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A Categorial Semantic Representation of Quantum Event Structures

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The overwhelming majority of the attempts in exploring the problems related to quantum logical structures and their interpretation have been based on an underlying set-theoretic syntactic language. We propose a transition in the involved syntactic language to tackle these problems from the set-theoretic to the category-theoretic mode, together with a study of the consequent semantic transition in the logical interpretation of quantum event structures. In the present work, this is realized by representing categorically the global structure of a quantum algebra of events (or propositions) in terms of sheaves of local Boolean frames forming Boolean localization functors. The category of sheaves is a topos providing the possibility of applying the powerful logical classification methodology of topos theory with reference to the quantum world. In particular, we show that the topos-theoretic representation scheme of quantum event algebras by means of Boolean localization functors incorporates an object of truth values, which constitutes the appropriate tool for the definition of quantum truth-value assignments to propositions describing the behavior of quantum systems. Effectively, this scheme induces a revised realist account of truth in the quantum domain of discourse. We also include an Appendix, where we compare our topos-theoretic representation scheme of quantum event algebras with other categorial and topos-theoretic approaches.

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  1. It is instructive to note that in an arbitrary topos, the existence of a classifying object or a subobject classifier Ω takes the role of the set {0,1}≅{false,true} of truth values. If B is an object in the topos, and A denotes a subobject of B, then, there is a monic arrow (monomorphism) AB, generalizing categorically the inclusion of a subset into a larger set. Like in the familiar topos, Sets, of sets and functions, we can also characterize A as a subobject of B by an arrow from B to the subobject classifier Ω. Intuitively, this “characteristic arrow”, BΩ, describes how A “lies in” B; in Sets, this arrow is the characteristic function χ S :X→{0,1} classifying whether a point χX lies in S or not. In general, the elements of the subobject classifier, understood as the arrows 1→Ω, are the truth values, just like “false” and “true”, the elements of {false,true}, are the truth values available in Sets.


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One of us (VK) acknowledges support from the research program “Thalis” co-financed by the European Union (ESF) and the Hellenic Research Council (project 70-3-11604).

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Correspondence to Elias Zafiris.



1.1 A.1 Comparison With Other Categorial Approaches

It is instructive to attempt a brief comparison of our topos-theoretic representation scheme with other categorial and topos-theoretic approaches. The current interest in applying methods of topos theory in the logical foundations of quantum physics was initiated by the work of Isham and Butterfield [9, 10], who provided a topos-theoretic reformulation of the Kochen-Specker theorem. For this purpose, they considered the partially ordered set of commutative von Neumann subalgebras of the non-commutative algebra of all bounded operators on a quantum Hilbert space as a “category of contexts” where the only arrows are inclusions. This “category of contexts” served as a base category for defining the topos of presheaves of sets over the poset of commutative subalgebras. The reformulation of the Kochen-Specker theorem took place by defining a special presheaf, called the spectral presheaf, and showing that the latter has no global sections. We note that the action of the spectral presheaf on each commutative von Neumann subalgebra gives its maximal ideal spectrum (Gelfand spectrum). Alternatively, the former “category of contexts” may be replaced by the poset of all Boolean subalgebras of the non-Boolean lattice of projection operators on a quantum Hilbert space. Similarly, the action of the spectral presheaf in this case (called the dual presheaf) on each Boolean subalgebra gives its Stone spectrum (that is the set of all its homomorphisms to the 2-valued Boolean algebra {0,1}). In this case, the statement of the Kochen-Specker theorem is equivalent to the assertion that the dual presheaf has no global sections.

This topos-theoretic research initiative has been extended, elaborated and developed further by Döring and Isham (DI) (e.g., [6] and references therein). The central principle of their work is that the construction of a theory of physics is equivalent to finding a representation in “a topos of a certain formal language” that is attached to the system. In particular, regarding quantum theory, their proposal is to use the formal language associated with the topos of presheaves of sets over the poset of commutative von Neumann subalgebras (or the poset of Boolean subalgebras) and mimic the classical topos, set-theoretic formulation of physical theories. This analogy is pursued up to the point of constructing a topos-theoretic framework of quantum kinematics (dynamical ideas have not yet been addressed in their framework). This difficult task required the following: (i) The association of physical quantities with morphisms in the topos of presheaves from a “state-object” to a “quantity-value” object; (ii) the definition of an appropriate “object of truth-values” in this topos, and iii, the construction of the so called “daseinisation map” for projections (and self-adjoint operators), which is used as a translation mechanism from the Hilbert space formalism to the topos formalism. Regarding (i) the “state object” is identified as the spectral presheaf, while regarding (ii) the “object of truth-values” in the topos of presheaves is identified with the subobject classifier, which assigns to each context the Heyting algebra of all sieves on that context. Regarding (iii) the “daseinisation map” transforms a projection operator to a clopen subobject of the spectral presheaf by approximating it, for each context of the base poset, by the smallest projection greater than or equal to it. In this way, propositions are represented by the clopen subobjects of the spectral presheaf (a Heyting algebra representation). At a next stage, the procedure of daseinisation is extended to self-adjoint operators by considering their spectral families and approximating with respect to the spectral order. This method comes in two versions depending on the procedure of approximating self-adjoint operators from above or from below in the spectral order. If for each context, the best approximations to a self-adjoint operator from above and below become evaluated at a state, they define an interval of real numbers, which is interpreted as the unsharp value of this operator at the selected context in that state. By essentially building on this insight, (DI) construct a “quantity-value” object, which is a presheaf different from the real number object of the (DI) topos of presheaves. In this way, the daseinisation of a self-adjoint operator is described as a natural transformation from the “state-object” to the “quantity-value” object. In classical physics, for every state of a system a proposition acquires a definite truth value (true/false) or equivalently each state defines a homomorphism from the Boolean algebra of propositions to the two-valued Boolean algebra. In the (DI) case, a state is not represented by a global section of the spectral presheaf (“state object”) due to the topos version of Kochen-Specker’s theorem, but by a probability measure on the spectral presheaf. This forces (DI) to define the truth value of a proposition at a state as the sieve (downwards closed set) of contexts for which the probability of its daseinisation at each such context is 1.

Let us now attempt a brief comparison of our approach with the (DI) approach. Initially, it is useful to focus on the different conceptual aspects involved in the utilization of topos-theoretic ideas in the foundations of quantum physics. (DI) use the notion of topos as a semantical framework of intuitionistic propositional or predicate logic in its function to serve as a linguistic representation (that is the topos is “a topos of a certain formal language”) attached to a system. Precise criteria of this attachment are not provided, which would justify the reasons of adherence to an intuitionistic framework. Rather, the scheme is built on the strong analogy provided by the notion of an “elementary topos” (that is the logical embodiment of the topos concept) as a generalized model of set theory being equipped with a subobject classifier (that is a distributive Heyting algebra classifying object, which forces the intuitionistic semantics) generalizing the classifying function of the Boolean two-valued object in the universe of sets. Behind this analogy there is the philosophical claim of “neo-realism”. This is also conceived in a purely logical manner by (DI), on the basis of the claim that a new form of realism in physics is restored if both the propositional structure and the truth values structure of the “linguistic representation” of a physical system are distributive and “almost Boolean”. In comparison, our approach uses the notion of topos in the sense of a generalized geometric environment, which makes it possible to constitute the structural information content of “complex objects” (like quantum event algebras) from the non-trivial localization properties of observables, which are used in order to probe (or technically cover) these “complex objects”. More precisely, the proposed crucial notion of topos in physics is the one associated with the conceptual framework of Grothendieck topoi. Every Grothendieck topos can be represented as a category of sheaves for some Grothendieck topology on a base category of “contexts”. Moreover, every Grothendieck topos is also an elementary topos, and thus equipped with an internal classifying object of truth values. Thus, in our perspective the “linguistic representation” is a consequence of the above mathematical fact and not the ultimate aim of formulating a physical theory in elementary topos logical terms in order to restore some form of traditional realism. (DI) avoid any reference to the notion of observables, mainly because of the possible instrumentalist connotations of this term, and use instead the term “physical quantity”. Still observables denote physical quantities that, in principle, can be measured and the constitution of quantum observables from interconnected families of local Boolean observables (with respect to an appropriate Grothendieck topology) reveals the non-trivial (unsharp) localization properties in the quantum realm. Thus, it is precisely these non-trivial localization properties that necessitate the constitution of quantum objects via factorization through a Grothendieck topos (a “superstructure of measurement”, viz. a “category of sheaves” in Grothendieck’s words) over a base category of Boolean localizing measurement contexts. In the topos scheme of (DI), which follows an inverse conceptual direction by attempting to reduce “quantum objects” (for instance self-adjoint operators) to “objects or arrows in a topos” (for instance a topos-conceived physical quantity), instead of constituting or inducing “quantum objects” by factorization through “objects or arrows in a topos” reflecting Boolean localization properties, the localization problem is not avoided but appears in another guise in the elaborate construction of the “quantity-value” object. It is important to stress that our conception of the functional role of topos in quantum mechanics is still realist (although in a different sense in comparison to “neo-realism”) since the consideration of Boolean localizing contexts forms a pre-condition of quantum physical experience, as we have explained previously.

The above brings into focus two other important issues in the attempted comparison between these two topos approaches to the foundations of quantum physics. The first refers to the role of “Boolean contexts” or “commutative contexts” as the objects of the base category and the other refers to the idea of translation between “quantum objects” and “topos objects”. Let us start with the comparison referring to the issue of “contexts”. The idea of a “context” describes an algebra of commuting physical quantities, or equivalently, a complete Boolean algebra of commuting projection operators (the idempotent elements of a “commutative context”). In the framework of (DI) the contexts are partially ordered by inclusion forming a poset which serves as the base category of presheaves. The contexts are called heuristically “local” since no topology is defined on the base category. Note that since the base category is a poset the consideration of the Alexandroff topology of upper or lower sets in the order does not make any difference at the topos level since every presheaf is a sheaf for this topology on a poset. In any case, since they consider a topos as “a topos of a certain formal language” attached to a quantum system, the consideration of the topos of presheaves over this partial order, being naturally equipped with a subobject classifier (the Heyting algebra of all sieves at each context), is adequate for their purpose to provide truth values of propositions (after the procedure of daseinisation) in an “almost Boolean” truth values object in this topos. Their intention is to use all these partially ordered “local” contexts simultaneously in order to capture the information of “quantum objects” (not homomorphically) in terms of truth valuations in the subobject classifier. A natural question arising in this setting is if the orthomodular lattice of all projections in a global non-commutative von Neumann algebra is determined by the partially ordered set of its Boolean subalgebras of projections, that is, by the poset of its “Boolean contexts”. This is not the case since at least the inclusions of the “Boolean contexts” together with the order relation should be taken into account. Still, it seems that this does not appear as a problem in the topos approach of (DI), because they are only interested in a non-homomorphic translation of projections into their daseinised approximations with respect to the partially ordered “contexts”, followed by another non-homomorphic mapping (of Heyting algebras) into the subobject classifier. In comparison, in our approach the specification of the base category of “Boolean contexts” plays a major role and is different from a poset. Initially, we define as a base category the category of complete Boolean algebras with morphisms all the corresponding homomorphisms (the technicality of considering σ-Boolean algebras is forced upon the requirement of having a well defined theory of observables according to standard measure-theoretic arguments). The choice of the category of complete Boolean algebras as a base category is justified by the fact that given any set of pairwise commuting self-adjoint operators, there exists a complete Boolean algebra which contains all the projection operators generating the spectral decomposition of these operators. Thus, complete Boolean algebras play the functional role of logical frames relative to which we are able to coordinatize the measurements of the observables corresponding to these self-adjoint operators. The semantic connotation of “Boolean contexts” as “Boolean logical frames” for covering the global non-Boolean lattice of projections poses the necessity to make precise the meaning of what is “local” in the base category. For this purpose, we define an appropriate Grothendieck topology on the (opposite) category of complete Boolean algebras (the sub-canonical topology of epimorphic families of Boolean covers), which boils down to the notion of Boolean localization functors forming a partially ordered set by inclusion. The notion of Boolean covers as probing frames of a quantum event algebra requires further explanation for the aims of the comparison. For this reason we point out that the spectral presheaf, the so called “state-object” of (DI) is different from our corresponding spectral presheaf, which is called functor of Boolean frames of a quantum event algebra. The (DI) spectral presheaf, at each “Boolean context” gives the set of Boolean homomorphisms from that context to the two-valued context (the Stone spectrum of the “Boolean context”). In our case, the functor of Boolean frames, at each “Boolean context” gives the set of quantum homomorphisms from the “modeled Boolean context” (that is the quantum event algebra image of the “Boolean context” under the action of the modeling functor) to a fixed quantum event algebra. These “modeled Boolean contexts” are the generators of covering families of a quantum event algebra, that is families of “Boolean covers” or “Boolean logical frames” of a quantum event algebra localizing it. Thus, it is convenient to think of these “Boolean covers” in terms of covering Boolean coordinate patches of a global quantum event algebra, so that there might be many with the same image. Notice also that, in contradistinction to the case where they are related only by inclusion, there may be many homomorphisms between each pair of them. Finally, instead of pairwise intersections we have to look at their fibered products (which define the pullback compatibility conditions for Boolean covers in some Boolean localization functor of a quantum event algebra). The upshot of this difference boils down to the following consequences: First, the homomorphism from a “modeled Boolean context” to some fixed quantum event algebra always factors in a homomorphic way through the inductive limit (colimit) in the category of elements of the functor of Boolean frames of the quantum event algebra. Second, the functor of Boolean frames becomes a sheaf with respect to compatible Boolean covering families in the defined topology (Boolean localization functors). Third, by restriction to such Boolean localization functors, a quantum event algebra can be represented isomorphically by the inductive limit in the category of elements of its functor (sheaf) of Boolean frames. Fourth, the whole structural information of a global quantum event algebra is constituted sheaf-theoretically (up to isomorphism) and inversely preserved by this inductive limit construction (restricted to Boolean covers in the topology). Fifth, the same idea can be implemented in an analogous way for the categories of quantum observables and quantum probabilities by passage to the corresponding slice categories of the base category of quantum event algebras. Hence, there is no need to introduce separately notions of “quantity-value” objects and “quasi-states”. Sixth, the Grothendieck topos of sheaves on the defined site is the geometric localization environment via which it becomes possible to constitute “quantum objects” contextually (from the local to the global level) by probing them through interconnected families of Boolean frames. Seventh, by reflection of the localization topos the category of quantum event algebras itself becomes equipped with a classifying object, which can be used for truth valuations of quantum propositions in analogy to the classical case. Eighth, the exact analogue of the spectral logical object in the localization topos assigns to each “modeled Boolean context” the set of quantum homomorphisms from this context to the quantum classifying algebra (instead of the Stone spectrum).

The final issue of our comparison refers to the idea of translation between “quantum objects” and “topos objects”. In the (DI) framework the translation is implemented from the “quantum side” to the “topos side” through the procedure of daseinisation of projectors (and self-adjoint operators). This is a procedure of order-theoretic approximation of each projector in the global non-Boolean lattice (representing a proposition about the value of a physical quantity) by some projector in each “classical context” of the base poset, such that all the “classical contexts” are taken into account simultaneously. The order-theoretic approximation procedure may be conducted either from above (outer daseinisation) or from below (inner daseinisation) with distinct physical interpretations. For example, in the outer case each approximating projector (with respect to a “classical context”) denotes the strongest consequence in that context of the original projector. In a nutshell, (outer) daseinisation produces an order embedding of the global non-distributive lattice of projections into a distributive lattice (complete Heyting algebra of clopen subobjects of the spectral presheaf), which does not preserve the conjunction and the negation operations of the quantum lattice as well as the law of excluded middle. Conceptually, daseinisation in its functional role as a translation from “quantum objects” to “topos objects” is interpreted as a means to “bring-a-quantum-property-into-existence” (inspired from Heidegger’s Dasein) by “hurling it into the collection of all possible classical snapshots of the world” in the words of (DI). In comparison, we think of the process of translation between “quantum objects” and “topos objects” in a different way. The key idea is the existence of a categorical adjunction (pair of adjoint functors) between the topos of presheaves (over the base category of complete Boolean algebras) and the category of quantum event algebras. The adjunction provides a bidirectional functorial correlation between this topos and the category of quantum event algebras, where the right adjoint is the functor of Boolean frames (of a quantum event algebra) and the left adjoint is the inductive limit of an object in the topos (taken in the category of its elements). Thus, in comparison to daseinisation, which translates (not homomorphically) a “quantum object” to a “topos object”, the adjunction is a bidirectional and functorial translation mechanism of encoding and decoding information from “topos objects” to “quantum objects” and inversely, by preserving the structural form of the correlated categories. The crucial part of the adjunction is the construction of the left adjoint, by means of which we obtain a homomorphism from the inductive limit of a “topos object” to a “quantum object”. In particular, the counit of the adjunction, evaluated at a quantum event algebra, is a quantum homomorphism from the inductive limit in the category of elements of the functor of Boolean frames to a quantum event algebra, which can be made into a quantum isomorphism by restriction to a Boolean localization functor. In this way, the global structural information of a quantum event algebra can be approximated homomorphically or (in the latter case) completely constituted (up to isomorphism) by means of gluing together the observable information collected in all compatible Boolean frames in the form of appropriate equivalence classes (by the inductive limit construction).

Moreover, the “Boolean frames-quantum adjunction” provides the key conceptual and technical device to show that the category of quantum event algebras is equipped with a classifying object, which should be used for the valuation of quantum propositions by analogy to the classical case, where the two-valued Boolean algebra plays this role. For this purpose we use the unit of the adjunction evaluated at the subobject functor (a “topos object”) and show that it becomes representable in the category of quantum event algebras by a classifying object in this category (a “quantum object”), which is again constructed by an inductive limit operation (in the category of elements of the subobject functor). Intuitively, this quantum classifying object contains the information of equivalence classes of truth valuations with respect to all compatible Boolean frames belonging to a Boolean localization functor of a quantum event algebra. In comparison, the truth value object of (DI) is the subobject classifier in their topos of presheaves over the poset of “classical contexts” (a “topos object”). In their case there does not exist a homomorphism (of Heyting algebras) from the (clopen) subobjects of their spectral presheaf to the subobject classifier of this topos, which would provide the analogy with the classical case. This is so because a state is not represented by a global element of their spectral presheaf due to the Kochen-Specker theorem, but by a probability measure on the spectral presheaf. Thus, the truth value of a proposition (at a state) is identified with the downwards closed set of “classical contexts” for which the probability of its daseinisation at each such context is 1. Nevertheless, from an inverse viewpoint, the truth of a “daseinized proposition” in a “classical context” does not convey any information about the truth of the original quantum proposition. In comparison, in our approach the truth of a proposition in a Boolean frame makes it equivalent to all other propositions in all Boolean frames being compatible with it with respect to a Boolean localization functor of a quantum event algebra by the explicit truth-value criterion.

Conclusively, in our approach the “Boolean frames-quantum adjunction” is a theoretical platform for probing the quantum domain of discourse via a localization topos by: [I] Decoding the global information contained in quantum event structures inductively via equivalence classes of partially compatible processes of localization in Boolean logical frames realized as physical contexts for measurement of observables, and [II] classifying quantum information in terms of contextual truth valuations with respect to these Boolean logical frames. We claim that the functioning of this bidirectional translation platform is fundamental philosophically for a novel realist understanding of the part-whole relation and the corresponding contextualist account of truth suited to the quantum domain.

We continue our comparison by commenting briefly on a similar topos-theoretic approach to that of Döring and Isham (DI), which has been developed by Heunen, Landsman and Spitters (HLS) (e.g., [8] and references therein). The similarity is based on the following facts: [I] They also use the notion of topos as a semantical framework of intuitionistic predicate logic in its function to serve as a linguistic representation (that is the topos is “a topos of a certain formal language”) attached to a quantum system. [II] The choice of the base category of their topos scheme is closely related to the one by (DI), meaning that it is also a partially ordered set of “classical contexts”, the essential difference being that they are not commutative von Neumann algebras but more general star algebras over the complex numbers. Regarding these structural similarities, our comparison comments referring to the (DI) scheme pertain to this scheme as well. Repeating concisely, the difference pertains to the following: (i) The distinct notions of an elementary topos in comparison to a Grothendieck localization topos (realized as a category of sheaves for an appropriate Grothendieck topology) as a foundation to probe the content of a physical theory, and (ii) the choice of the partial order relation among “classical contexts” as an adequate base category to capture the complexity of quantum logic, in contradistinction to the category of complete Boolean algebras and homomorphisms together with their function as Boolean logical frames in quantum logic.

Notwithstanding the above similarities there are considerable differences between the topos approaches of these two groups. They can be very concisely summarized as follows: (i) The (HLS) topos approach uses a covariant functorial perspective, which is based on the topos of co-presheaves on the partial order of “classical contexts”. (ii) The conceptual and philosophical underpinning of the topos scheme serves different purposes and is interpreted in distinctively different ways: in the (DI) case it is interpreted as a framework of “neo-realism” in the sense of resembling classical physics in an “almost Boolean” way, whereas in the (HLS) case it is interpreted as a framework making precise Bohr’s “doctrine of classical concepts” invoking explicitly the notions of experiments, measurement and observables. This is also reflected in the terminology (for example, (DI) speak of physical quantities, whereas (HLS) speak of observables), and it is somehow strange to us that (HLS) also use the term “daseinisation” of (DI) in order to describe the approximation procedure, although the meaning of this term is at odds with Bohr’s doctrine. (iii) The essential point of the (HLS) topos approach is that there exists an internal commutative star algebra (or an internal Boolean algebra) within the topos of co-presheaves over the poset of “classical contexts”, so their topos comes equipped together with an “internal commutative algebraic object”, which is not the case in the (DI) approach.

For our comparison purposes, we focus on the aspect [iii] above, marking the basic technical difference between the (HLS) and (DI) approaches in relation to ours. An initial remark is that the “internal commutative algebraic object” is introduced in the topos by means of a tautological covariant functor, which assigns to each object in the poset of “classical contexts” itself, seen as a set. So, it is this tautological covariant functor which serves as an “internal commutative algebraic object” in the topos of (HLS). Then, the use of the constructive version of the Gelfand duality theorem of Banaschewski and Mulvey [3], generalizing Gelfand duality internally in topoi, allows (HLS) to define the internal Gelfand spectrum of this “internal commutative algebraic object” in their topos, which is a frame (and thus a Heyting algebra in the topos) to act as the topos intuitionistic logical surrogate of quantum logic. The process of passing from a non-commutative star algebra to an internal commutative star algebra via a tautological functor in the topos of covariant functors over the poset of “classical contexts” is called “Bohrification” by (HLS). Now, the internal observables are given by the self-adjoint elements in the “internal commutative algebraic object” and the internal states by the linear functionals to the constant functor of complex numbers in the topos. Moreover, there exists an internal complete Boolean algebra in the topos formed by the idempotent internal observables. In a nutshell, (HLS) using these “internal objects” define embeddings of the standard “quantum objects” into “topos objects”, set up an analogous approximation procedure (inner daseinisation) for projections (and self-adjoint elements), and manage to embed the standard quantum logic into a “topos object” analogous to the “clopen subfunctors of the spectral presheaf” of the (DI) approach, which is not an “internal Boolean algebra” “topos object”. This is, similarly to the (DI) case, an order embedding to a Heyting algebra object in a topos, which does not preserve disjunctions and the negation operator. In comparison, in our approach we have not considered the existence of any analogous “internal Boolean algebra object” in our topos (which is different from the topos of (HLS) both in terms of the base category and the fact that we use a topos of (pre)sheaves and not a topos of co-presheaves). It is not clear if the existence of such an “internal commutative object” has been somehow forced by the employment of a tautological functor (together with the choice of the base category as a poset) or is a more general phenomenon. At least (HLS) do not provide any other instance, except the tautological case, and do not make any further remark concerning this issue. In the physical state of affairs, apart from the functionality of the “internal commutative object” in order to define observables and states internally in their topos—thus bypassing the issues with the “quantity value object” and “quasi-states” in the (DI) approach—they do not use an appropriate “internal Boolean algebra” “topos object” for valuations of quantum propositions, and therefore such an internal object is not relevant for logical classification internally in their topos. In our case, focusing on the viewpoint of a topos as a Grothendieck localization topos of sheaves, we may further make use of the notion of “Boolean localization” implied by a result of Barr and Diaconescu [16], according to which for any Grothendieck topos of sheaves there exists a Boolean cover, that is a geometric morphism from the topos of sheaves over a complete Boolean algebra to this topos. This theorem, applied in our case, provides also an adjunction between the topos of sheaves over a complete Boolean algebra and the category of quantum event algebras. This notion of “Boolean localization” as pertaining to our approach will be explored in detail elsewhere.

An interesting further development in the “Bohrification program” of (HLS) is the work of van der Berg and Heunen (BH) [17], who make the claim that this program is most naturally developed in the context of partial algebras, a concept introduced in quantum mechanical considerations by Kochen and Specker. They show that every partial Boolean algebra is the inductive limit of its total subalgebras, viz. the commeasurable Boolean subalgebras. Note that in the proof of this result they use a partial Boolean algebra together with a prescribed poset of total subalgebras as well as the inclusions of the total subalgebras into the partial algebra. This is, in fact, another form of a well-known theorem in the theory of orthomodular lattices, called “Kalmbach’s Bundle Lemma” [11], as (BH) also point out. As we have also stressed previously in our remarks to the (DI) approach, this result shows that the partial order relation of “classical contexts” is not adequate to capture the structural information of quantum logic, and at least, the inclusion functions of the “Boolean contexts” to the quantum lattice should be also taken into account. In comparison, our approach to the specification of a quantum event algebra via the left adjoint functor of the “Boolean frames-quantum adjunction” is more general. In our case, the Boolean algebras of the base category do not form a poset and actually they are not even required to be subalgebras of a quantum event algebra. Moreover, the inductive limit is taken in the category of elements of the functor of Boolean frames of a quantum event algebra. It is instructive to remark that the partial order relation of a quantum event algebra in this way is induced by lifting morphisms from the base category of Boolean algebras to the fibers of the category of elements subject to the pullback compatibility conditions.

Finally, we would like to comment briefly on a currently emerging research program by Abramsky and Brandenburger (AB) [1], who have proposed the modeling of contextuality and non-locality using the framework of sheaf theory. Their setting is quite general by using weaker assumptions than standard quantum theory, and their aim is to explicate the introduced sheaf-theoretic notions by applying them on empirical models in a clear and simple way. An interesting aspect of this approach is that the phenomena of contextuality and non-locality are detached from their quantum-theoretic origins since they become applicable in a much wider spectrum through their association with sheaf-theoretic notions. In particular, the central claim of (AB) is that the phenomena of contextuality and non-locality should be modeled in sheaf-theoretic terms as giving rise to obstructions to the existence of global sections. More precisely, they show that the existence of a global section, gluing together uniquely a compatible family of elements in a presheaf pertaining to the empirical modeling of a system, is equivalent to the realization of this system by a factorizable hidden-variable model. Their empirical model of a system involves a measurement space (a finite and discrete set), a finite covering of the measurement space (called a measurement covering consisting of a family of subsets, corresponding to measurement contexts, where a measurement context is a set of measurements that can be performed jointly), a finite set of outcomes, a presheaf of events assigning to each measurement context its set of outcomes (being trivially a sheaf over a discrete space), and a presheaf of distributions assigning to each measurement context its set of distributions on the sections defined over this context (such that the operation of restriction in the presheaf corresponds to taking the marginal of a distribution). Then, for a measurement covering, a compatible family of elements of the presheaf of distributions (thus a sheaf of distributions with respect to this measurement covering) defines a no-signalling empirical model corresponding to this measurement covering. Of particular interest for our purposes is the quantum representation of these empirical models. In this case, a measurement covering consists of measurement contexts, which are identified as sets of maximal commuting subsets of the set of all observables on a fixed Hilbert space (i.e., the set of all observables on a fixed Hilbert space define the measurement space of a quantum empirical model according to (AB)). In comparison to our sheaf-theoretic model, we notice the following: Instead of the set of all observables on a fixed Hilbert space, we take into account the global quantum event and observable structure explicitly, thus our measurement space at the level of events is a quantum event algebra (a quantum logic) and at the level of observables is a partial commutative algebra (a quantum observable algebra). The measurement covering of (AB) by sets of “maximal commutative contexts” corresponds to a Boolean covering consisting of maximal complete Boolean algebras of projections, where each one of them generates the spectral resolution of each “maximal commutative context”. Now, in this setting of a quantum empirical model, (AB) define a quantum representation by a state (density operator) on the fixed Hilbert space. Then, for each “maximal commutative context” in the measurement covering, the state defines a probability distribution on the set of commuting observables belonging to this context, by the standard “trace rule”, and thus defines a presheaf of probability distributions on the measurement space with respect to the measurement covering. This is analogous in our case to the presheaf functor of Boolean measure theoretic (probabilistic) frames of a quantum state with respect to a Boolean covering of a quantum event algebra, which we have shown that it is a sheaf [21]. The pertinent question in the setting of (AB) is if their presheaf of probability distributions is a sheaf with respect to the considered measurement covering. (AB) show that this is actually the case, namely, families of distributions are compatible on overlaps of measurement contexts in the covering, and thus can be glued together. The important conceptual insight of (AB) is that this result implies a “generalized no-signalling theorem” in quantum mechanics, which incorporates the standard no-signalling theorem of Bell-type scenarios corresponding to special cases of measurement coverings.

1.2 A.2 The Left Adjoint Colimit Construction

The left adjoint \(\mathbf{L} : {{\bf Sets}^{{\mathcal{B}}^{op}}} \to {\mathcal{L}}\) of the Boolean realization functor of \({\mathcal{L}}\) is defined for each presheaf P in \({{\bf Sets}^{{\mathcal{B}}^{op}}}\) as the colimit (inductive limit)

We can provide an explicit form of the left adjoint functor by expressing the above colimit as a coequalizer of a coproduct using standard category-theoretic arguments. For this purpose, if we consider the category of elements of the presheaf of Boolean algebras P, that is \(\bf{{\int}}({\mathbf{P}},{\mathcal{B}})\), as an index category \(\mathcal{I}\), then the colimit of the functor \({{\mathbf{M}} \circ \bf{{\int}}_{\mathbf{P}}}: \mathcal{I} \rightarrow {\mathcal{L}}\) is expressed as follows:

figure j

where χ is the coequalizer of the arrows ζ and η. In the diagram above the second coproduct is over all the objects (B,p) with pP(B) of the category of elements, while the first coproduct is over all the maps \(v : ({\acute{B}},{\acute{p}}) \rightarrow (B,p)\) of that category, so that \(v : {\acute{B}} \rightarrow B\) and the condition \(p \cdot v=\acute{p}\) is satisfied.

In order to analyze in more detail the colimit in the category of elements of P induced by the functor of local Boolean frames M, and because of the fact that \(\mathcal{L}\) is a concrete category, we may consider the forgetful functor from \(\mathcal{L}\) to \(\bf Sets\). Then, the coproduct ⨆(B,p) M(B) is a coproduct of sets, which is equivalent to the product P(BM(B) for \(B \in \mathcal{B}\). The coequalizer is thus equivalent to the definition of the tensor product \({\mathbf{P}} {\otimes}_{\mathcal{B}} {\mathbf{M}}\) of the set valued functors \(\mathbf{P} : {\mathcal{B}}^{op} \rightarrow {\bf Sets}\) and \(\mathbf{M} : {\mathcal{B}} \rightarrow {\bf Sets}\). We call this construction the functorial tensor product decomposition of the colimit in the category of elements of P induced by the functor of local Boolean frames  M:

figure k

According to the above diagram, for elements pP(B), \(v : {\acute{B}} \to B\) and \(\acute{q} \in {\mathbf{M}}({\acute{B}})\) the following equations hold:

$$\zeta (p,v, \acute{q})=(p \cdot v, \acute{q}), \qquad \eta(p,v, {\acute{q}})= \bigl(p, v (\acute{q})\bigr) $$

symmetric in P and M. Hence the elements of the set \({\mathbf{P}} \,{\otimes}_{\mathcal{B}}\, {\mathbf{M}}\) are all of the form χ(p,q). This element can be written as:

$$\chi(p,q)=p \otimes q, \quad p \in {\mathbf{P}}(B), q \in {\mathbf{M}}(B). $$

Thus, if we take into account the definitions of ζ and η above, we obtain:

$$p \cdot v \otimes \acute{q}=p \otimes v({\acute{q}}), \quad p \in {\mathbf{P}}(B), \acute{q} \in {\mathbf{M}}(\acute{B}), v : {\acute{B}} \rightarrow B. $$

We conclude that the set \({\mathbf{P}} \,{\otimes}_{\mathcal{B}}\, {\mathbf{M}}\) is actually the quotient of the set ⨆ B P(BM(B) by the smallest equivalence relation generated by the above equations. The equivalence classes of this relation can be further endowed with the structure of a quantum event algebra, thus completing the construction of the left adjoint colimit in \(\mathcal{L}\) via the category of \(\bf Sets\).

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Zafiris, E., Karakostas, V. A Categorial Semantic Representation of Quantum Event Structures. Found Phys 43, 1090–1123 (2013).

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