary: The joint extension to A⊗ B of operations on A and B guaranteed by Proposition 5.2 is just the tensor product of the two operations, which is again an operation [4][p. 190], (see also Proposition 9. in [32]).
Note that C * -algebras A, B are not just Op Alg -independent in A ⊗ B, they are Op Alg -independent in A ⊗ B in the product sense: the tensor product of two operations factorizes over the components. Op Algindependence in the product sense is a very strong independence property. It is known to be strictly stronger than Op Alg -independence simpliciter: Op Alg -independence in C of commuting C * -subalgebras A, B of C in the product sense is equivalent to C * -independence of A, B in the product sense (Proposition 10, [32]) but C * -independence of A, B is strictly weaker than C * -independence of A, B in the product sense [34] (cf. Proposition 1. in [32]). operations, one obtains operational W * -independence (cf. [29]). One also can define the product versions of these specific independence concepts by considering Op Alg -independence in the product sense with respect to the respective subclasses of operations. One has then the notions of C * -and W * -independence in the product sense, and operational C * -and W * -independence in the product sense. Specifications of further sub-types of independence obtains by considering particular operations such as conditional expectations, or Kraus operations (see [29]). The logical relation of these independence concepts emerges then as a non-trivial problem, some of which are still open [29]. Viewed from the perspective of the resulting hierarchy of independence notions, Op Alg -independence serves as a general, categorial frame in which independence can be formulated and analyzed.

The difference between Op
Given the concept of Op Alg -independence, it is natural to consider it as a possible condition to impose it on the covariant functor F representing quantum field theory in order to express causal locality in terms of it. To do so we recall first the definition of the functor F describing quantum field theory.
6 Op Alg -independence as locality condition in categorial quantum field theory The functor F representing a general covariant quantum field theory is between two categories: (i) (Man, hom Man ), the category of spacetimes with isometric embeddings of spacetimes as morphisms; and (ii) (Alg, hom Alg ), the category of C * -algebras with injective C * -algebra homomorphisms as morphisms.
The category (Man, hom Man ) is specified by the following stipulations (see [8] for more details): 1. Covariance: F (M 2 , g 2 ) are Op Alg -independent in F (M, g) in the sense of Definition 5.1.
The axiom system specified by Definition 6.1 differs from the one originally proposed in [8] by the addition of the Op Alg -independence condition. Following the terminology introduced in [31], we call the original axiom system in [8] BASIC, to distinguish it from the one given by Definition 6.1, which we call OPIND. One also can strengthen OPIND by requiring in 4. in Definition 6.1 that the objects F (M 1 , g 1 ) and F (M 2 , g 2 ) are Op Alg -independent in F (M, g) in the product sense. We call the resulting axiom system OPIND × .
Other stipulations on the functor are also possible and have been formulated: The axiom system BASIC was amended by Brunetti and Fredenhagen by replacing the Einstein Causality condition by an axiom that requires a tensorial property of F (Axiom 4 in [6]; also see [15]). To formulate this axiom one first extends (Man, hom Man ) to a tensor category (Man ⊗ , hom ⊗ Man ). This tensor category has, by definition, as its objects finite disjoint unions of objects from Man, and the empty set as unit object. The morphisms h ⊗ in hom ⊗ Man are embeddings of unions of disjoint spacetimes that are hom Manhomomorphisms when restricted to the (connected) elements of the disjoint union of spacetimes and have the feature that the images under h ⊗ of disjoint spacetimes are spacelike. The functor F is then required to be extendable to a tensor functor F ⊗ between (Man ⊗ , hom ⊗ Man ) in a natural way. We call the resulting axiom system TENSOR.
One obtains yet another axiom system if one requires a categorial version of the split property. This 2. σ-continuity of the F (ψ ′ ) with respect to the inclusion R ⊂ R ′ , where ψ ′ : (M, g M ) → (L, g L ) and we have We call BASIC+SPLIT the axiom system that requires of the covariant functor F to have weak additivity and the categorial split property, in addition to Einstein Locality and Time Slice axiom.
As these different conditions imposed on the functor F show, one can articulate the concept of physical locality understood as independence of the algebras of observables of spatio-temporaly local physical systems localized in causally disjoint spacetime regions in more than one way. Thus the question or relation of the different axiom systems arise, and one also can ask: which one of the axiom systems is the most adequate.
The problem of the relation of the axiom systems was raised in [31], where it was argued that the implications in the following diagram depicting the logical relations hold. Here we comment on the reverse of the indicated implications below.

BASIC + SPLIT
We have seen in Section 1 that BASIC does not entail OPIND. The technical obstacle prohibiting the reverse of the implication OPIND × ⇒ TENSOR to hold trivially is that operations on C * -subalgebras of a C * -algebra C need not be extendable to C. Hence, although C * -subalgebras A, B are Op Algindependent in the tensor product A ⊗ B, this does not entail without further conditions that A, B are Op Alg -independent in a C * -algebra C containing A ⊗ B as a C * -subalgebra. Injectivity of C would entail this; however, it is not clear to us whether the C * -algebras F (M, g) are injective in general -or at least for some specific, typical spacetime regions such as double cones.
The reverse of the implication OPIND × ⇒ OPIND is unlikely to hold, given that operational C * -independence in the product sense is a strictly stronger independence condition than operational C *independence -but we do not have a rigorous proof of OPIND ⇒ OPIND × in terms of a model of the axioms displaying the non-implication.
In view of the logical (in)dependencies of the axiom systems depicted in the chart, the conclusion we propose is that the most natural independence condition to stipulate to hold for the functor F in order to express physical locality is Op Alg -independence. This condition has a very natural physical interpretation and it does not require more than what is contained in the notion of subsystem independence as copossibility. So, if some physically relevant models existed which violate TENSOR but satisfy OPIND, that model would still be entirely acceptable from the perspective of a causal behavior of the quantum filed theory represented by the functor satisfying OPIND.