Contextual semantics in quantum mechanics from a categorical point of view

Abstract

The category-theoretic representation of quantum event structures provides a canonical setting for confronting the fundamental problem of truth valuation in quantum mechanics as exemplified, in particular, by Kochen–Specker’s theorem. In the present study, this is realized on the basis of the existence of a categorical adjunction between the category of sheaves of variable local Boolean frames, constituting a topos, and the category of quantum event algebras. We show explicitly that the latter category is equipped with an object of truth values, or classifying object, which constitutes the appropriate tool for assigning truth values to propositions describing the behavior of quantum systems. Effectively, this category-theoretic representation scheme circumvents consistently the semantic ambiguity with respect to truth valuation that is inherent in conventional quantum mechanics by inducing an objective contextual account of truth in the quantum domain of discourse. The philosophical implications of the resulting account are analyzed. We argue that it subscribes neither to a pragmatic instrumental nor to a relative notion of truth. Such an account essentially denies that there can be a universal context of reference or an Archimedean standpoint from which to evaluate logically the totality of facts of nature.

Appendix

Category theory provides a general theoretical framework for dealing with systems, formalized through appropriate mathematical structures, by putting the emphasis on their mutual relations and transformations. The central focus of the categorical way of rethinking basic notions can be described as a shift in the emphasis of what is considered to be fundamental for the formation of structures. In the set-theoretic mode of thinking, structures of any conceivable form are defined as sets of elements endowed with appropriate relations. In the category-theoretic mode, the emphasis is placed on the transformations among the objects of a category and the processes preserving them by means of appropriate structural constraints on the collection of these transformations. In this sense, the notion of structure does not refer exclusively to a fixed universe of sets of pre-determined elements, but acquires a variable reference. The basic categorical notions used systematically in the present paper are summarized as follows.

Categories: A category ${\mathcal {C}}$ is an aggregate consisting of the following:

• 1. A class $Ob({\mathcal {C}})$, whose elements $A, B, \ldots$ are called objects. For each object A an element $id_{A}: A \rightarrow A$ is distinguished; it is called the identity morphism for A.

• 2. A class $Hom({\mathcal {C}})$, whose elements $f, g, \ldots$ are called morphisms or arrows. Each morphism $f: A \rightarrow B$ is associated with a pair of objects, known as its domain and codomain respectively. Note that in category theory this arrow notation is used even if f is not a function in the normal set-theoretic sense. The expression $Hom_{{\mathcal {C}}}(A, B)$ denotes the Hom-class of all morphisms from A to B.

• 3. A binary operation $\circ$, called composition of morphisms, such that, for given arrows $f: A \rightarrow B$ and $g: B \rightarrow E$, that is, with codomain of f equal to the domain of g, then f and g can be composed to give an arrow $g \circ f: A \rightarrow E$. Generally, for any three objects A, B, and E in ${\mathcal {C}}$ the set mapping is defined as:

  $$\begin{aligned} {Hom}_{{\mathcal {C}}}(B,E) \times {Hom}_{{\mathcal {C}}}(A,B) \rightarrow {Hom}_{{\mathcal {C}}}(A,E). \end{aligned}$$

  The operation of composition is associative, $h \circ (g \circ f) = (h \circ g) \circ f$, for all $f: A \rightarrow B$, $g: B \rightarrow E$, $h: E\rightarrow D$, satisfying also the property of identity, $f \circ id_{A}=f=id_{B} \circ f$, for all $f: A \rightarrow B$.

For an arbitrary category ${\mathcal {C}}$ the opposite category ${{\mathcal {C}}}^{op}$ is defined in the following way. The objects are the same, but ${Hom}_{{{\mathcal {C}}}^{op}}(A,B)={Hom}_{{\mathcal {C}}}(B,A)$, namely, all arrows are inverted. A category ${\mathcal {C}}$ is called small if the classes of its objects and morphisms form genuine sets.

Functors: Let ${\mathcal {C}}$, ${\mathcal {D}}$ be categories. A covariant functor $\mathbf {F}: {\mathcal {C}} \rightarrow {\mathcal {D}}$ is a class mapping that:

• 1. Associates to each object $A \in {\mathcal {C}}$ an object ${\mathbf {F}}(A) \in {{\mathcal {D}}}$.

• 2. Associates to each morphism $f: A \rightarrow B \in {\mathcal {C}}$ a morphism ${\mathbf {F}}(f): {\mathbf {F}}(A) \rightarrow {\mathbf {F}}(B) \in {\mathcal {D}}$.

• 3. Makes these assignments by preserving identity morphisms and compositions, i.e., ${\mathbf {F}}(id_A)=id_{{\mathbf {F}}(A)}$, and ${\mathbf {F}}(g \circ f)={\mathbf {F}}(g) \circ {\mathbf {F}}(f).$

A contravariant functor $\hat{\mathbf {F}}: {{\mathcal {C}}} \rightarrow {\mathcal {D}}$ is, by definition, a covariant functor $\mathbf {F}: {{\mathcal {C}}}^{op} \rightarrow {{\mathcal {D}}}$. A functor, therefore, is a type of mapping between categories that associates to every object of one category an object of another category and to every morphism in the first category a morphism in the second by preserving domains and codomains, identity morphisms, and compositions. A functor $\mathbf {F}: {\mathcal {C}} \rightarrow {\mathcal {D}}$ thus gives a sort of ‘picture’ of category ${\mathcal {C}}$ in ${\mathcal {D}}$ by preserving the structure of ${\mathcal {C}}$.

Natural transformations: Let ${\mathcal {C}}$, ${\mathcal {D}}$ be categories, and let further $\mathbf {F}$, $\mathbf {G}$ be functors from the category ${\mathcal {C}}$ to the category ${\mathcal {D}}$. A natural transformation $\tau : \mathbf {F} \rightarrow \mathbf {G}$ is a mapping that assigns to each object A in ${\mathcal {C}}$ a morphism $\tau _A : {\mathbf {F}}(A) \rightarrow {\mathbf {G}}(A)$ in ${\mathcal {D}}$, called the component of $\tau$ at A, such that for every arrow $f: A \rightarrow B$ in ${\mathcal {C}}$ the following diagram in ${\mathcal {D}}$ commutes:

That is, for every arrow $f: A \rightarrow B$ in ${\mathcal {C}}$, we have:

$$\begin{aligned} {{\mathbf {G}}(f)} \circ {\tau _A}={\tau _B} \circ {{\mathbf {F}}(f)}. \end{aligned}$$

Thus, natural transformations define structure preserving mappings of functors. Pictorially, one can think of this situation as follows: if the parallel functors $\mathbf {F} : {\mathcal {C}} \rightarrow {\mathcal {D}}$ and $\mathbf {G} : {\mathcal {C}} \rightarrow {\mathcal {D}}$ are thought of as projecting a ‘picture’ of category ${\mathcal {C}}$ in ${\mathcal {D}}$, then a natural transformation is a way to transform globally or systematically the ‘picture’ defined by $\mathbf {F}$ onto the ‘picture’ defined by $\mathbf {G}$.

Natural isomorphisms: A natural transformation $\tau : {\mathbf {F}} \rightarrow {\mathbf {G}}$ is called a natural isomorphism if every component $\tau _A$ is invertible.

Adjoint functors: Let $\mathbf {F}: {\mathcal {C}} \rightarrow {\mathcal {D}}$ and $\mathbf {G}: {\mathcal {D}} \rightarrow {\mathcal {C}}$ be functors. We say that $\mathbf {F}$ is left adjoint to $\mathbf {G}$ (correspondingly, $\mathbf {G}$ is right adjoint to $\mathbf {F}$), if there exists a bijective correspondence between the arrows ${\mathbf {F}}(C) \rightarrow D$ in ${\mathcal {D}}$ and $C \rightarrow {\mathbf {G}}(D)$ in ${\mathcal {C}}$, which is natural in both C and D:

This means that the objects of the categories ${\mathcal {C}}$ and ${\mathcal {D}}$ are related with each other through natural transformations. Then, the above pair of adjoint functors constitutes a categorical adjunction. The latter concept is of fundamental logical and mathematical importance expressing a generalization of equivalence of categories.

Diagrams: A diagram $\mathbf {X}\!=\!(\{X_i \}_{i \in I}, \{F_{ij}\}_{i,j \in I})$ in a category ${\mathcal {C}}$ is defined as an indexed family of objects $\{X_i \}_{i \in I}$ and a family of morphisms sets $\{F_{ij}\}_{i,j \in I} \!\subseteq \! {Hom_{{\mathcal {C}}}}({X_i},{X_j})$.

Cocones: A cocone of the diagram $\mathbf {X}=(\{X_i \}_{i \in I}, \{F_{ij}\}_{i,j \in I})$ in a category ${\mathcal {C}}$ consists of an object X in ${\mathcal {C}}$, and for every $i \in I$, a morphism $f_i : X_i \rightarrow X$, such that $f_i=f_j \circ f_{ij}$ for all j $\in$ I, that is, for every i, j $\in$ I, and for every $f_{ij}$ $\in$ $F_{ij}$ the diagram below commutes:

Colimits: A colimit of the diagram $\mathbf {X}=(\{X_i \}_{i \in I}, \{F_{ij}\}_{i,j \in I})$ is a cocone with the property that for every other cocone given by morphisms $f_{\acute{i}} : X_i \rightarrow {\acute{X}}$, there exists exactly one morphism $f:X \rightarrow {\acute{X}}$, such that $f_{\acute{i}}=f \circ f_i$, for all i $\in$ I (universality property).

Reversing the arrows in the above definitions of cocone and colimit of a diagram $\mathbf {X}=(\{X_i \}_{i \in I}, \{F_{ij}\}_{i,j \in I})$ in a category ${\mathcal {C}}$ results in the dual notions called cone and limit of $\mathbf {X}$, respectively.

Pullbacks: Let $f: A \rightarrow C$ and $g: B \rightarrow C$ be morphisms with common codomain in a category ${\mathcal {C}}$. A pullback of morphisms f and g is a triple $(P, \, p_{1}, \, p_{2})$ consisting of an object P and morphisms $p_{1}: P \rightarrow A$ and $p_{2}: P \rightarrow B$, in ${\mathcal {C}}$, such that $f \circ p_{1} = g \circ p_{2}$, i.e., the following diagram commutes:

In addition, the pullback $(P, \, p_{1}, \, p_{2})$ must be universal with respect to this diagram. That is, for every other such triple $(Q, q_{1}, q_{2})$ in ${\mathcal {C}}$, there exists a unique morphism $u: Q \rightarrow P$ such that $q_{1} = p_{1} \circ u$ and $q_{2} = p_{2} \circ u$. As with all categorical universal constructions, the pullback of two morphisms, if it exists, is uniquely defined up to an isomorphism. The dual notion of a pullback is that of a pushout.

Subobject functors: A subobject functor $\mathbf{Sub}: {{\mathcal {C}}}^{op} \rightarrow \mathbf{Sets}$ in any category ${\mathcal {C}}$ with finite limits is constructed as follows:

• 1. For an object A of ${\mathcal {C}}$, $\mathbf{Sub}(A)$ is the set of equivalence classes of subobjects of A, i.e., the set of equivalence classes of monic arrows $\mu : M \rightarrow A$.

• 2. Given a morphism $f: B \rightarrow A$ and a monic arrow $\mu : M \rightarrow A$, $\mathbf{Sub}(f): \mathbf{Sub}(A) \rightarrow \mathbf{Sub}(B)$ is obtained by pulling back $\mu$ along f as in the following diagram:

Sieves: A sieve on an object A of a category ${{\mathcal {C}}}$ is a collection S of morphisms with codomain A in ${{\mathcal {C}}}$ such that, if $f: B \rightarrow A$ belongs to S and $g: C \rightarrow B$ is any morphism, then $f \circ g: C \rightarrow A$ belongs also to S.

Generalized covering systems: A generalized topological covering system (or a localization system) on a category ${\mathcal {C}}$ is a function $\mathbf {J}$ which assigns to each object A of ${\mathcal {C}}$ a collection ${\mathbf {J}}(A)$ of sieves on A, called covering A-sieves, such that:

• 1. For each ${\mathcal {C}}$-object A, the maximal A-sieve {f: cod$(f) = A$} belongs to ${\mathbf {J}}(A)$ (maximality condition).

• 2. If S belongs to ${\mathbf {J}}(A)$, then, for any arrow $h: C \rightarrow A$, the pullback sieve $h^{*}(S) = \{g: D \rightarrow C,\, (h \circ g) \in S\}$, for any object D of ${{\mathcal {C}}}$, belongs to ${\mathbf {J}}(C)$ (stability condition).

• 3. If S belongs to ${\mathbf {J}}(A)$, and if for each arrow $h: C_{h} \rightarrow A$ in S there is a sieve $R_{h}$ belonging to ${\mathbf {J}}(C_{h})$, then the set of all composites $h \circ f$, with $h \in S$ and $f \in R_{h}$, belongs to ${\mathbf {J}}(A)$ (transitivity condition).

A generalized topological covering system on ${\mathcal {C}}$, satisfying the previously stated conditions, is equivalent to the notion of a Grothendieck topology on ${\mathcal {C}}$. If ${\mathcal {C}}$ is a (small) category equipped with a Grothendieck topology $\mathbf {J}$, then the pair $({\mathcal {C}}, \mathbf {J})$ defines a site.

Presheaves: A presheaf ${\mathbf {P}}$ on a small category ${{\mathcal {C}}}$ is a function that:

• 1. Assigns to each ${{\mathcal {C}}}$-object A a set ${\mathbf {P}}(A)$.

• 2. Assigns to each ${{\mathcal {C}}}$-morphism $f: B \rightarrow A$ a set function, ${\mathbf {P}}(f): {\mathbf {P}}(A) \rightarrow {\mathbf {P}}(B)$.

• 3. Makes these assignments by preserving identity morphisms and compositions, i.e., ${\mathbf {P}}({id}_{A}) = {id}_{{\mathbf {P}}(A)};$ and, if $g: C \rightarrow B$ and $f: B \rightarrow A$, then ${\mathbf {P}}(f \circ g) = {\mathbf {P}}(g) \circ {\mathbf {P}}(f)$.

Intuitively, a presheaf is a collection of sets that vary in a ‘meshing’ way between objects $A, B, \ldots$ of the category ${{\mathcal {C}}}$. In terms of contravariant and covariant functors, a presheaf on ${{\mathcal {C}}}$ is a contravariant functor from category ${{\mathcal {C}}}$ to the category Sets of normal sets. Equivalently, it is a covariant functor from the opposite category ${{\mathcal {C}}}^{op}$ to Sets. The collection of all presheaves on ${{\mathcal {C}}}$ forms a category, denoted $\mathbf{Sets}^{{{\mathcal {C}}}^{op}}$, in which morphisms between presheaves ${\mathbf {P}}$ and ${\mathbf {Q}}$ are natural transformations $\tau : {\mathbf {P}} \rightarrow {\mathbf {Q}}$.

Sheaves: Given a site $({\mathcal {C}}, \mathbf {J})$, a sheaf is defined to be a presheaf $\mathbf {P}$ such that for all objects A of C and all covering sieves S on A, any natural transformation $\tau : S \rightarrow \mathbf {P}$ has a unique extension to $Hom(-, A)$, that is there is an isomorphism between the set of natural transformations induced by the inclusion $S \mapsto Hom(-, A)$,

$$\begin{aligned} Hom(S, \mathbf {P}) \cong Hom(Hom(-, A), \mathbf {P}). \end{aligned}$$

Observe that a sheaf is systematically connected to the covering sieves of the objects of C. Its most characteristic property is that it provides a unique way to “glue” together functions that are defined locally. In this respect, through the notion of a sheaf, one can systematically move from the local level to the global, or inversely, the global can be constructed from the local in a systematic manner.

Topoi: A topos, ${\mathcal {T}}$, is a type of a category that is equivalent to the category of sheaves of sets on a topological space, or more generally, on a site. For the objects of a topos, which are sheaves of sets, the usual constructions of the category, Sets, of sets can be defined in a generalized categorical manner. For this reason topoi may serve as alternative models of set theory, especially, in view of the fact that each topos completely defines its own mathematical framework. The most important properties of a topos, ${\mathcal {T}}$, include the following:

• 1. There is an initial object ${0}_{\mathcal {T}}$ in ${\mathcal {T}}$ such that, for any object A in the topos, there is a unique arrow ${0}_{{\mathcal {T}}} \rightarrow A$. It is the analogue of the empty set, $\emptyset$.

• 2. There is a terminal object ${1}_{{\mathcal {T}}}$ in ${\mathcal {T}}$ such that, for any object A in the topos, there is a unique arrow $A \rightarrow {1}_{{\mathcal {T}}}$. It is the analogue of the singleton set $\{*\}$.

  For any object A in the topos, an arrow ${1}_{{\mathcal {T}}} \rightarrow A$ is called a global element of A.

• 3. For any objects A, B in ${\mathcal {T}}$, there is a product $A \times B$ in the topos. It is the analogue of the Cartesian product in set theory. A topos always has pull-backs, and the product is just a special case of this.

  For any objects A, B in ${\mathcal {T}}$, there is a co-product $A \sqcup B$ in the topos. It is the analogue of the disjoint union in set theory. A topos always has push-outs, and the co-product is just a special case of this.

• 4. There is exponentiation, associating to each pair of objects A, B in ${\mathcal {T}}$ an object ${A}^{B}$, which is the topos analogue of the set of functions $f: B \rightarrow A$ between sets B and A in set theory. The characteristic property of exponentiation is that, given any object X, there is an isomorphism

  $$\begin{aligned} {Hom}_{{\mathcal {T}}} (X, {A}^{B}) \cong {Hom}_{{\mathcal {T}}} (X \times B, A) \end{aligned}$$

  that is natural in A and X. In set theory, the relevant statement is that a parameterized family of functions $\chi \mapsto {f}_\chi : A \rightarrow B$, $\chi \in X$, is equivalent to a single function $F : X \times A \rightarrow B$ defined by $F(\chi , \alpha ):= {f}_\chi (\alpha )$ for all $\chi \in X$, $\alpha \in A$.

• 5. There is a subobject classifier ${\Omega }_{{\mathcal {T}}}$.