“What is a Thing?”: Topos Theory in the Foundations of Physics

Abstract

The goal of this article is to summarise the first steps in developing a fundamentally new way of constructing theories of physics. The motivation comes from a desire to address certain deep issues that arise when contemplating quantum theories of space and time. In doing so we provide a new answer to Heidegger’s timeless question “What is a thing?”.

Our basic contention is that constructing a theory of physics is equivalent to finding a representation in a topos of a certain formal language that is attached to the system. Classical physics uses the topos of sets. Other theories involve a different topos. For the types of theory discussed in this article, a key goal is to represent any physical quantity A with an arrow $\breve{A}_\phi:\varSigma_\phi\rightarrow{\cal R}_\phi$ where Σ _φ and ${\cal R}_\phi$ are two special objects (the “state object” and “quantity-value object”) in the appropriate topos,τ _φ .

We discuss two different types of language that can be attached to a system, S. The first, $\mathcal{PL}(S)$, is a propositional language; the second, $\mathcal{L}({S})$, is a higher-order, typed language. Both languages provide deductive systems with an intuitionistic logic. With the aid of $\mathcal{PL}(S)$ we expand and develop some of the earlier work¹ on topos theory and quantum physics. A key step is a process we term “daseinisation” by which a projection operator is mapped to a sub-object of the spectral presheaf ${\underline{\varSigma}}$—the topos quantum analogue of a classical state space. The topos concerned is $\textbf{Sets}{\cal V}({\cal H})^{\textrm{op}}$: the category of contravariant set-valued functors on the category (partially ordered set) ${\cal V}({\cal H})$ of commutative sub-algebras of the algebra of bounded operators on the quantum Hilbert space ${\cal H}$.

There are two types of daseinisation, called “outer” and “inner”: they involve approximating a projection operator by projectors that are, respectively, larger and smaller in the lattice of projectors on ${\cal H}$.

We then introduce the more sophisticated language $\mathcal{L}({S})$ and use it to study “truth objects” and “pseudo-states” in the topos. These objects play the role of states: a necessary development as the spectral presheaf has no global elements, and hence there are no microstates in the sense of classical physics.

One of the main mathematical achievements is finding a topos representation for self-adjoint operators. This involves showing that, for any bounded, self-adjoint operator ${\hat A}$, there is a corresponding arrow ${\breve{\delta}^o(\hat{A})}:{\underline{\varSigma}}\rightarrow{\underline{{\mathbb{R}}^{\succeq}}}$ where ${\underline{{\mathbb{R}}^{\succeq}}}$ is the quantity-value object for this theory. The construction of ${\breve{\delta}^o(\hat{A})}$ is an extension of the daseinisation of projection operators.

The object ${\underline{{\mathbb{R}}^{\succeq}}}$ can serve as the quantity-value object if only outer daseinisation of self-adjoint operators is used in the construction of arrows ${\breve{\delta}^o(\hat{A})}:{\underline{\varSigma}}\rightarrow{\underline{{\mathbb{R}}^{\succeq}}}$. If both inner and outer daseinisation are used, then a related presheaf ${\underline{\mathbb{R}^{\leftrightarrow}}}$ is the appropriate choice. Moreover, in order to enhance the applicability of the quantity-value object, one can consider a topos analogue of the Grothendieck extension of a monoid to a group, applied to ${\underline{{\mathbb{R}}^{\succeq}}}$ (resp. ${\underline{\mathbb{R}^{\leftrightarrow}}}$). The resulting object, ${k({\underline{{\mathbb{R}}^{\succeq}}})}$ (resp. $k({\underline{\mathbb{R}^{\leftrightarrow}}})$), is an abelian group-object in τ _φ .

Finally we turn to considering a collection of systems: in particular, we are interested in the relation between the topos representation of a composite system, and the representations of its constituents. Our approach to these matters is to construct a category of systems and to find coherent topos representations of the entire category.

From the range of the basic questions of metaphysics we shall here ask this one question: What is a thing? The question is quite old. What remains ever new about it is merely that it must be asked again and again. [40]

Martin Heidegger

By CJI and collaborators.

Appendices

Appendix 1: Some Theorems and Constructions Used in the Main Text

1.1 Results on Clopen Sub-Objects of Σ

Theorem 15

The collection, ${\textrm{Sub}_{\textrm{cl}}({\underline{\varSigma}})}$, of all clopen sub-objects of ${\underline{\varSigma}}$ is a Heyting algebra.

Proof

First recall how a Heyting algebra structure is placed on the set, ${\textrm{Sub}({{\underline{\varSigma}}})}$, of all sub-objects of ${\underline{\varSigma}}$.

1.2 The “∨”- and “∧”-operations.

Let $\underline{S},\underline{T}$ be two sub-objects of ${\underline{\varSigma}}$. Then the “∨” and “∧” operations are defined by

$$(\underline{S}\vee\underline{T})_V :=\underline{S}_V\cup \underline{T}_V$$

((13.472))

$$(\underline{S}\wedge \underline{T})_V :=\underline{S}_V\cap \underline{T}_V$$

((13.473))

for all contexts V. It is easy to see that if $\underline{S}$ and $\underline{T}$ are clopen sub-objects of ${\underline{\varSigma}}$, then so are $\underline{S}\vee\underline{T}$ and $\underline{S}\wedge \underline{T}$.

1.3 The Zero and Unit Elements

The zero element in the Heyting algebra ${\textrm{Sub}({{\underline{\varSigma}}})}$ is the empty sub-object $\underline{0}:=\{\varnothing_{V}\mid V\in{\textrm{Ob}({{{\cal V}{(\cal H)}}}\})}$, where ∅ _V is the empty subset of ${\underline{\varSigma}}_V$. The unit element in ${\textrm{Sub}({{\underline{\varSigma}}})}$ is ${\underline{\varSigma}}$. It is clear that both $\underline{0}$ and ${\underline{\varSigma}}$ are clopen sub-objects of ${\underline{\varSigma}}$.

1.4 The “⇒”-operation.

The most interesting part is the definition of the implication $\underline{S}\Rightarrow\underline{T}$. For all $V\in{\textrm{Ob}({{{\cal V}{(\cal H)}}})}$, it is given by

$$\begin{array}{lll} (\underline{S}\Rightarrow\underline{T})_V&:=&\{\lambda\in{\underline{\varSigma}}_V\mid\forall\, V^{\prime}\subseteq V, \ \hbox{if} \\ &{}&\hspace{1cm} {\underline{\varSigma}}(i_{V^{\prime}V})(\lambda)\in \underline{S}_{V^\prime} \ \hbox{then} \ {\underline{\varSigma}}(i_{V^{\prime}V})(\lambda)\in \underline{T}_{V^{\prime}}\}\end{array}$$

((13.474))

$$\begin{array}{rll} &=&\{\lambda\in{\underline{\varSigma}}_V\mid \forall V^{\prime} \subseteq V, \ \hbox{if} \\ &{}&\hspace{1cm} \lambda|_{V^{\prime}}\in \underline{S}_{V^\prime} \hbox{ then }\lambda|_{V^{\prime}}\in\underline{T}_{V^{\prime}}\}.\end{array}$$

((13.475))

Since $\neg\underline{S}:=\underline{S}\Rightarrow\underline{0}$, the expression for negation follows from the above as

$$(\neg \underline{S})_V =\{\lambda\in{\underline{\varSigma}}_V\mid\forall\, V^{\prime}\subseteq V,\ {\underline{\varSigma}}(i_{V^{\prime}V})(\lambda)\notin \underline{S}_{V^{\prime}}\}$$

((13.476))

$$=\{\lambda\in{\underline{\varSigma}}_V\mid\forall\,V^{\prime}\subseteq V, \, \lambda|_{V^{\prime}}\notin\underline{S}_{V^{\prime}}\}.$$

((13.477))

We rewrite the formula for negation as

$$(\neg \underline{S})_V=\bigcap_{V^{\prime}\subseteq V}\big\{\lambda\in{\underline{\varSigma}}_V\mid \lambda|_{V^{\prime}}\in \underline{S}_{V^{\prime}}^c\big\}$$

((13.478))

where $\underline{S}_{V^{\prime}}{}^c$ denotes the complement of $\underline{S}_{V^{\prime}}$ in ${\underline{\varSigma}}_{V^{\prime}}$. Clearly, $\underline{S}_{V^{\prime}}{}^c$ is clopen in ${\underline{\varSigma}}_{V^{\prime}}$ since $\underline{S}_{V^{\prime}}$ is clopen. Since the restriction ${\underline{\varSigma}}(i_{V^{\prime}V}):{\underline{\varSigma}}_V\rightarrow{\underline{\varSigma}}_{V^{\prime}}$ is continuous and surjective,¹³⁴ it is easy to see that the inverse image ${\underline{\varSigma}}(i_{V^{\prime}V})^{-1}(\underline{S}_{V^{\prime}}{}^{c})$ is clopen in ${\underline{\varSigma}}_V$. Clearly,

$${\underline{\varSigma}}(i_{V^{\prime}V} )^{-1}(\underline{S}_{V^{\prime}}{}^{c})=\big\{\lambda\in{\underline{\varSigma}}_V\mid \lambda|_{V^{\prime}}\in \underline{S}_{V^{\prime}}{}^{c}\big\}$$

((13.479))

and so, from (13.478) we have

$$(\neg \underline{S})_V=\bigcap_{V^{\prime}\subseteq V} {\underline{\varSigma}}(i_{V^{\prime}V})^{-1}(\underline{S}_{V^{\prime}}{}^{c})$$

((13.480))

The problem is that we want $(\neg\underline{S})_V$ to be a clopen subset of ${\underline{\varSigma}}_V$. Now the right hand side of (13.480) is the intersection of a family, parameterised by $\{V^\prime\mid V^\prime\subseteq V\},$ of clopen sets. Such an intersection is always closed, but it is only guaranteed to be open if $\{V^\prime\mid V^\prime\subseteq V\}$ is a finite set, which of course may not be the case.

If $V^{\prime\prime}\subseteq V^{\prime}$ and $\lambda|_{V^{\prime\prime}}\in \underline{S}_{V^{\prime\prime}}{}^{c}$, then $\lambda|_{V^{\prime}}\in \underline{S}_{V^{\prime}}{}^{c}$. Indeed, if we had $\lambda|_{V^{\prime}}\in \underline{S}_{V^{\prime}}$, then $(\lambda|_{V^{\prime}})|_{V^{\prime\prime}}= \lambda|_{V^{\prime\prime}}\in \underline{S}_{V^{\prime\prime}}$ by the definition of a sub-object, so we would have a contradiction. This implies ${\underline{\varSigma}}(i_{V^{\prime\prime}V} )^{-1}(\underline{S}_{V^{\prime\prime}}{}^{c})\subseteq {\underline{\varSigma}}(i_{V^{\prime}V})^{-1}(S_{V^{\prime}}{}^{c})$, and hence the right hand side of (13.480) is a decreasing net of clopen subsets of ${\underline{\varSigma}}_V$ which converges to something, which we take as the subset of ${\underline{\varSigma}}_V$ that is to be $(\neg\underline{S})_V$.

Here we have used the fact that the set of clopen subsets of ${\underline{\varSigma}}_V$ is a complete lattice, where the minimum of a family $(U_i)_{i\in I}$ of clopen subsets is defined as the interior of $\bigcap_{i\in I}U_i$. This leads us to define

$$(\neg \underline{S})_V :=\hbox{int}\bigcap_{V^{\prime}\subseteq V} {\underline{\varSigma}}(i_{V^{\prime}V} )^{-1}(\underline{S}_{V^{\prime}}{}^{c})$$

((13.481))

$$=\hbox{int}\bigcap_{V^{\prime}\subseteq V}\big\{\lambda\in {\underline{\varSigma}}_V\mid \lambda|_{V^{\prime}}\in(S_{V^{\prime}}{}^{c})\big\}$$

((13.482))

as the negation in ${\textrm{Sub}_{\textrm{cl}}({\underline{\varSigma}})}$. This modified definition guarantees that $\neg\underline{S}$ is a clopen sub-object. A straightforward extension of this method gives a consistent definition of $\underline{S}\Rightarrow\underline{T}$.

This concludes the proof of the theorem.

The following theorem shows the relation between the restriction mappings of the outer presheaf ${{\underline{{O}}}}$ and those of the spectral presheaf ${\underline{\varSigma}}$. We basically follow de Groote’s proof of Proposition 3.22 in [38] and show that this result, which uses quite a different terminology, actually gives the desired relation.

Theorem 16

Let $V,V'\in{\rm Ob}({{{\cal V}{(\cal H)}}})$ such that $V'\subset V$. Then

$$S_{{{\underline{{O}}}}(i_{V^{\prime}V})({\delta^o(\hat{{P}})_{{V}}})}= {\underline{\varSigma}} (i_{V^{\prime} V})(S_{{\delta^o(\hat{{P}})_{{V}}}}).$$

((13.483))

Proof

First of all, to simplify notation, we can replace ${\delta^o(\hat{{P}})_{{V}}}$ by $\hat{P}$ (which amounts to the assumption that $\hat{P}\in{\mathcal{P}(V)}$. This does not play a role for the current argument). By definition, ${{\underline{{O}}}}(i_{V^{\prime}V})\big(\hat{P}\big)={\delta^o(\hat{{P}})_{{V^\prime}}}$, so we have to show that $S_{{\delta^o(\hat{{P}})_{{V^\prime}}}}={\underline{\varSigma}} (i_{V^{\prime}V})(S_{\hat{P}})$ holds.

If $\lambda\in S_{\hat{P}}$, then $\lambda(\hat{P})=1$, which implies $\lambda(\hat Q)=1$ for all $\hat Q\geq\hat{P}$. In particular, $\lambda({\delta^o(\hat{{P}})_{{V'}}})=1$, so ${\underline{\varSigma}} (i_{V^{\prime}V})(\lambda)= \lambda|_{V'}\in S_{{\delta^o(\hat{{P}})_{{V^\prime}}}}$. This shows that ${\underline{\varSigma}} (i_{V^{\prime}V})(S_{\hat{P}})\subseteq S_{{\delta^o(\hat{{V^\prime}})_{{P}}}}$.

To show the converse inclusion, let $\lambda'\in S_{{\delta^o(\hat{{P}})_{{V'}}}}$, which means that $\lambda'({\delta^o(\hat{{P}})_{{V'}}})=1$. We have $\hat{P}\in{{{\underline{{O}}}}(i_{V'V})}^{-1} ({\delta^o(\hat{{V'}})_{{P}}})$. Let

$$F_{\lambda'}:=\{\hat Q\in\mathcal{P}(V')\mid\lambda'(\hat Q)=1\} =\lambda'^{-1}(1)\cap\mathcal{P}(V').$$

((13.484))

As shown in Sect. 13.8.3, $F_{\lambda'}$ is an ultrafilter in the projection lattice $\mathcal{P}(V')$.¹³⁵ The idea is to show that $F_{\lambda'}\cup\hat{P}$ is a filter base in ${\mathcal{P}(V)}$ that can be extended to an ultrafilter, which corresponds to an element of the Gel’fand spectrum of V.

Let us assume that $F_{\lambda'}\cup\hat{P}$ is not a filter base in ${\mathcal{P}(V)}$. Then there exists some $\hat Q\in F_{\lambda'}$ such that

$$\hat Q\wedge\hat{P}=\hat Q\hat{P}=\hat 0,$$

((13.485))

which implies $\hat{P}\leq\hat 1-\hat Q$, so

$${{\underline{{O}}}}(i_{V'V})(\hat{P})={\delta^o(\hat{{V'}})_{{P}}}\leq{{\underline{{O}}}}(i_{V'V}) (\hat 1-\hat Q)=\hat 1-\hat Q$$

((13.486))

and hence we get the contradiction

$$1=\lambda'({\delta^o(\hat{{P}})_{{V'}}})\leq\lambda'(\hat 1-\hat Q)=0.$$

((13.487))

By Zorn’s lemma, the filter base $F_{\lambda'}\cup\hat{P}$ is contained in some (not necessarily unique) maximal filter base in ${\mathcal{P}(V)}$. Such a maximal filter base is an ultrafilter and thus corresponds to an element λ of the Gel’fand spectrum ${\underline{\varSigma}}_V$ of V. Since $\hat{P}$ is contained in the ultrafilter, we have $\lambda(\hat{P})=1$, so $\lambda\in S_{\hat{P}}$. By construction, ${\underline{\varSigma}} (i_{V^{\prime}V})(\lambda)=\lambda|_{V'}=\lambda'\in S_{{\delta^o(\hat{{P}})_{{V'}}}}$, the element of ${\underline{\varSigma}}_{V'}$ we started from. This shows that $S_{{\delta^o(\hat{{P}})_{{V^\prime}}}}\subseteq{\underline{\varSigma}} (i_{V^{\prime}V}) (S_{\hat{P}})$, and we obtain

$$S_{{\delta^o(\hat{{P}})_{{V^\prime}}}}={\underline{\varSigma}}(i_{V^{\prime}V})(S_{\hat{P}}).$$

((13.488))

It is well-known that every state $\lambda'\in{\underline{\varSigma}}_{V'}$ is of the form $\lambda'={\underline{\varSigma}}(i_{V'V})(\lambda)=\lambda|_{V'}$ for some $\lambda\in{\underline{\varSigma}}_V$. This implies

$${\underline{\varSigma}}(i_{V'V})^{-1}(S_{{\delta^o(\hat{{P}})_{{V'}}}})=S_{{\delta^o(\hat{{P}})_{{V'}}}} \subseteq{\underline{\varSigma}}_V.$$

((13.489))

Note that on the right hand side, $S_{{\delta^o(\hat{{P}})_{{V'}}}}$ (and not $S_{\hat{P}}$, which is a smaller set in general) shows up.

De Groote has shown in [38] that for any unital abelian von Neumann algebra V, the clopen sets $S_{\hat Q}$, $\hat Q\in{\mathcal{P}(V)}$, form a base of the Gel’fand topology on ${\underline{\varSigma}}_V$. Formulas (13.488) and (13.489) hence show that the restriction mappings

$$\begin{array}{lll} {\underline{\varSigma}}(i_{V^{\prime}V}):{\underline{\varSigma}}_V &\rightarrow& {\underline{\varSigma}}_{V'} \\ \lambda &\mapsto& \lambda|_{V'} \end{array}$$

of the spectral presheaf are open and continuous. Using continuity, it is easy to see that ${\underline{\varSigma}} (i_{V^{\prime}V})$ is also closed: let $C\subseteq{\underline{\varSigma}}_{V}$ be a closed subset. Since ${\underline{\varSigma}}_{V}$ is compact, C is compact, and since ${\underline{\varSigma}} (i_{V^{\prime}V})$ is continuous, ${\underline{\varSigma}} (i_{V^{\prime}V})(C)\subseteq{\underline{\varSigma}}_{V^{\prime}}$ is compact, too. However, ${\underline{\varSigma}}_{V^{\prime}}$ is Hausdorff, and so ${\underline{\varSigma}}(i_{V^{\prime}V})(C)$ is closed in ${\underline{\varSigma}}_{V^{\prime}}$.

1.5 The Grothendieck k-Construction for an Abelian Monoid

Let us briefly review the Grothendieck construction for an abelian monoid M.

Definition 23

A group completion of M is an abelian group $k(M)$ together with a monoid map $\theta:M\rightarrow k(M)$ that is universal. Namely, given any monoid morphism $\phi:M\rightarrow G$, where G is an abelian group, there exists a unique group morphism $\phi^\prime: k(M)\rightarrow G$ such that φ factors through φ ^′; i.e., we have the commutative diagram

with $\phi=\phi^\prime\circ\theta$

It is easy to see that any such $k(M)$ is unique up to isomorphism.

To prove existence, first take the set of all pairs $(a,b)\in M\times M$, each of which is to be thought of heuristically as $a-b$. Then, note that if inverses existed in M, we would have $a-b=c-d$ if and only if $a+d=c+b$. This suggests defining an equivalence relation on $M\times M$ in the following way:

$$(a,b)\equiv (c,d) \makebox{ iff} \exists g \in M \hbox{such that } a+d+g=b+c+g.$$

((13.490))

Definition 24

The Grothendieck completion of an abelian monoid M is the pair $(k(M),\theta)$ defined as follows:

• (i) $k(M)$ is the set of equivalence classes $[a,b]$, where the equivalence relation is defined in (13.490). A group law on $k(M)$ is defined by

  $$\mathrm{(i)}\ [a,b]+[c,d]:=[a+c,b+d],$$

  ((13.491))
  $$\mathrm{(ii)}\ 0_{k(M)}:=[0_M,0_M],$$

  ((13.492))
  $$\mathrm{(iii)}\ -[a,b]:=[b,a],$$

  ((13.493))

  where 0 _M is the unit in the abelian monoid M.

• (ii) The map $\theta:M\rightarrow k(M)$ is defined by

  $$\theta(a):=[a,0]$$

  ((13.494))

  for all $a\in M$.

It is straightforward to show that (i) these definitions are independent of the representative elements in the equivalence classes; (ii) the axioms for a group are satisfied; and (iii) the map θ is universal in the sense mentioned above.

It is also clear that k is a functor from the category of abelian monoids to the category of abelian groups. For, if $f:M_1\rightarrow M_2$ is a morphism between abelian monoids, define $k(f):k(M_1)\rightarrow k(M_2)$ by $k(f)[a,b]:=[f(a),f(b)]$ for all $a,b\in M_1$.

1.6 Functions of Bounded Variation and Γ $\mathbb{R}^{\succeq}$

These techniques will now be applied to the set, $\varGamma{\underline{{\mathbb{R}}^{\succeq}}}$, of global elements of ${\underline{{\mathbb{R}}^{\succeq}}}$. We could equally well consider $\varGamma{\underline{\mathbb{R}}^{\leftrightarrow}}$ and its k-extension, but this would just make the notation more complex, so in this and the following Subsections, we will mainly concentrate on $\varGamma{\underline{{\mathbb{R}}^{\succeq}}}$ (resp. ${\underline{{\mathbb{R}}^{\succeq}}}$). The results can easily be extended to $\varGamma{\underline{\mathbb{R}}^{\leftrightarrow}}$ (resp. $\underline{\mathbb{R}^{\leftrightarrow}}$).

It was discussed in Sect. (13.8.2) how global elements of ${\underline{\mathbb{R}^{\leftrightarrow}}}$ are in one-to-one correspondence with pairs $(\mu,\nu)$ consisting of an order-preserving and an order-reversing function on the category ${\cal V}({\cal H})$; i.e., with functions $\mu:{\textrm{Ob}({{\cal V}{(\cal H)}})}\rightarrow{\mathbb{R}}$ such that, for all ${V_1,V_2\in{\textrm{Ob}({{{\cal V}{(\cal H)}}})}}$, if $V_2\subseteq V_1$ then $\mu(V_2)\leq\mu(V_1)$ and $\nu:{\textrm{Ob}({{{\cal V}{(\cal H)}}})}\rightarrow\mathbb{R}$ such that, for all $V_1,V_2\in{\textrm{Ob}({{{\cal V}{(\cal H)}}})}$, if $V_2\subseteq V_1$ then $\nu(V_2)\geq\nu(V_1)$; see (13.291). The monoid law on $\varGamma{\underline{\mathbb{R}}^{\leftrightarrow}}$ is given by (13.295).

Clearly, global elements of ${\underline{{\mathbb{R}}^{\succeq}}}$ are given by order-reversing functions $\nu:{{\cal V}{(\cal H)}}\rightarrow\mathbb{R}$, and $\varGamma{\underline{{\mathbb{R}}^{\succeq}}}$ is an abelian monoid in the obvious way. Hence the Grothendieck construction can be applied to give an abelian group $k(\varGamma{\underline{{\mathbb{R}}^{\succeq}}})$. This is defined to be the set of equivalence classes $[\nu,{\kappa}]$ where $\nu,{\kappa}\in\varGamma{\underline{{\mathbb{R}}^{\succeq}}}$, and where $(\nu_1,{\kappa}_1)\equiv(\nu_2,{\kappa}_2)$ if, and only if, there exists $\alpha\in\varGamma{\underline{{\mathbb{R}}^{\succeq}}}$, such that

$$\nu_1+{\kappa}_2+\alpha={\kappa}_1+\nu_2+\alpha$$

((13.495))

Since $\varGamma{\underline{{\mathbb{R}}^{\succeq}}}$ has a cancellation law, we have $(\nu_1,{\kappa}_1)\equiv(\nu_2,{\kappa}_2)$ if, and only if,

$$\nu_1+{\kappa}_2={\kappa}_1+\nu_2.$$

((13.496))

Intuitively, we can think of $[\nu,{\kappa}]$ as being “$\nu-{\kappa}$”, and embed $\varGamma{\underline{{\mathbb{R}}^{\succeq}}}$ in $k(\varGamma{\underline{{\mathbb{R}}^{\succeq}}})$ by $\nu\mapsto[\nu,0]$. However, $\nu,{\kappa}$ are $\mathbb{R}$-valued functions on ${\textrm{Ob}({{\cal V}{(\cal H)}})}$ and hence, in this case, the expression “$\nu-{\kappa}$” also has a literal meaning: i.e., as the function $(\nu-{\kappa})(V):=\nu(V)-{\kappa}(V)$ for all $V\in{\textrm{Ob}({{{\cal V}{(\cal H)}}})}$.

This is not just a coincidence of notation. Indeed, let $F\big({\textrm{Ob}({{{\cal V}{(\cal H)}}})},\mathbb{R}\big)$ denote the set of all real-valued functions on ${\textrm{Ob}({{{\cal V}{(\cal H)}}})}$. Then we can construct the map,

$$\begin{array}{lll} j:k(\varGamma{\underline{{\mathbb{R}}^{\succeq}}})&\rightarrow & F\big({\textrm{Ob}({{{\cal V}{(\cal H)}}})},\mathbb{R}\big) \\ {[}\nu,{\kappa}]&\mapsto& \nu-{\kappa} \end{array}$$

((13.497))

which is well-defined on equivalence classes.

It is easy to see that the map in (13.497) is injective. This raises the question of the image in $F\big({\textrm{Ob}({{{\cal V}{(\cal H)}}})},\mathbb{R}\big)$ of the map j: i.e., what types of real-valued function on ${\textrm{Ob}({{{\cal V}{(\cal H)}}})}$ can be written as the difference between two order-reversing functions?

For functions $f:\mathbb{R}\rightarrow\mathbb{R}$, it is a standard result that a function can be written as the difference between two monotonic functions if, and only if, it has bounded variation. The natural conjecture is that a similar result applies here. To show this, we proceed as follows.

Let $f:{\textrm{Ob}({{{\cal V}{(\cal H)}}})}\rightarrow\mathbb{R}$ be a real-valued function on the set of objects in the category ${\cal V}({\cal H})$. At each $V\in{\textrm{Ob}({{{\cal V}{(\cal H)}}})}$, consider a finite chain

$$C:=\{V_0,V_1,V_2,\ldots,\ V_{n-1},V\mid V_0\subset V_1\subset V_2\subset\cdots\subset V_{n-1}\subset V\}$$

((13.498))

of proper subsets, and define the variation of f on this chain to be

$$V_f(C):=\sum_{j=1}^n|f(V_j)-f(V_{j-1})|$$

((13.499))

where we set $V_n:=V$. Now take the supremum of $V_f(C)$ for all such chains C. If this is finite, we say that f has a bounded variation and define

$$I_f(V):=\sup_C V_f(C)$$

((13.500))

Then it is clear that (i) $V\mapsto I_f(V)$ is an order-preserving function on ${\textrm{Ob}({{{\cal V}{(\cal H)}}})}$; (ii) $f-I_f$ is an order-reversing function on ${\textrm{Ob}({{{\cal V}{(\cal H)}}})}$; and (iii) $-I_f$ is an order-reversing function on ${\textrm{Ob}({{{\cal V}{(\cal H)}}})}$. Thus, any function, f, of bounded variation can be written as

$$f\equiv(f-I_f)-(-I_f)$$

((13.501))

which is the difference of two order-reversing functions; i.e., f can be expressed as the difference of two elements of $\varGamma{\underline{{\mathbb{R}}^{\succeq}}}$.

Conversely, it is a straightforward modification of the proof for functions ${f:\mathbb{R}\rightarrow\mathbb{R}}$, to show that if $f:{\textrm{Ob}({{{\cal V}{(\cal H)}}})}\rightarrow\mathbb{R}$ is the difference of two order-reversing functions, then f is of bounded variation. The conclusion is that $k(\varGamma{\underline{{\mathbb{R}}^{\succeq}}})$ is in bijective correspondence with the set, ${\textrm{BV}({\textrm{Ob}({{\cal V}{(\cal H)}})\mathbb{R})}}$, of functions $f:{\textrm{Ob}({{{\cal V}{(\cal H)}}})}\rightarrow\mathbb{R}$ of bounded variation.

1.7 Taking Squares in $k(\varGamma\underline{\mathbb{R}^{\succeq})}$

We can now think of $k(\varGamma{\underline{{\mathbb{R}}^{\succeq}}})$ in two ways: (i) as the set of equivalence classes $[\nu,{\kappa}]$, of elements $\nu,{\kappa}\in\varGamma{\underline{{\mathbb{R}}^{\succeq}}}$; and (ii) as the set, $\textrm{BV}(\textrm{Ob}({\cal V}({\cal H})),{\mathbb{R}})$, of differences $\nu-{\kappa}$ of such elements.

As expected, $\textrm{BV}(\textrm{Ob}({\cal V}({\cal H})),{\mathbb{R}})$ is an abelian group. Indeed: suppose $\alpha =\nu_1-{\kappa}_1$ and $\beta=\nu_2-{\kappa}_2$ with $\nu_1,\nu_2,{\kappa}_1,{\kappa}_2\in \varGamma{\underline{{\mathbb{R}}^{\succeq}}}$, then

$$\alpha+\beta=(\nu_1+\nu_2)-({\kappa}_1+{\kappa}_2)$$

((13.502))

Hence $\alpha+\beta$ belongs to $\textrm{BV}(\textrm{Ob}({\cal V}({\cal H})),{\mathbb{R}})$ since $\nu_1+\nu_2$ and ${\kappa}_1 +{\kappa}_2$ belong to $\varGamma{\underline{{\mathbb{R}}^{\succeq}}}$.

The definition of $[\nu,0]^2$.

We will now show how to take the square of elements of $k(\varGamma{\underline{{\mathbb{R}}^{\succeq}}})$ that are of the form $[\nu,0]$. Clearly, ν ² is well-defined as a function on $\textrm{Ob}({\cal V}({\cal H}))$, but it may not belong to $\varGamma{\underline{{\mathbb{R}}^{\succeq}}}$. Indeed, if $\nu(V) <0$ for any V, then the function $V\mapsto\nu^2(V)$ can get smaller as V gets smaller, so it is order-preserving instead of order-reversing.

This suggests the following strategy. First, define functions ν ₊ and ν ₋ by

$$\nu_+(V):= \left\{\begin{array}{ll} \nu(V) & \hbox{\ if}\ \nu(V)\geq 0 \\ 0 & \hbox{\ if} \nu(V)< 0 \end{array} \right.$$

((13.503))

and

$$\nu_-(V):= \left\{\begin{array}{ll} 0 & \hbox{if} \nu(V)\geq 0 \\ \nu(V) & {\hbox{\ if} \nu(V) <0.} \end{array} \right.$$

((13.504))

Clearly, $\nu(V)=\nu_+(V)+\nu_-(V)$ for all $V\in\textrm{Ob}({\cal V}({\cal H}))$. Also, for all V, $\nu_+(V)\nu_-(V)=0$, and hence

$$\nu(V)^2=\nu_+(V)^2+\nu_-(V)^2$$

((13.505))

However, (i) the function $V\mapsto\nu_+(V)^2$ is order-reversing; and (ii) the function $V\mapsto\nu_-(V)^2$ is order-preserving. But then $V\mapsto -\nu_-(V)^2$ is order-reversing. Hence, by rewriting (13.505) as

$$\nu(V)^2=\nu_+(V)^2-(-\nu_-(V)^2)$$

((13.506))

we see that the function $V\mapsto \nu^2(V):=\nu(V)^2$ is an element of $\textrm{BV}(\textrm{Ob}({\cal V}({\cal H})),{\mathbb{R}})$.

In terms of $k(\varGamma{\underline{{\mathbb{R}}^{\succeq}}})$, we can define

$$[\nu,0]^2:=[\nu_+^2, -\nu_-^2]$$

((13.507))

which belongs to $k(\varGamma{\underline{{\mathbb{R}}^{\succeq}}})$. Hence, although there exist $\nu\in\varGamma{\underline{{\mathbb{R}}^{\succeq}}}$ that have no square in $\varGamma{\underline{{\mathbb{R}}^{\succeq}}}$, such global elements of ${\underline{{\mathbb{R}}^{\succeq}}}$ do have squares in the k-completion, $k(\varGamma{\underline{{\mathbb{R}}^{\succeq}}})$. On the level of functions of bounded variation, we have shown that the square of a monotonic (order-reversing) function is a function of bounded variation.

On the other hand, we cannot take the square of an arbitrary element $[\nu,{\kappa}]\in\varGamma{\underline{{\mathbb{R}}^{\succeq}}}$, since the square of a function of bounded variation need not be a function of bounded variation.¹³⁶

1.8 The Object $k(\underline{\mathbb{R}^{\succeq}})$ in the Topos $\textbf{Sets}^{{\cal V}({\cal H})^{op}}$

1.8.1 The Definition of $k(\underline{\mathbb{R}^{\succeq}})$

The next step is to translate these results about the set $k(\varGamma{\underline{{\mathbb{R}}^{\succeq}}})$ into the construction of an object ${k({\underline{{{\mathbb{R}}}^{\succeq}}})}$ in the topos $\textbf{Sets}{\cal V}({\cal H})^{\textrm{op}}$. We anticipate that, if this can be done, then $k(\varGamma{\underline{{\mathbb{R}}^{\succeq}}})\simeq\varGamma{k({\underline{{{\mathbb{R}}}^{\succeq}}})}$.

As was discussed in Sect. (13.8.2), the presheaf ${\underline{{\mathbb{R}}^{\succeq}}}$ is defined at each stage V by

$${\underline{{\mathbb{R}}^{\succeq}}}_V:=\{\nu:\downarrow\!\!V\rightarrow\mathbb{R} \mid\nu\in\mathcal{OR}(\downarrow\!\!V,\mathbb{R})\}.$$

((13.508))

If $i_{V^\prime V}:V^\prime\subseteq V$, then the presheaf map from ${\underline{{\mathbb{R}}^{\succeq}}}_V$ to ${\underline{{\mathbb{R}}^{\succeq}}}_{V^\prime}$ is just the restriction of the order-reversing functions from $\downarrow\!\! V$ to $\downarrow\!\! V^\prime$.

The first step in constructing ${k({\underline{{{\mathbb{R}}}^{\succeq}}})}$ is to define an equivalence relation on pairs of functions, $\nu,{\kappa}\in{\underline{{\mathbb{R}}^{\succeq}}}_V$, for each stage V, by saying that $(\nu_1,{\kappa}_1) \equiv (\nu_2,{\kappa}_2)$ if, and only, there exists $\alpha\in{\underline{{\mathbb{R}}^{\succeq}}}_V$ such that

$$\nu_1(V^\prime)+{\kappa}_2(V^\prime)+\alpha(V^\prime) = {\kappa}_1(V^\prime)+\nu_2(V^\prime)+\alpha(V^\prime)$$

((13.509))

for all $V^\prime\subseteq V$.

Definition 25

The presheaf ${k({\underline{{{\mathbb{R}}}^{\succeq}}})}$ is defined over the category ${\cal V}({\cal H})$ in the following way.

• (i) On objects $V\in\textrm{Ob}({\cal V}({\cal H}))$:

  $${k({\underline{{{\mathbb{R}}}^{\succeq}}})}_V:=\{[\nu,{\kappa}]\mid\nu,{\kappa}\in\mathcal{OR}(\downarrow\!\!V,\mathbb{R})\},$$

  ((13.510))

  where $[\nu,{\kappa}]$ denotes the k-equivalence class of $(\nu,{\kappa})$.

• (ii) On morphisms $i_{V^\prime V}:V^\prime\subseteq V$: The arrow ${k({\underline{{{\mathbb{R}}}^{\succeq}}})}(i_{V^\prime V}):{k({\underline{{{\mathbb{R}}}^{\succeq}}})}_V\rightarrow{k({\underline{{{\mathbb{R}}}^{\succeq}}})}_{V^\prime}$ is given by $\big({k({\underline{{{\mathbb{R}}}^{\succeq}}})}(i_{V^\prime V})\big)([\nu,{\kappa}]):=[\nu|_{V^\prime},{\kappa}|_{V^\prime}]$ for all $[\nu,{\kappa}]\in{k({\underline{{{\mathbb{R}}}^{\succeq}}})}_V$.

It is straightforward to show that ${k({\underline{{{\mathbb{R}}}^{\succeq}}})}$ is an abelian group-object in the topos $\textbf{Sets}{\cal V}({\cal H})^{\textrm{op}}$. In particular, an arrow $+:{k({\underline{{{\mathbb{R}}}^{\succeq}}})}\times{k({\underline{{{\mathbb{R}}}^{\succeq}}})}\rightarrow{k({\underline{{{\mathbb{R}}}^{\succeq}}})}$ is defined at each stage V by

$$+_V\big([\nu_1,{\kappa}_1],[\nu_2,{\kappa}_2]\big): = [\nu_1+\nu_2,{\kappa}_1+{\kappa}_2]$$

((13.511))

for all $\big([\nu_1,{\kappa}_1],[\nu_2,{\kappa}_2]\big)\in{k({\underline{{{\mathbb{R}}}^{\succeq}}})}_V\times{k({\underline{{{\mathbb{R}}}^{\succeq}}})}_V$. It is easy to see that (i) $\varGamma{k({\underline{{{\mathbb{R}}}^{\succeq}}})}\simeq k(\varGamma{\underline{{\mathbb{R}}^{\succeq}}})$; and (ii) ${\underline{{\mathbb{R}}^{\succeq}}}$ is a sub-object of ${k({\underline{{{\mathbb{R}}}^{\succeq}}})}$ in the topos $\textbf{Sets}{\cal V}({\cal H})^{\textrm{op}}$.

1.8.2 The Presheaf $k(\underline{\mathbb{R}^{\succeq}})$ as the Quantity-Value Object

We can now identify ${k({\underline{{{\mathbb{R}}}^{\succeq}}})}$ as a possible quantity-value object in $\textbf{Sets}{\cal V}({\cal H})^{\textrm{op}}$. To each bounded, self-adjoint operator ${\hat A}$, there is an arrow $[{\breve{\delta}^o(\hat{A})}]:{\underline{\varSigma}}\rightarrow{k({\underline{{{\mathbb{R}}}^{\succeq}}})}$, given by first sending ${{\hat A}}\in{B\mathcal{(H)}}_\textrm{sa}$ to ${\breve{\delta}^o(\hat{A})}$ and then taking k-equivalence classes. More precisely, one takes the monic $\iota:{\underline{{\mathbb{R}}^{\succeq}}}\hookrightarrow{k({\underline{{{\mathbb{R}}}^{\succeq}}})}$ and then constructs $\iota\circ{\breve{\delta}^o(\hat{A})}:{\underline{\varSigma}}\rightarrow{k({\underline{{{\mathbb{R}}}^{\succeq}}})}.$

Since, for each stage V, the elements in the image of $[{\breve{\delta}^o(\hat{A})}]_V= (\iota\circ{\breve{\delta}^o(\hat{A})})_V$ are of the form $[\nu,0]$, $\nu\in{\underline{{\mathbb{R}}^{\succeq}}}_V$, their square is well-defined. From a physical perspective, the use of ${k({\underline{{{\mathbb{R}}}^{\succeq}}})}$ rather than ${\underline{{\mathbb{R}}^{\succeq}}}$ renders possible the definition of things like the ‘intrinsic dispersion’, $\nabla({{\hat A}}):={\breve{\delta}^o(\hat{A^2})}-{\breve{\delta}^o(\hat{A})}^2$; see (13.306).

1.8.3 The Square of an Arrow $[{\breve{\delta}^o(\hat{A})}]$

An arrow $[{\breve{\delta}^o(\hat{A})}]:{\underline{\varSigma}}\rightarrow{k({\underline{{{\mathbb{R}}}^{\succeq}}})}$ is constructed by first forming the outer daseinisation ${\breve{\delta}^o(\hat{A})}$ of ${\hat A}$, which is an arrow from ${\underline{\varSigma}}$ to ${\underline{{\mathbb{R}}^{\succeq}}}$, and then composing with the monic arrow from ${\underline{{\mathbb{R}}^{\succeq}}}$ to ${k({\underline{{{\mathbb{R}}}^{\succeq}}})}$. Since only outer daseinisation is used, for each $V\in{\cal V}({\cal H})$ and each $\lambda\in{\underline{\varSigma}}_V$ one obtains an element of ${k({\underline{{{\mathbb{R}}}^{\succeq}}})}_V$ of the form $[{\delta^o(\hat{A})}_V(\lambda),0]$. We saw how to take the square of these functions, and applying this to all $\lambda\in{\underline{\varSigma}}_V$ and all $V\in{\cal V}({\cal H})$, we get the square $[{\breve{\delta}^o(\hat{A})}]^2$ of the arrow $[{\breve{\delta}^o(\hat{A})}]$.

If we consider an arrow of the form ${\breve{\delta}(\hat{A})}:{\underline{\varSigma}}\rightarrow{\underline{\mathbb{R}^{\leftrightarrow}}}$, then the construction involves both inner and outer daseinisation, see Theorem 7. For each V and each $\lambda\in{\underline{\varSigma}}_V$, we obtain a pair of functions $(\delta^i(\hat{A})_V(\lambda),{\delta^o(\hat{A})}_V(\lambda))$, which are both not constantly 0 in general. There is no canonical way to take the square of these in ${\underline{\mathbb{R}^{\leftrightarrow}}}_V$. Going to the k-extension $k({\underline{\mathbb{R}^{\leftrightarrow}}})$ of ${\underline{\mathbb{R}^{\leftrightarrow}}}$ does not improve the situation, so we cannot define the square of an arrow ${\breve{\delta}(\hat{A})}$ (or $[{\breve{\delta}(\hat{{A}})}]$ in general.

Appendix 2: A Short Introduction to the Relevant Parts of Topos Theory

2.1 What is a Topos?

It is impossible to give here more than the briefest of introductions to topos theory. At the danger of being highly imprecise, we restrict ourselves to mentioning some aspects of this well-developed mathematical theory and give a number of pointers to the literature. The aim merely is to give a very rough idea of the structure and internal logic of a topos.

There are a number of excellent textbooks on topos theory, and the reader should consult at least one of them. We found the following books useful: [64, 34, 66, 55, 11, 59].

Topos theory is a remarkably rich branch of mathematics which can be approached from a variety of different viewpoints. The basic area of mathematics is category theory; where, we recall, a category consists of a collection of objects and a collection of morphisms (or arrows).

In the special case of the category of sets, the objects are sets, and a morphism is a function between a pair of sets. In general, each morphism f in a category is associated with a pair of objects.¹³⁷ known as its “domain” and “codomain”, and is written as $f:B\rightarrow A$ where B and A are the domain and codomain respectively. Note that this arrow notation is used even if f is not a function in the normal set-theoretic sense. A key ingredient in the definition of a category is that if $f:B\rightarrow A$ and $g:C\rightarrow B$ (i.e., the codomain of g is equal to the domain of f) then f and g can be ‘composed’ to give an arrow $f\circ g:C\rightarrow A$; in the case of the category of sets, this is just the usual composition of functions.

A simple example of a category is given by any partially-ordered set (“poset”) ${\cal C}$: (i) the objects are defined to be the elements of ${\cal C}$; and (ii) if $p,q\in\cal C$, a morphism from p to q is defined to exist if, and only if, $p\preceq q$ in the poset structure. Thus, in a poset regarded as a category, there is at most one morphism between any pair of objects $p,q\in\cal C$; if it exists, we shall write this morphism as $i_{pq}:p\rightarrow q$. This example is important for us in form of the “category of contexts”, ${\cal V}({\cal H})$, in quantum theory. The objects in ${\cal V}({\cal H})$ are the commutative, unital¹³⁸ von Neumann sub-algebras of the algebra, ${B\mathcal{(H)}}$, of all bounded operators on the Hilbert space ${\cal H}$.

Topoi as Mathematical Universes

Every (elementary) topos τ can be seen as a mathematical universe. As a category, a topos τ possesses a number of structures that generalise constructions that are possible in the category, Sets, of sets and functions.¹³⁹ Namely, in Sets, we can construct new sets from given ones in several ways. Specifically, let S, T be two sets, then we can form the cartesian product $S\times T$, the disjoint union $S\amalg T$ and the exponential S ^T—the set of all functions from T to S.

These constructions turn out to be fundamental, and they can all be phrased in an abstract, categorical manner, where they are called the “product”, “co-product” and “exponential”, respectively. By definition, in a topos τ, these operations always exist. The first and second of these properties are called “finite completeness” and “finite co-completeness”, respectively.

One consequence of the existence of finite limits is that each topos, τ, has a terminal object, denoted by 1 _τ . This is characterised by the property that for any object A in the topos τ, there exists exactly one arrow from A to 1 _τ . In Sets, any one-element set $1=\{*\}$ is terminal.¹⁴⁰

Of course, Sets is a topos, too, and it is precisely the topos which usually plays the role of our mathematical universe, since we construct our mathematical objects starting from sets and functions between them. As a slogan, we have: a topos τ is a category with “certain crucial” properties that are similar to those in Sets. A very nice and gentle introduction to these aspects of topos theory is the book [64]. Other good sources are [34, 65].

In order to “do mathematics”, one must also have a logic, including a deductive system. Each topos comes equipped with an internal logic, which is of intuitionistic type. We will now very briefly sketch the main characteristics of intuitionistic logic and the mathematical structures in a topos that realise this logic.

The Sub-object Classifier

Let X be a set, and let $P(X)$ be the power set of X; i.e., the set of subsets of X. Given a subset $K\in P(X)$, one can ask for each point $x\in X$ whether or not it lies in K. Thus there is the characteristic function $\chi_K:X\rightarrow \{0,1\}$ of K, which is defined as

$$\chi_{K}(x):=\left\{\begin{array}{ll}1 & \hbox{if} x\in K \\ 0 & \hbox{if} x\not in K \end{array}\right.$$

((13.512))

for all $x\in X$; cf. (13.95). The two-element set $\{0,1\}$ plays the role of a set of truth values for propositions (of the form “$x\in K$”). Clearly, 1 corresponds to “true”, 0 corresponds to “false”, and there are no other possibilities. This is an argument about sets, so it takes place in, and uses the logic of, the topos Sets of sets and functions. Sets is a Boolean topos, in which the familiar two-valued logic and the axiom (*) hold. (This does not contradict the fact that the internal logic of topoi is intuitionistic, since Boolean logic is a special case of intuitionistic logic.)

In an arbitrary topos, τ, there is a special object Ω _τ , called the sub-object classifier, that takes the role of the set $\{0,1\}\simeq \{\mathrm{false,true}\}$ of truth values. Let B be an object in the topos, and let A be a sub-object of B. This means that there is a monic $A\rightarrow B$,¹⁴¹ (this is the categorical generalisation of the inclusion of a subset K into a larger set X). As in the case of Sets, we can also characterise A as a sub-object of B by an arrow from B to the sub-object classifier Ω _τ ; in Sets, this arrow is the characteristic function $\chi _{K}:X\rightarrow\{0,1\}$ of (13.512). Intuitively, this ‘characteristic arrow’ from B to Ω _τ describes how A ‘lies in’ B. The textbook definition is:

Definition 26

In a category τ with finite limits, a sub-object classifier is an object Ω _τ , together with a monic $\mathrm{true} :1_\tau\rightarrow\varOmega_\tau$, such that to every monic $m:A\rightarrow B$ in τ there is a unique arrow $\chi_A:B\rightarrow\varOmega_\tau$ which, with the given monic, forms a pullback square

In Sets, the arrow $\mathrm{true}:1\rightarrow\{0,1\}$ is given by $\mathrm{true}(*)=1$. In general, the sub-object classifier, Ω _τ , need not be a set, since it is an object in the topos τ, and the objects of τ need not be sets. Nonetheless, there is an abstract notion of elements (or points) in category theory that we can use. Then the elements of Ω _τ are the truth values available in the internal logic of our topos τ, just like “false” and “true”, the elements of $\{\mathrm{false, true}\}$, are the truth values available in the topos Sets.

To understand the abstract notion of elements, let us consider sets for a moment. Let $1=\{*\}$ be a one-element set, the terminal object in Sets. Let S be a set and consider an arrow e from 1 to S. Clearly, (i) $e(*)\in S$ is an element of S; and (ii) the set of all functions from 1 to S corresponds exactly to the set of all elements of S.

This idea can be generalised to any category that has a terminal object 1. More precisely, an element of an object A is defined to be an arrow from 1 to A in the category. For example, in the definition of the sub-object classifier the arrow “$\mathrm{true}:1_\tau\rightarrow\varOmega_\tau$” is an element of Ω _τ . It may happen that an object A has no elements, i.e., there are no arrows $1_\tau\rightarrow A$. It is common to consider arrows from sub-objects U of A to A as generalised elements.

As mentioned above, the elements of the sub-object classifier, understood as the arrows $1_\tau\rightarrow\varOmega_\tau$, are the truth values. Moreover, the set of these arrows forms a Heyting algebra (see, for example, Sect. 8.3 in [34]). This is how (the algebraic representation of) intuitionistic logic manifests itself in a topos. Another, closely related fact is that the set, ${\textrm{Sub}(A)}$, of sub-objects of any object A in a topos forms a Heyting algebra.

The Definition of a Topos

Let us pull together these various remarks and list the most important properties of a topos, τ, for our purposes:

• 1. There is a terminal object 1 _τ in τ. Thus, given any object A in the topos, there is a unique arrow $A\rightarrow 1_\tau$.

  For any object A in the topos, an arrow $1_\tau\rightarrow A$ is called a global element of A. The set of all global elements of A is denoted $\varGamma A$.

  Given $A,B\in{\textrm{Ob}({\tau})}$, there is a product $A\times B$ in τ. In fact, a topos always has pull-backs, and the product is just a special case of this.¹⁴²

• 2. There is an initial object 0 _τ in τ. This means that given any object A in the topos, there is a unique arrow $0_\tau\rightarrow A$.

  Given $A,B\in{\textrm{Ob}({\tau})}$, there is a co-product $A\sqcup B$ in τ. In fact, a topos always has push-outs, and the co-product is just a special case of this.¹⁴³

• 3. There is exponentiation: i.e., given objects $A,B$ in τ we can form an object A ^B, which is the topos analogue of the set of functions from B to A in set theory. The definitive property of exponentiation is that, given any object C, there is an isomorphism

  $${\textrm{Hom}_{\tau}\big({C},{A^B}\big)}\simeq {\textrm{Hom}_{\tau}\big({C\times B},{A}\big)}$$

  ((13.513))

  that is natural in A and C; i.e., it is ‘well-behaved’ under morphisms of the objects involved.

• 4. There is a sub-object classifier Ω _τ .

2.2 Presheaves on a Poset

To illustrate the main ideas, we will first give a few definitions from the theory of presheaves on a partially ordered set (or “poset”); in the case of quantum theory, this poset is the space of “contexts” in which propositions are asserted. We shall then use these ideas to motivate the definition of a presheaf on a general category. Only the briefest of treatments is given here, and the reader is referred to the standard literature for more information [34, 66].

A presheaf (also known as a varying set) $\underline{X}$ on a poset ${\cal C}$ is a function that assigns to each $p\in\cal C$, a set $\underline{X}_p$; and to each pair $p\preceq q$ (i.e., $i_{pq}:p\rightarrow q$), a map $\underline{X}_{qp}:\underline{X}_q\rightarrow \underline{X}_p$ such that (i) $\underline{X}_{pp}:\underline{X}_p\rightarrow\underline{X}_p$ is the identity map $\mathrm{id}_{{\underline{X}_p}}$ on $\underline{X}_p$, and (ii) whenever $p\preceq q\preceq r$, the composite map $\underline{X}_r\stackrel{\underline{X}_{rq}}{\longrightarrow} \underline{X}_q\stackrel{\underline{X}_{qp}}{\longrightarrow} \underline{X}_p$ is equal to $\underline{X}_r\stackrel{\underline{X}_{rp}}{\longrightarrow} \underline{X}_p$, i.e.,

$$\underline{X}_{rp}= \underline{X}_{qp}\circ\underline{X}_{rq}.$$

((13.514))

The notation $\underline{X}_{qp}$ is shorthand for the more cumbersome $\underline{X}(i_{pq})$; see below in the definition of a functor.

An arrow, or natural transformation $\eta:\underline{X}\rightarrow \underline{Y}$ between two presheaves $\underline{X},\underline{Y}$ on ${\cal C}$ is a family of maps $\eta_p:\underline{X}_p\rightarrow \underline{Y}_p$, $p\in\cal C$, that satisfy the intertwining conditions

$$\eta_p\circ\underline{X}_{qp}=\underline{Y}_{qp}\circ\eta_q$$

((13.515))

whenever $p\preceq q$. This is equivalent to the commutative diagram

((13.516))

It follows from these basic definitions, that a sub-object of a presheaf $\underline{X}$ is a presheaf $\underline{K}$, with an arrow $i:\underline{K}\rightarrow \underline{X}$ such that (i) $\underline{K}_p\subseteq \underline{X}_p$ for all $p\in\cal C$; and (ii) for all $p\preceq q$, the map $K_{qp}:\underline{K}_q\rightarrow \underline{K}_p$ is the restriction of $\underline{X}_{qp}:\underline{X}_q\rightarrow \underline{X}_p$ to the subset $\underline{K}_q\subseteq\underline{X}_q$. This is shown in the commutative diagram

((13.517))

where the vertical arrows are subset inclusions.

The collection of all presheaves on a poset $\mathcal{C}$ forms a category, denoted $\textbf{Sets}^{{\mathcal{C}^{\textrm{op}}}}$. The arrows/morphisms between presheaves in this category the arrows (natural transformations) defined above.

2.3 Presheaves on a General Category

The ideas sketched above admit an immediate generalization to the theory of presheaves on an arbitrary “small” category $\mathcal{C}$ (the qualification “small” means that the collection of objects is a genuine set, as is the collection of all arrows/morphisms between any pair of objects). To make the necessary definition we first need the idea of a “functor”:

The Idea of a Functor

A central concept is that of a “functor” between a pair of categories ${\cal C}$ and ${\cal D}$. Broadly speaking, this is an arrow-preserving function from one category to the other. The precise definition is as follows.

Definition 27

• 1. A covariant functor F from a category ${\cal C}$ to a category ${\cal D}$ is a function that assigns

  • (a) to each ${\cal C}$-object A, a ${\cal D}$-object F _A ;

  • (b) to each ${\cal C}$-morphism $f:B\rightarrow A$, a ${\cal D}$-morphism $F(f):F_B\rightarrow F_A$ such that $F({\textrm{id}}_A)=\mathrm{id}_{F_A}$; and, if $g:C\rightarrow B$, and $f:B\rightarrow A$ then

    $$F(f\circ g)=F(f)\circ F(g).$$

    ((13.518))

• 2. A contravariant functor X from a category ${\cal C}$ to a category ${\cal D}$ is a function that assigns

  • (a) to each ${\cal C}$-object A, a $\cal D$-object X _A ;

  • (b) to each ${\cal C}$-morphism $f:B\rightarrow A$, a ${\cal D}$-morphism $X(f):X_A\rightarrow X_B$ such that $X({\textrm{id}}_A)={\textrm{id}}_{X_A}$; and, if $g:C\rightarrow B$, and $f:B\rightarrow A$ then

    $$X(f\circ g)= X(g)\circ X(f).$$

    ((13.519))

The connection with the idea of a presheaf on a poset is straightforward. As mentioned above, a poset ${\cal C}$ can be regarded as a category in its own right, and it is clear that a presheaf on the poset ${\cal C}$ is the same thing as a contravariant functor $\underline{X}$ from the category ${\cal C}$ to the category Sets of normal sets. Equivalently, it is a covariant functor from the “opposite” category¹⁴⁴ $\mathcal{C}^\mathrm{op}$ to Sets. Clearly, (13.514) corresponds to the contravariant condition (13.519). Note that mathematicians usually call the objects in ${\cal C}$ “stages of truth”, or just “stages”. For us they are “contexts”, “classical snap-shops”, or “world views”.

Presheaves on an Arbitrary Category ${\cal C}$

These remarks motivate the definition of a presheaf on an arbitrary small category ${\cal C}$: namely, a presheaf on ${\cal C}$ is a covariant functor¹⁴⁵ $\underline{X}:\mathcal{C}^{\textrm{op}}\rightarrow{\textbf{Sets}}$ from $\mathcal{C}^{\textrm{op}}$ to the category of sets. Equivalently, a presheaf is a contravariant functor from ${\cal C}$ to the category of sets.

We want to make the collection of presheaves on ${\cal C}$ into a category, and therefore we need to define what is meant by a “morphism” between two presheaves $\underline{X}$ and $\underline{Y}$. The intuitive idea is that such a morphism from $\underline{X}$ to $\underline{Y}$ must give a “picture” of $\underline{X}$ within $\underline{Y}$. Formally, such a morphism is defined to be a natural transformation $N:\underline{X}\rightarrow\underline{Y}$, by which is meant a family of maps (called the components of N) $N_A:\underline{X}_A\rightarrow\underline{Y}_A$, $A\in{\textrm{Ob}(\mathcal{C})}$, such that if $f:B\rightarrow A$ is a morphism in ${\cal C}$, then the composite map $\underline{X}_{A} \stackrel{N_A}\longrightarrow\underline{Y}_A\stackrel{\underline{Y}(f)} \longrightarrow\underline{Y}_B$ is equal to $\underline{X}_A \stackrel{\underline{X}(f)}\longrightarrow\underline{X}_B\stackrel{N_B} \longrightarrow \underline{Y}_A$. In other words, we have the commutative diagram

((13.520))

of which (13.516) is clearly a special case. The category of presheaves on ${\cal C}$ equipped with these morphisms is denoted $\textbf{Sets}^{{\mathcal{C}^{\textrm{op}}}}$.

The idea of a sub-object generalizes in an obvious way. Thus we say that $\underline{K}$ is a sub-object of $\underline{X}$ if there is a morphism in the category of presheaves (i.e., a natural transformation) $\iota:\underline{K}\rightarrow\underline{X}$ with the property that, for each A, the component map $\iota_A:\underline{K}_A\rightarrow\underline{X}_A$ is a subset embedding, i.e., $\underline{K}_A\subseteq \underline{X}_A$. Thus, if $f:B\rightarrow A$ is any morphism in ${\cal C}$, we get the analogue of the commutative diagram (13.517):

((13.521))

where, once again, the vertical arrows are subset inclusions.

The category of presheaves on ${\cal C}$, ${\textbf{Sets}}^{\mathcal{C}^\mathrm{op}}$, forms a topos. We do not need the full definition of a topos; but we do need the idea, mentioned in Sect. 13.17.2, that a topos has a sub-object classifier Ω, to which we now turn.

Sieves and The Sub-object Classifier ${\underline\varOmega}$

Among the key concepts in presheaf theory is that of a “sieve”, which plays a central role in the construction of the sub-object classifier in the topos of presheaves on a category ${\cal C}$.

A sieve on an object A in ${\cal C}$ is defined to be a collection S of morphisms $f:B\rightarrow A$ in ${\cal C}$ with the property that if $f:B\rightarrow A$ belongs to S, and if $g:C\rightarrow B$ is any morphism with co-domain B, then $f\circ g:C\rightarrow A$ also belongs to S. In the simple case where ${\cal C}$ is a poset, a sieve on $p\in\cal C$ is any subset S of ${\cal C}$ such that if $r\in S$ then (i) $r\preceq p$, and (ii) $r'\in S$ for all $r'\preceq r$; in other words, a sieve is nothing but a lower set in the poset.

The presheaf $\underline{\Omega}:\mathcal{C}\rightarrow {\textbf{Sets}}$ is now defined as follows. If A is an object in ${\cal C}$, then $\underline{\Omega}_{A}$ is defined to be the set of all sieves on A; and if $f:B\rightarrow A$, then $\underline{\Omega}(f):\underline{\Omega}_A\rightarrow\underline{\Omega}_B$ is defined as

$${\underline{\varOmega}}(f)(S):= \{h:C\rightarrow B\mid f\circ h\in S\}$$

((13.522))

for all $S\in\underline{\Omega}_A$; the sieve $\underline{\Omega}(f)(S)$ is often written as $f^{\ast}(S)$, and is known as the pull-back to B of the sieve S on A by the morphism $f:B\rightarrow A$.

It should be noted that if S is a sieve on A, and if $f:B\rightarrow A$ belongs to S, then from the defining property of a sieve we have

$$f^{\ast}(S):=\{h:C\rightarrow B\mid f\circ h\in S\}= \{h:C\rightarrow B\}=:\ \downarrow\!\!B$$

((13.523))

where $\downarrow\!\!B$ denotes the principal sieve on B, defined to be the set of all morphisms in ${\cal C}$ whose codomain is B.

If ${\cal C}$ is a poset, the pull-back operation corresponds to a family of maps $\underline{\Omega}_{qp}:\underline{\Omega}_q\rightarrow\underline{\Omega}_p$ (where $\underline{\Omega}_p$ denotes the set of all sieves/lower sets on p in the poset) defined by $\underline{\Omega}_{qp}=\underline{\Omega}(i_{pq})$ if $i_{pq}:p\rightarrow q$ (i.e., $p\preceq q$). It is straightforward to check that if $S\in\underline{\Omega}_q$, then

$$\underline{\Omega}_{qp}(S):=\downarrow\!{p}\cap S$$

((13.524))

where $\downarrow\!{p}:=\{r\in\mathcal{C}\mid r\preceq p\}$.

A crucial property of sieves is that the set $\underline{\Omega}_{A}$ of sieves on A has the structure of a Heyting algebra. Specifically, the unit element $1_{\underline{\Omega}_A}$ in $\underline{\Omega}_{A}$ is the principal sieve $\downarrow\!\!A$, and the null element $0_{\underline{\Omega}_A}$ is the empty sieve ø. The partial ordering in $\underline{\Omega}_{A}$ is defined by $S_1\preceq S_2$ if, and only if, $S_1\subseteq S_2$; and the logical connectives are defined as:

$$S_1\wedge S_2:=S_1\cap S_2$$

((13.525))

$$S_1\vee S_2:=S_1\cup S_2$$

((13.526))

$$S_1\Rightarrow S_2:=\{f:B\rightarrow A\mid \forall g:C\rightarrow B \hbox{if} f\circ g\in S_1 \hbox{then} f\circ g\in S_2\}$$

((13.527))

As in any Heyting algebra, the negation of an element S (called the pseudo-complement of S) is defined as $\neg S:=S\Rightarrow 0$; so that

$$\neg S:=\{f:B\rightarrow A\mid \hbox{for all} g:C\rightarrow B, f\circ g\not\in S \}.$$

((13.528))

It can be shown that the presheaf $\underline{\Omega}$ is a sub-object classifier for the topos $\textbf{Sets}^{{\mathcal{C}^{\textrm{op}}}}$. That is to say, sub-objects of any object $\underline{X}$ in this topos (i.e., any presheaf on ${\cal C}$) are in one-to-one correspondence with morphisms $\chi:\underline{X}\rightarrow {\underline{\varOmega}}$. This works as follows. First, let ${\underline{K}}$ be a sub-object of $\underline{X}$ with an associated characteristic arrow $\chi_{\underline{K}}:\underline{X}\rightarrow{\underline{\varOmega}}$. Then, at any stage A in ${\cal C}$, the ‘components’ of this arrow, $\chi_{\underline{K}A}:\underline{X}_A\rightarrow\underline{\Omega}_A$, are defined as

$$\chi_{\underline{K} A}(x):=\{f:B\rightarrow A\mid \underline{X}(f)(x)\in \underline{K}_B\}$$

((13.529))

for all $x\in \underline{X}_A$. That the right hand side of (13.529) actually is a sieve on A follows from the defining properties of a sub-object.

Thus, in each “branch” of the category ${\cal C}$ going “down” from the stage A, $\chi{\underline{K}}_A(x)$ picks out the first member B in that branch for which $\underline{X}(f)(x)$ lies in the subset $\underline{K}_B$, and the commutative diagram (13.521) then guarantees that $\underline{X}(h\circ f)(x)$ will lie in $\underline{K}_C$ for all $h:C\rightarrow B$. Thus each stage A in ${\cal C}$ serves as a possible context for an assignment to each $x\in \underline{X}_A$ of a generalised truth value—a sieve belonging to the Heyting algebra $\underline{\Omega}_{A}$. This is the sense in which contextual, generalised truth values arise naturally in a topos of presheaves.

There is a converse to (13.529): namely, each morphism $\chi:\underline{X}\rightarrow{\underline{\varOmega}}$ (i.e., a natural transformation between the presheaves $\underline{X}$ and ${\underline{\varOmega}}$) defines a sub-object $\underline{K}^\chi$ of $\underline{X}$ via

$$\underline{K}^\chi_A:=\chi_A^{-1}\{1_{\Omega_A}\}.$$

((13.530))

at each stage A.

Global Elements of a Presheaf

For the category of presheaves on ${\cal C}$, a terminal object $\underline{1}:\mathcal{C}\rightarrow {\textbf{Sets}}$ can be defined by $\underline{1}_A:=\{*\}$ at all stages A in ${\cal C}$; if $f:B\rightarrow A$ is a morphism in ${\cal C}$ then $\underline{1}(f):\{*\}\rightarrow\{*\}$ is defined to be the map $*\mapsto *$. This is indeed a terminal object since, for any presheaf $\underline{X}$, we can define a unique natural transformation $N:\underline{X}\rightarrow\underline{1}$ whose components $N_A:\underline{X}(A)\rightarrow\underline{1}_A=\{*\}$ are the constant maps $x\mapsto *$ for all $x\in\underline{X}_A$.

As a morphism $\gamma:\underline{1}\rightarrow\underline{X}$ in the topos $\textbf{Sets}^{\mathcal{C}^{\textrm{op}}}$, a global element corresponds to a choice of an element $\gamma_A\in\underline{X}_A$ for each stage A in ${\cal C}$, such that, if $f:B\rightarrow A$, the ‘matching condition’

$$\underline{X}(f)(\gamma_A)=\gamma_B$$

((13.531))

is satisfied.