Appendix A Topoi and Logic

Transcription

1 ppendix Topoi and Logic In this section, we will explore the tight connection between topos theory and logic. In particular, to each topos there is associated a language for expressing the internal language of the topos. The converse is also true: given a language one can define a corresponding topos.. First Order Languages language, in its most raw definition, comprises a collection of atomic variables, and a collection of primitive operations called logical connectives, whose role is to combine together such primitive variables transforming them into formulas or sentences. Moreover, in order to reason with a given language, one also requires rules of inference, i.e. rules which allow you to generate other valid sentences from the given ones. The semantics or meaning of the logical connectives, however, is not given by the formulae and sentences themselves, but it is defined through a so called evaluation map, which is a map from the set of atomic variables and sentences to a set of truth values. Such a map enables one to determine when a formula is true and, thus, defines its semantics/meaning. In this perspective it turns out that the meaning of the logical connectives is given in terms of some set of objects which represent the truth values. The logic that a given language will exhibit will depend on what the set of truth values is considered to be. In fact, what was said above is a very abstract characterisation of what a language is. To actually use it as a deductive system of reasoning, one needs to define a mathematical context in which to represent this abstract language. In this way the elementary and compound propositions will be represented by certain mathematical objects and the set of truth values will itself be identified with an algebra. For example, in standard classical logic, the mathematical context used is Sets and the algebra of truth values is the Boolean algebra of subsets of a given set. C. Flori, First Course in Topos Quantum Theory, Lecture Notes in Physics 868, DOI 0.007/ , Springer-Verlag Berlin Heidelberg

2 398 Topoi and Logic However, in a general topos, the internal logic/algebra will not be Boolean but will be a generalisation of it, namely a Heyting algebra. In order to get a better understanding of what a language is, we will start with a very simple language called propositional language P(l)..2 Propositional Language The propositional language P(l) contains a set of symbols and a set of formation rules. Symbols of P(l) (i) n infinite list of symbols α 0,α,α 2,...called primitive propositions. (ii) set of symbols,,, which, for now, have no explicit meaning. (iii) Brackets ), (. Formation Rules (i) Each primitive proposition α i P(l)is a sentence. (ii) If α is a sentence, then so is α. (iii) If α and α 2 are sentences, then so are α α 2, α α 2 and α α 2. Note also that P(l) does not contain the quantifiers and. This is because it is only a propositional language. To account for quantifiers one has to go to more complicated languages called higher-order languages, which will be described later. The inference rule present in P(l)is the modus ponens (the rule of detachment ) which states that, from α i and α i α j, the sentence α j may be derived. Symbolically this is written as α i,α i α j. (.) α i We will see, later on, what exactly the above expression means. In order to use the language P(l) one needs to represent it in a mathematical context. The choice of such a context will depend on what type of system we want to reason about. For now we will consider a classical system, thus the mathematical context in which to represent the language P(l) will be Sets. InSets, the truth object (object in which the truth values lie) will be the Boolean set {0, }, thus the truth values will undergo a Boolean algebra. This, in turn, implies that the logic of the language P(l), as represented in Sets, will be Boolean. The rigorous definition of a representation of the language P(l)is as follows: Definition. Given a language P(l)and a mathematical context τ, a representation of P(l)in τ is a map π from the set of primitive propositions to elements in the algebra in question: α π(α).

3 .2 Propositional Language 399 s we will see, the specification of the algebra will depend on what type of theory we are considering, i.e. classical or quantum. In classical physics, propositions are represented by the Boolean algebra of all (Borel) subsets of the classical state space, thus, given a representation π, we can define the semantics of P(l)as follows: π(α i α j ) := π(α i ) π(α j ) π(α i α j ) := π(α i ) π(α j ) π(α i α j ) := π(α i ) π(α j ) π( α i ) := ( π(α i ) ) (.2) where, on the left hand side, the symbols {,,, } are elements of the language P(l), while on the right hand side they are the logical connectives in algebra, in which the representation takes place. It is in such an algebra that the logical connectives acquire meaning. For the classical case, since the algebra in which a representation lives, is the Boolean algebra of subsets, the logical connectives on the right hand side of (.2) are defined in terms of set-theoretic operations. In particular, we have the following associations: π(α i ) π(α j ) := π(α i ) π(α j ) π(α i ) π(α j ) := π(α i ) π(α j ) ( φ(α i ) ) := π(α i ) c π(α i ) π(α j ) := π(α i ) c π(α j ). (.3) (.4) So far, we have seen how logical connectives are represented in the topos Sets. However, it is possible to give a general definition of logical connectives in terms of arrows. Such a definition would then be valid for any topos. To retrieve the logical connectives for the classical case, in which the topos is Sets, we then simply replace, in the definitions that will follow, the general truth object Ω with the set {0, }=2. Logical connectives in a general topos τ are defined as follows: Negation We will now describe how to represent negation as an arrow in a given topos τ. Let us assume that the τ -arrow representing the value true is : Ω, which is the arrow used in the definition of the sub-object classifier. Given such an arrow, negation is identified with the unique arrow :Ω Ω, such that the following

4 400 Topoi and Logic diagram is a pullback: Ω Ω Where is the topos analogue of the arrow false in Sets, i.e. is the character of! : 0 : 0!! Ω Conjunction Conjunction is identified with the following arrow: :Ω Ω Ω which is the character of the product arrow, : Ω Ω, such that the following diagram is a pullback:, Ω Ω id Ω

5 .2 Propositional Language 40 where, is defined as follows: I, Ω pr Ω Ω I Ω pr 2 Disjunction Disjunction is identified with the arrow :Ω Ω Ω (.5) which is the character of the image of the arrow [, idω, id Ω, ] : Ω + Ω Ω Ω (.6) such that the following diagram commutes: Ω + Ω [, Ω, Ω, ] Ω Ω! Ω+Ω Ω Implication Given two arrows their equaliser is some map Ω Ω pr Ω e : ( := { x,y x y in Ω }) Ω Ω (.7) such that e = pr e. Implication is then defined as the character of e,i.e.asthemap : Ω Ω Ω (.8)

6 402 Topoi and Logic such that the following diagram is a pullback: e Ω Ω! e Ω In order to complete the definition of a propositional language in a general topos τ, we also need to define the valuation functions (which give us the semantics) in terms of arrows in that topos. We recall from the definition of the sub-object classifier that a truth value in a general topos τ is given by a map Ω (in Sets we have {0, }=2 = Ω). The collection of such arrows τ(,ω)represents the collection of all truth values. Thus, a valuation map in a general topos is defined to beamapv :{π(α i )} τ(,ω)such that the following equalities hold: V ( ( π(α i ) )) = V ( π(α i ) ) (.9) V ( π(α i ) π(α j ) ) = V ( π(α i ) ),V ( π(α j ) ) (.0) V ( π(α i ) π(α j ) ) = V ( π(α i ) ),V ( π(α j ) ) (.) V ( π(α i ) π(α j ) ) = V ( π(α i ) ),V ( π(α j ) ). (.2).2. Example in Classical Physics We have stated above that classical physics uses the topos Sets. We now want to represent, in Sets, the propositional language P(l), as defined for a classical system S. So now the elementary propositions will be propositions pertaining the physical system S. From now on, we will denote a language referred to a particular system S by P (l)(s). Since S is a (classical) physical system, the elementary propositions which P (l)(s) will contain will be of the form Δ meaning the quantity which represents some physical observable, has value in a set Δ. We now define the representation map for this language as follows: π cl : P (l)(s) O(S) Δ { s S Ã(s) Δ } = à (Δ) (.3) where S is the classical state space, O(S) is the Boolean algebra of subsets of S which lives in Sets and à : S R is the map from the state space to the reals, which identifies the physical quantity. We now define the truth values for propositions.

7 .3 The Higher Order Type Language l 403 Normally, such truth values are state-dependent, i.e. they depend on the state with respect to which we are preforming the evaluation. In classical physics, states are simply identified with elements s of the state space S. Thus, for all s S, we define the truth value of the proposition à (Δ) as follows: v( Δ; s) = { iff s à (Δ) 0 otherwise. (.4) Therefore the truth values lie in the Boolean algebra Ω ={0, }. It is interesting to note that the application of the propositional language P(l)for quantum theory, fails. This is because in quantum theory propositions are identified with projection operators, thus the representation map will be π q :{α i } P(H) Δ π q ( Δ) := Ê[ Δ] (.5) where Ê[ Δ] is the projection operator which projects onto the subset Δ of the spectrum of Â. Now the problem with this construction is that the set of all projection operators undergoes a logic which is not distributive, but the logic of the propositional language is distributive with respect to the logical connectives and. Therefore, such a representation will not work. To solve this problem we need to introduce a higher order language which we will examine in the next section..3 The Higher Order Type Language l We will now define a more complex language called higher order type language and denoted by l. Such a language consists of a set of symbols and terms. Symbols. collection of sorts or types. If T,T 2,...,T n, n, are type symbols, then so is T T 2 T n.ifn = 0 then T T 2 T n =. 2. If T is a type symbol, then so is PT. 3. Given any type T there are countable many variables of type T. 4. There is a special symbol. 5. set of function symbols for each pair of type symbols, together with a map which assigns to each function its type. This assignment consists of a finite, nonempty list of types. For example, if we have the pair of type symbols (T,T 2 ), the associated function symbol will be F l (T,T 2 ). n element f F l (T,T 2 ) has type T,T 2. This is indicated by writing f : T T 2. PT indicates the collection of all subobjects of T.

8 404 Topoi and Logic 6. set of relation symbols R i together with a map which assigns the type of the arguments of the relation. This consists of a list of types. For example, a relation taking an argument x T of type T and an argument x 2 T 2 of type T 2 is denoted as R = R(x,x 2 ) T T 2. Terms. The variables of type T are terms of type T, T. 2. The symbol is a term of type. 3. term of type Ω is called a formula. If the formula has no free variables, then we call it a sentence. 4. Given a function symbol f : T T 2 and a term t of type T, then f(t) is a term of type T Given t,t 2,...,t n which are terms of type T,T 2,...,T n, respectively, then t,t 2,...,t n is a term of type T T 2 T n. 6. If x is a term of type T T 2 T n, then for i n, x i is a term of type T i. 7. If ω is a term of type Ω and α is a variable of type T, then {α ω} is a term of type PT. 8. If x,x 2 are terms of the same type, then x = x 2 is a term of type Ω. 9. If x,x 2 are terms of type T and PT respectively, then x x 2 is a term of type Ω. 0. If x,x 2 are terms of type PT and PPT respectively, then x x 2 is a term of type Ω.. If x,x 2 are both terms of type PT, then x x 2 is a term of type Ω. The entire set of formulas in the language l are defined, recursively, through repeated applications of formation rules, which are the analogues of the standard logical connectives. In particular, we have atomic formulas and composite formulas. The former are:. The set of relation symbols. 2. Equality terms defined above. 3. Truth is an atomic formula with no free variables. 4. False is an atomic formula with no of free variables. We can now build more complicated formulas through the use of the logical connectives,, and. These are the composite formulas:. Given two formulas α and β, then α β is a formula for which, the set of free variables is defined to be the union of the free variables in α and β. 2. Given two formulas α and β, then α β is a formula for which, the set of free variables is defined to be the union of the free variables in α and β. 3. Given a formula α its negation α is still a formula with the same number of free variables. 4. Given two formulas α and β, then α β is a formula with free variables given by the union of the free variables in α and β.

9 .4 Representation of l in a Topos 405 It is interesting to note that the logical operations just defined can actually be expressed in terms of the primitive symbols as follows:. :=( = ). 2. α β := ( α, β =, ) = =, =. 3. α β := (α = β). 4. α β := ((α β) α) := ( α, β =, = α). 5. xα := ({x : α}={x : }) := ww = ({w : w}={w : }). 7. α := α. 8. α β := w[(α w β w) w]. 9. xα := w[ x(α w) w]. 3.4 Representation of l in a Topos We now want to show how a representation of the first order language l takes place in a topos. The main idea is that of identifying each of the terms in l an arrow in a topos. In particular we have: Definition.2 Given a topos τ the interpretation/representation (M) of the language l in τ consists of the following associations:. To each type T l an object T τ M τ. 2. To each relation symbol R T T 2 T n a sub-object R τ M T τ M T τ M 2 T τ M n. 3. To each function symbol f : T T 2 T n X a τ -arrow f τ M : T τ M T τ M 2 T τ M n X. 4. To each constant c of type T a τ -arrow c τ M : τ M T τ M. 5. To each variabe x of type T a τ -arrow x τ M : T τ M T τ M. 6. The symbol Ω is represented by the sub-object classifier Ω τ M. 7. The symbol is represented by the terminal object τ M. Now that we understand how the basic symbols of the abstract language l are represented in a topos, we can proceed to understand also how the various terms and formulas are represented. Needless to say, these are all defined in a recursive manner. Given a term t(x,x 2,...,x n ) of type Y with free variables x i of type T i,i.e. t(x,x 2,...,x n ) : T T n Y, then the representative in a topos of this term 2 xα means: for all x with the property α, while {x : y} indicates the set of all x, such that y 3 xα means: there exists an x with the property α.

10 406 Topoi and Logic would be a τ -map t(x,x 2,...,x n ) : T τ M T τ M n Y τ M. (.6) Formulas in the language are interpreted as terms of type Ω. In the topos τ this type Ω is identified with the sub-object classifier Ω τ M. In particular, a term of type Ω of the form φ(t,t 2,...,t n ) with free variables t i of type T i is represented by an arrow φ(t,t 2,...,t n ) τ M : T τ M T τ M n Ω τ M. On the other hand, a term φ of type Ω with no free variables is represented by a global element φ : τ M Ω τ M. s we will see, these arrows will represent the truth values. The reason why, in a topos, formulas are identified with arrows with codomain Ω rests on the fact that sub-objects, of a given object in a topos, are in bijective correspondence with maps from that object to the sub-object classifier. In fact, by construction, formulas single out sub-objects of a given object X in terms of a particular relation which they satisfy, i.e. they define elements of Sub(X). Such sub-objects are in bijective correspondence with maps X Ω. In particular, given a formula φ(x,...,x n ) with free variables x i of type T i, which in the language l is associated with the subset {x i φ} i T i, we obtain the topos representation { (x,...,x n ) φ } τ M T τ M T τ M n (.7) which, through the Omega xiom (see xion 8.4), gets identified with the map { (x,...,x n ) φ } τ M T τ M T τ χ {(x,...,xn) φ} τ M M n Ω τ M. (.8) To understand how formulas are represented in a topos τ, let us consider the formula stating that two terms are the same, i.e. t(x,x 2,...,x n ) = t (x,x 2,...,x n ).The representation of such a formula in a topos τ is identified with the equalizer of the two τ -arrows representing the terms t(x,x 2,...,x n ) and t (x,x 2,...,x n ).In particular, we have { x,x 2,...,x n t = t } τ M T τ M T τ M n tτm t τ M Y τm Instead, if we consider a relation R(t,...,t n ) of terms t i of type Y i with variables x j of type T j, then the formula pertaining this relation {(x...x n ) R(t,...,t n )} is represented, in τ, by pulling back the sub-object R τ M Y τ M Y τ M n (representing the relation R(t,...,t n )) along the term arrow t τ M,...,t τ M n :T τ M T τ M n

11 .4 Representation of l in a Topos 407 Y τ M Y τ M n : {(x,x 2,...,x n ) R(t,...,t n )} τ M R τ M T τ M T τ M τ t M,...,t τ M n n Y τ M Y τ M n The atomic formulas meaning truth and false ( and, respectively) will be represented in a topos τ by the greatest and lowest elements of the Heyting algebra of the sub-objects of any object in the topos. For example we have that {x...x n } τ M = T τ M T τ M 2 T τ M n (.9) {x...x n } τ M = τ M. (.20) So far we have established how to represent formulas in a topos. Next, we will explain how to represent logical connectives between formulas in a topos. In particular, given a collection of formulas represented as sub-objects of the type object i T i, the logical connectives between these are represented by the corresponding operations in the Heyting algebra of sub-objects of the object i T τ M i in τ. Since we are dealing with sub-objects, we can also represent the logical connectives in terms of τ -arrows with codomain Ω as follows: For example, consider two formulas φ, ρ, of type Ω with free variables x and x 2 of type T τ M and T τ M 2, respectively. The conjunction φ ρ is represented by the arrow φ ρ : T τ M T τ φ,ρ M 2 Ω τ M Ω τ M Ω τ M. (.2) Similarly, we have φ ρ : T τ M T τ M φ,ρ 2 Ω τ M Ω τ M Ω τ M (.22) φ ρ : T τ M T τ φ,ρ M 2 Ω τ M Ω τ M Ω τ M (.23) ρ : T τ M ρ 2 Ω τ M Ω τ M. (.24) We now give two very important notions: that of a theory in l and that of a model of l. Definition.3 Given a language l, a theory T in l is a set of formulas which are called the axioms of T. Definition.4 model of a theory is a representation M in which all the axioms of T are valid. Such axioms are, then, represented by the arrow true : Ω.

12 408 Topoi and Logic n example of this is given by the theory of abelian groups which can be seen as model of a theory in a given language. The language required will only contain one type of elements G, no relations, two function symbols +:G G G :G G (.25) (.26) and a constant 0. representation of this language, which will lead us to the theory of groups, will be defined in the topos Sets. Such a representation of G will be identified as a set G M, on which the function symbols act upon: and + M : G M G M G M (.27) g,g 2 g g 2 (.28) M : G M G M g g. (.29) (.30) The constant 0 will be an element 0 M G M. Such a representation will be a model for the theory of abelian groups if the function symbols satisfy the axioms of abelian groups, i.e. if the following should hold in the representation M: (g + g 2 ) + g 3 = g + (g 2 + g 3 ) (.3) g + g 2 = g 2 + g (.32) g + 0 = g (.33) g + ( g ) = 0. (.34) Given two models M and M of a theory T in a language l, we say that these two models are homomorphic if there is a homomorphism of the respective interpretations of the model, i.e. for each symbol type X in l, these maps are homomorphisms: H X : X M X M (.35) where X M and X M are the representations of the symbol type X of l in the representation M and M, respectively. Such a map is called a homomorphism if it respects all relation symbols, function symbols and constants. In the example of abelian groups, model homomorphisms would simply be group homomorphisms. The definition of homomorphic representations gives rise to a category I, whose objects are all possible representations of a given language l in a topos τ, and whose morphisms are the above mentioned homomorphisms of representations. Given such a category, each theory T gives rise to a full subcategory of I called Mod(T,τ),

13 .5 Language l for a Theory of Physics and Its Representation in a Topos τ 409 whose objects are models of the theory T in the topos τ, and whose morphisms are homomorphisms of models. In this section, we have seen how, given a first order type language l, it is possible to represent such a language in a topos τ. However, interestingly enough, the converse is also true, namely: given a topos τ, it has associated to it an internal first order language l, which enables one to reason about τ in a set theoretic way, i.e. using the notion of elements. Definition.5 Given a topos τ, its internal language l(τ) has a type symbol for each object τ, a function symbol f : 2 n B for each map f : 2 n B in τ and a relation R 2 n for each sub-object R 2 n in τ..5 Language l for a Theory of Physics and Its Representation in a Topos τ We will now try to construct a physics theory for a system S. The construction of such a theory is defined by an interplay between a language l(s), associated to the system S, a topos and the representation of the theory in the topos. In particular, we can say that a theory of the system S is defined by choosing a representation/model, M, of the language l(s) in a topos τ M. The choice of both topos and representation depend on the kind of theory being used, i.e. if it is classical or quantum theory. s we have seen above, since each topos τ has an internal language l(τ) associated to it, constructing a theory of physics consists in translating the language, l(s), of the system to the local language l(τ) of the topos. s a first step in constructing a theory of physics we need to specify, exactly, what l(s) is. In particular we need to analyse which primitive type terms and formulas should be present in l(s) for it to be a language that will enable us to talk about the physical system S..5. The Language l(s) of a System S The minimum set of type symbols and formulas, which are needed for a language to be used as a language to talk about a physical system S, are the following:. The state space object and the quantity value object. These objects are represented in l(s) by the ground type symbols Σ and R. 2. Physical quantities. Given a physical quantity, it is standard practice to represent such a quantity in terms of a function from the state space to the quantity value object. Thus, we require l(s) to contain the set function symbols F l(s) (Σ, R) of signature Σ R, such that the physical quantity is : Σ R.

14 40 Topoi and Logic 3. Values. We would like to have values of physical quantities. These are defined in l(s) as terms of type R with free variables s of type Σ, i.e.theyaretheterms (s) R, where : Σ R F l(s) (Σ, R). 4. Propositions. Imagine we would like to talk about collections of states of the system with a particular property. Such a collection is represented in terms of sub-objects of the state space, which comprises the states with that particular property in question. Thus we have terms Q ={s (s) Δ} which are of type PΣ with a free variable Δ of type P R. 5. Truth values. We generally would like to talk about values of physical quantities for a given state of the system, thus we require the presence of formulas of the type (s) Δ, where Δ is a variable of type P R and s is a variable of type Σ. Such a formula is a term of type Ω. formulaw with no free variables, called a sentence, is a special element of Ω which is represented, in a topos, by a global element of Ω,i.e. [w]: Ω. (.36) These, as we will see later on, will represent truth values for propositions about the system. 6. States. There are three options for describing a state, which we will analyse separately. (i) Microstate option. The microstate option is the one used in classical physics where a state is identified with an element of the state space. Hence in the context of the language l(s), a micro-state is a term t of type Σ,i.e.t Σ. To understand how the micro-state option is utilised to evaluate proposition consider the term (s) Δ, this is a term of type Ω with free variables s and Δ of type Σ and P(R), respectively. On the other hand {s (s) Δ} is a term of type P(Σ) with free variable Δ of type P(R). Given a state t Σ we can then form a term of type Ω as follows: t {s (s) Δ}.This term has free variables t and Δ of type Σ and P R, respectively. Intuitively, (t) Δ represents the proposition stating: the value of, given the state t, lies in the range Δ. However semantically 4 we have the following equivalence t { s (s) Δ } (t) Δ. (.37) Therefore the proposition the value of, given the state t, lies in the range Δ becomes the term (t) Δ of type Ω with free variable t Σ and Δ R. (ii) Pseudo-state option. This method consists in defining a term w of type P(Σ). Then, given the term {s (s) Δ}, which is of type PΣ with a 4 Note that there are two distinct notions of equivalence: (i) syntactical (ii) semantical. The first one is defined in terms of inference rules as discussed in Sect..6, while two propositions are semantically equivalence whenever, in each topos τ, they are represented by the same element in Ω τ.

15 .5 Language l for a Theory of Physics and Its Representation in a Topos τ 4 free variable Δ of type P R, we want to know whether the elements in w have the property (s) Δ, i.e. we want to know whether the proposition (s) Δ is true, given the pseudo-state w. To this end we need to check the assertion w { s (s) Δ }. (.38) This is a term of type Ω. (iii) Truth object option. This method consists in defining a term T of type P(P(Σ)). The simplest choice is a variable of type P(P(Σ))defined as T : P ( P(Σ) ) P ( P(Σ) ). (.39) term of type Ω is then obtained by { s (s) Δ } T (.40) which has as free variable Δ P(R) and whatever free variables are contained in T. Intuitively we can think of T as a collection of subsets of the state space that have a particular property which we know to be true. Then we consider another subset of the state space, namely Q S and we would like to know if the collection of states in Q have the property (s) Δ. Since we know that there is a T to which all collection of sets of states with the property (s) Δ belong, we simply check if {s (s) Δ} T. 7. ny axioms added to the language have to be represented by the arrow true : Ω..5.2 Representation of l(s) in a Topos Given a topos τ with representation M, we now want to know how l(s) is represented in τ.. State space and quantity value object. The objects Σ and R are represented by the objects Σ τ M and R τ M in τ, which take the role of the state object and the quantity value object. 2. Physical quantities. Physical quantities are defined in terms of τ -arrows between the τ -objects Σ τ M and R τ M. We will generally require the representation to be faithful, i.e. the map τ M is one-to-one. 3. Values. Values are represented in τ by terms of type R τ M,i.e. τ M (s) R τ M where τ M : Σ τ M R τ M. 4. Truth values. formula (s) Δ is a term of type Ω, thus it is represented in a topos τ by an arrow [ (s) Δ ] τm : Σ τ M P R τ M Ω τ M.

16 42 Topoi and Logic Such an arrow gets factored as follows: [(s) Δ] τ M = e R τ M [(s)] τ M, [Δ] τ M (.4) where e R τ M : R τ M P R τ M Ω τ M is the evaluation map, [(s)] τ M : Σ τ M R τ M is the arrow representing the physical quantity and [Δ] τ M : P R τ M P R τ M is simply the identity arrow. Putting the two results together we have Σ τ M P R τ M [(s)]τ M [Δ] τ M R τ M P R τ e M R τ M Ω τ M. (.42) Truth values are terms of type Ω with no free variables. Hence in the topos τ, they will be represented by elements of the sub-object classifier Ω τ M, i.e. global elements γ τ M : τ M Ω τ M, γ τ M ΓΩ τ M. 5. Propositions. proposition is a term of type P(Σ), hence in a topos it will be defined as an element in P(Σ τ M ). In particular, consider a term of type P(Σ τ M ) with free variable Δ of type P(R τ M ). In a topos this is represented by an arrow [{ }] τm s (s) Δ : P R τ M PΣ τ M. (.43) Using this term of type P(Σ τ M ), which is represented in τ by the arrow [Ξ] τ M : τ M P(R τ M ), a proposition Δ is represented as: [{ }] τm s (s) Δ [Ξ] τ M : τ M P ( Σ τ ) M. (.44) 6. States. We will now analyse how the different types of states described above are represented in a topos. (a) Micro-state option. We have seen that a micro-state is essentially a term of type Σ, hence in a topos it is represented by a global element (if it exists) of Σ τ M,i.e. s : τ M Σ τ M. (.45) Moreover, given a term of type P(R), which is represented in τ by an arrow [Ξ] τm : τ M P(R τ M ) it is possible to define a map s,[ξ] : τ M Σ τ M P(R τ M ), which, if combined with the arrow [(s) Δ] τ M : Σ τ M P(R τ M ) Ω τ M gives τm s,[ξ] τm Σ τ M P ( R τ M ) [(s) Δ] τ M Ω τ M. (.46) This is the global element of Ω τ M representing the truth value of the proposition ((s) Δ). (b) Psuedo-state object. Pseudo-states are identified with terms of type P(Σ), so in a topos they are represented by elements w τ M : τ M P ( Σ τ M ). (.47)

17 .5 Language l for a Theory of Physics and Its Representation in a Topos τ 43 Given a proposition [{ s (s) Δ }] τm [Ξ] τ M : P ( Σ τ M ) (.48) we combine the two maps to give ( w τ M, [{ s (s) Ξ }] τ M [Δ] τ M ) : τ M P ( Σ τ M ) P ( Σ τ M ). (.49) Considering the arrow [w [{s (s) Δ}] τ M ]:P(Σ τ M ) P(Σ τ M ) Ω τ M, which represents the term (w {s (s) Δ}) of type Ω, we can define the truth value of the proposition (.48) given the pseudo-state (.47) as τ M (wτ M,[{s (s) Δ}] τ M [Ξ] τ M ) P ( Σ τ ) M P ( Σ τ ) [w [{s (s) Δ}] τ M ] M Ω τ M. (.50) (c) Truth object. truth object is a term T of type P(P(Σ))such that, given the proposition {s (s) Δ}, theterm({s (s) Δ} T) is of type Ω. Such a term has free variables Δ of type P(R) and T of type P(P(Σ)). Therefore, its representation in a topos τ is [{ } ] τm s (s) Δ T : P ( R τ ) M P ( P ( Σ τ )) M Ω τ M (.5) which can be factored as follows: [{ s (s) Δ } T ] τm = e P(Σ τ M ) ([{ s (s) Δ }] τ M [T] τ M ). (.52) Here e P(Σ τ M ) : P(Σ τ M ) P(P(Σ τ M )) Ω τ M is the evaluation map and [{ s (s) Δ }] τm : P ( R τ M ) P ( Σ τ M ) (.53) [T] τ M : P ( P ( Σ τ M )) id P ( P ( Σ τ M )). (.54) Given the above, the truth value of the proposition Δ is represented as v( Δ; T) : [{ s (s) Δ } T ] τ M [Δ] τ M, [T] τ M (.55) where [Δ τ M ], [T τ M ] : τ M P(R τ M ) P(P(Σ τ M )). For example in classical physics the topos in which we represent the language l(s) is Sets, thus the truth object Ω is simply the set {0, }. Inthis context we have v( Δ; T) := [{ s (s) Δ } T s] : P(R) P ( P(Σ) ) {0, } (.56) such that. { if {s Σ (s) Δ} T v( Δ; T)(Δ, T) = 0 otherwise { if (Δ) T = 0 otherwise. (.57) (.58)

18 44 Topoi and Logic.6 Deductive System of Reasoning for First Order Logic So far, we have defined the symbols and formation rules for the first order language l. However, in order to actually use l as a language that enables us to talk about things, we also require rules of inference. Such rules will allow us to derive true statements from other true statements. In order to describe this better we need to introduce the notion of a sequent. Definition.6 Given two formulae ψ and φ a sequent is an expression ψ x φ which indicates that φ is a logical consequence of ψ in the context x. 5 What this means is that any assignment of values of the variables in x which makes ψ true, will also make φ true. The deduction system will then be defined as a sequent calculus, i.e. a set of inference rules which will allow us to infer a sequent from other sequents. Symbolically, a rule of inference is written as follows: Γ (.59) ψ x φ which means that the sequent ψ x φ can be inferred by the collection of sequents Γ. We can also have a double inference as follows: Γ ψ x φ. This can be read in both directions, thus it means that ψ x φ can be inferred from the collection of sequents Γ, but also that the collection of sequents Γ can be inferred from ψ x φ. We will now define a list of inference rules. In the following, the symbol Γ will represent a collection of sequents, the letters γ,β,α will represent formulae, the letters σ,τ,...will represent terms of some type and α Γ represent the collections of formulas in both Γ and the formula α. Variables in a term can be either free or bounded. We say that a variable α of a term σ is bounded if it appears within a context of the form {α x σ }, otherwise it will be called free. The inference rules are: Thinning β Γ x α Γ x α. Cut Γ x α, α Γ x β. Γ x β 5 context x is a list of distinct variables. When applied to a formula α it indicates that α has variables only within that context.

19 .6 Deductive System of Reasoning for First Order Logic 45 For any free variable of α free in Γ or β. Substitution Γ x α Γ(x/σ) x α(x/σ) when σ is free in Γ and α.thetermγ(x/σ)indicates the term obtained from Γ by substituting σ (which is a term of some type) for each free occurrence of x. Extensionality Γ x x σ x ρ Γ x σ = ρ where x is not free in either Γ, σ or ρ. Equivalence α Γ x β β Γ x α Γ x α β Finite Conjunction The rules for finite conjunction are the following: α x α =, α β x β, α β x α. (.60) Note that we have used part of the definition of the logical connective if then. consequence of these rules is Proof. α x βα x γ α x γ β. (.6) α x β () γ β x β = (3) x γ = γ = β = x γ β (2) (4) γ β = x γ β (5) α x γ γ β x γ β (6) α β x α β (7) α x γ β This proof should be read from top to bottom and consists, as one can see, of a finite collection of sequents called a finite tree, in which the bottom vertex represents the conclusion of the proof. ll the sequents of the proof are correlated to each other in the following way:. sequent belonging to a node 6 which has nodes above it is derived by applying a rule of inference to the sequents belonging to the above nodes. 2. Every top most node is either a basic axiom or a premise of the proof. 6 node is an inference step: Γ Γ 2.

20 46 Topoi and Logic In the proof above we have that γ x γ = is derived by the thinning axiom, the equivalence axiom and the axioms γ x γ and x true as follows: Proof x true γ x () γ γ x (3) γ x γ = γ x γ γ x γ (2) (4) Where the lines (), (2) and (3) are an application of the thinning axiom, while line (4) is the application of the equivalence axiom where the equivalence α β := α = β was used. Going back to the proof of the conjunction axiom the remaining lines are derived as follows: (i) Line (2) is the definition of the logical connective. (ii) ll the other lines are derived from applications of the cut axiom. It should be noted that it is also possible to form a more general version of the conjunction axiom by replacing the single sequent α by a collection of sequents Γ, thus the conjunction xiom becomes: Γ x β Γ x γ. Γ x β γ Finite Disjunction The rules for finite disjunction consist of the following axioms: x αα x α ββ x α β (.62) and the following rule of inference: α x γ β x γ α β x γ (.63) whose generalisation is α Γ x γ β Γ x γ. α β Γ x γ Implication For implication we have the double inference rule β α x γ α x β γ.

21 .6 Deductive System of Reasoning for First Order Logic 47 gain the general form of which the above is a specification is β Γ x γ Γ x β γ. To see why that is the case we will prove the above generalisation, but only one way: Proof β x β β Γ x β () β Γ β γ Γ x β (2) x β β Γ x γ β Γ x β γ (4) Γ x β γ where in (4) we used the definition of implication given in Sect..3: α β := (α β) α. Negation For negation we only have one axiom (3). x α (.64) while the inference rules are (α Γ) x Γ x α and Γ x α ( α Γ) x. Universal Quantification We have the following double inference rule where y is not free in either β or α. gain the generalisation is α xy β α x yβ Γ xy β Γ x yβ. Existential Quantifier We have the double inference rule α xy β ( y)α x β

22 48 Topoi and Logic where y is a free variable in β. gain the generalization would be Distributive xiom α Γ xy β ( y)α Γ x β. ( α (β γ) ) x ( (α β) (α γ) ). (.65) Frobenius xiom ( α ( y)β ) x ( y)(α β) (.66) where y/ x. Law of Excluded Middle x α α. (.67) It should be noted that, for intuitionistic type of higer order languages the law of excluded middle does not hold. ll the rest does. With this we end our definition of higher order languages L, which are comprised of a set of term types, a set of logical connectives and a set of rules of inference which determine the logic.

23 ppendix B Worked out Examples B. Category Theory Example B. n iso arrow is always epic. In fact, consider an iso f such that g f = h f (f : a b and g,h : b c) g = g id b = g ( f f ) = (g f) f = (h f) f = h ( f f ) = h therefore f is right cancellable. Example B. Given a category C, isomorphism is an equivalence relation on C.This is defined as follows: Given any two objects,b C we say that they are equivalent iff there exists an iso i : B. This is a well defined equivalence relation since it satisfies the following properties:. Reflexivity. C, id : is an iso since id id = id. 2. Transitive. ssume that we have two isos i : B and i 2 : B C. We define the composite h := i 2 i. For this to be an iso it has to have an inverse. (i 2 i ) (i 2 i ) = i 2 i i i associativity 2 = i 2 id B i 2 identity = i 2 i 2 = id C. (B.) Similarly we can show that (i i 2 ) (i i 2 ) = id B 3. Symmetry. If we have an iso i : B then there exists a unique inverse i : B. Since i : B is unique with inverse i then i is the desired iso. Example B.2 In any category the following are true:. g f is monic if both g and f are monic. 2. If g f is monic then so is f. C. Flori, First Course in Topos Quantum Theory, Lecture Notes in Physics 868, DOI 0.007/ , Springer-Verlag Berlin Heidelberg

24 420 B Worked out Examples In fact we have that () If f : B and g : B C are monic then, given two arrows h, j : D such that (g f) h = (g f) j we have: (g f) h = (g f) j associativity g (f h) = g (f j) monic f h = f j monic h = j. (B.2) It follows that (g f)is monic. (2) Given f : B, and h, k : D, assume that f h = f k. Then consider g : B C such that (g f)is monic. It follows that implies that h = k. (g f) h = g (f h) = g (f k) = (g f) k (B.3) Example B.3 Given any category C op,amapf C op is monic in C op if and only if it is epic in C. In fact f : B is monic in C op iff for all g,h : C B in C op then f g = f h implies that g = h. (B.4) However from the definition of dual category the above holds iff for all g,h : B C in C we have g f = h f implies g = h (B.5) where now f : B. However, this is true iff f is epic in C. We then say that monic and epic are dual notions. Example B.4 The category (R, ) op has as:. Objects: r R. 2. Morphisms: given any two elements x,y (R, ) such that x y then there exists a unique arrow f : y x in (R, ) op. These morphisms undergo the following properties: (i) Composition: if x y and y z then in (R, ), x z. This implies that there exist three maps in (R, ) op, namely f : y x, g : z y and h : z x such that h := f g. (ii) ssociativity: given that f : x y, g : y z and k : z w in (R, ) we then have that k (g f ) = (k g ) f, therefore in (R, ) op we have f (g k ) = (f g ) k. Here x y means that there is a map f : x y in (R, ) op or equivalently a map f : y x in (R, ).

25 B. Category Theory 42 (iii) Identity element: in R we have that x x, by duality i x : x x is the identity morphism on x in (R, ) op. Example B.5 Given a category C with initial object 0, the following hold: (i) If 0 (i.e. there is an iso map between them) then is an initial object. (ii) If there exists a monic arrow f : 0, then f is an iso. Proof (i) n initial object 0 is such that, given any other object C there exists one and only one map i : 0. If such an arrow i is iso, then i : 0exists and is the unique inverse. Now consider any other object C C, we know that there exists a unique arrow f : 0 C. We then assume that we have two maps h, g : C, we then obtain the following diagram g C i h f 0 From the property of the initial object we have that h i = g i = f. (B.6) However since i is iso, it is right cancellable, therefore h = g. This implies that given any object C C there exists one and only one map f : C. Thus is an initial object. (ii) f : 0 is monic. Since 0 is the initial object we have the unique arrow i : 0. From the property of 0 being an initial object it follows that f i = id 0 (since there is one and only one arrow from any object to 0, including from 0 to itself). On the other hand f (i f)= f id C implies i f = id C (monic property). (B.7) Example B.6 Given a category C with terminal object, if there exists an arrow g : with domain the terminal object, then g must be monic. In fact, if we assume that there exists an arrow g :, since is the terminal object then, given any other object C, h : is unique, including. It follows that h g = id. Now consider two maps f,k : B such that g k = g f we then have f = id f = (h g) f = h (g f)= h (g k) = (h g) k = id k = k (B.8)

26 422 B Worked out Examples which implies that g is monic. Example B.7 Given two objects,b in some category C, consider the category Pair(, B), whose objects are triplets (P, p,p 2 ) where p : P and p 2 : P B. The morphisms in Pair(, B) are maps f : (P, p,p 2 ) (Q, q,q 2 ) (B.9) such that f : P Q is a morphism in C and q f = p ; q 2 f = p 2,i.e.the following diagram commutes P p 2 p f Q q q 2 B We then have that. Pair(, B) is a category. 2. ( B,π,π 2 ) is a product if it is a terminal object in Pair(, B). Proof. To show that Pair(, B) is a category, we need to show that the following properties hold: (a) Composition. Givenf : (P, p,p 2 ) (Q, q,q 2 ) and g : (Q, q,q 2 ) (S, s,s 2 ) we need to define composition. Thus we say that g f is the map which makes the following diagram commute: P p 2 p f Q q q 2 g s s 2 S B

27 B. Category Theory 423 (b) Identity morphism. Id (P,p,p 2 ) : (P, p,p 2 ) (P, p,p 2 ) P p 2 p id P P p p 2 B (c) ssociativity P p 2 p f Q q q 2 g s s 2 S B composed with S s 2 s h R B r r 2

28 424 B Worked out Examples gives P p 2 p f Q B q q 2 g s s 2 s s 2 S h R which is the same as composed with P p 2 p f Q q q 2 B Q q 2 q g S B s s 2 h r r 2 R

29 B. Category Theory 425 which gives P f 2 p p Q q 2 q g S B s s 2 h r r 2 R 2. If ( B,π,π 2 ) is a terminal object then, for all elements (P, p,p 2 ) Pair(, B) there exists one and only one arrow (P, p,p 2 ) ( B,π,π 2 ) (B.0) which, by definition implies that the following diagram commutes: P p 2 p f B π π 2 B But this is precisely the condition satisfied by the product. Example B.8 For any triple π B π 2 B the following propositions are equivalent.. Given any triple f C g B, there exists a unique morphism f,g :C B, such that π f,g =f and π 2 f,g =g. (B.)

30 426 B Worked out Examples 2. For any triple f C g B there exists a morphism f,g :C B, such that π f,g =f and π 2 f,g =g (B.2) and, moreover, for any h : C B, h = π h, π 2 h. (B.3) Proof (a) () (2). Given any h : C B, we need to show that h = π h, π 2 h. From () we have that π h C π 2 h B. (B.4) Thus there exists a unique arrow k : C B such that π k = π h and π k = π h. (B.5) Since this equation holds for k := h and from () k := π h, π 2 h, it follows that h = π h, π 2 h. (b) (2) (). For any triple f C g B, (2) implies that there exists a map f,g :C B such that π f,g =f and π 2 f,g =g. (B.6) What remains to show is that f,g is unique. To this end let k : C B be such that π k = f and π 2 k = g. (B.7) Then, from (2) it follows that k = π k,π 2 k = f,g. (B.8) Example B.9 Given the category C: 0, 2 (B.9) and the category D: 0 2

31 B. Category Theory 427 we define the functor F : C D as It follows that F : (B.20) 2 2. F(f : 0 ) := 0 ; F ( g : 2 ) := 2. (B.2) The only arrow which does not lie in the image of the F functor is 0 2 which is 0 2 which is the composite F(g) F(f). Thus the image of F is not a subcategory. Example B.0 We will now define the bi-variant Hom functor C(, ) : C op C Set and show that it is a functor. possible definition would be: C(, ) : C op C Sets (, B) C(, B) (f, g) C(f, g) (B.22) where f : C D and g : E F are such that (f, g) : (D, E) (C, F ) (B.23) and C(f, g) : C(D, E) C(C, F ) h g h f. (B.24) The requirements for C(, ) to be a well defined functor are C(id, id ) = id C(,) (B.25) which is trivially satisfied and C(g f,g f)= C(f, g) C(f, g) (B.26) where C(g f,g f): C(C, ) C(, C) h g f h g f (B.27)

32 428 B Worked out Examples while C(f, g) C(f, g) : C(C, ) C(B, B) C(, C) h f h g g f h g f. (B.28) Thus (B.22) is a well defined functor. Example B. The diagram id id f f B is a pullback iff f is monic. We start by assuming that f is monic. Given any pair of maps h, g : C, such that f id g = f id h then g = h, thus g will be the only map making the following diagram commute: C g g id g id f f B Hence the square is a pullback. On the other hand, if the diagram is a pullback then for any other h : C, such that f id g = f id h, by uniqueness of g it follows that g = h. Thus f is monic.

33 B. Category Theory 429 Example B.2 Given any category, if i B i f B g C is a pullback, then i is an equaliser of f and g. In fact, let us assume that the diagram is a pullback, E l h i B h i f B g C then f i = g i. Moreover for any arrow h : E B, such that f h = g h, there exists a unique l such that h = i l, thus i is an equaliser. On the other hand, if i is an equaliser, then f i = g i. Moreover from the universal property of an equaliser, given any arrow h : E B there exists a unique arrow l : E which makes the outer square of the above diagram commute. It follows that the inner square is a pullback. Example B.3 Given a category C with products and terminal object. For any two objects,b C the pullback of B is the product of and B. To see this

34 430 B Worked out Examples we start by assuming that D g f! B! is indeed a pullback. Therefore, for each pair of map h : E and k : E S such that! h =! k, there exists a unique l : E D such that the following diagram commutes: E l h D g k f! B! Thus h = g l, k = f l. Moreover since! is the unique arrow to the terminal object, the condition! h =! k is trivially satisfied (always satisfied), thus we end up with the following diagram: B E k h l D f g But this is precisely the definition of a product.

35 B. Category Theory 43 Example B.4 If is an object in a category with a terminal object, then! id is a product diagram. In fact, given the maps!:b and f : B we construct B! f h! id Obviously, the only arrow making the above diagram commute would be h := f. Example B.5 Consider a pair of morphisms f C g B. We then define a category Con(f, g) whose objects are (f, g)-cone defined as a triple (D,p,q), such that the following diagram commutes: D q B p g f C Given two (f, g)-cones (D,p,q) (D,p,q ) a morphism h : (D,p,q) (D,p,q ) between them is a map h : D D, such that the following diagram commutes: D p q h D B p q

36 432 B Worked out Examples pullback of f along g is defined via the diagram E q l D q B p p g f C where the map l is unique for a given E. However the top part of the diagram is simply an object (E, p,q ) in Con(f, g). Thus the pullback property tells us that for each object (E i,p i,q i ) Con(f, g) there exists a unique map l : (E i,p i,q i ) (D,p,q). This means precisely that (D,p,q)is a terminal object Example B.6 Given any map f : B then the characteristic functions of the identities id and id B are such that χ idb f = χ id. To see this, consider the diagram f B f id B B id B B χ idb true Ω where! B : B and! :. Since pullbackness implies commutativity, it follows that χ idb f = true! B f = true! = χ id.

37 B.2 Topos Quantum Theory 433 B.2 Topos Quantum Theory Example B.7 Let us consider a 4 dimensional Hilbert space C 4, with basis ψ = (, 0, 0, 0), ψ 2 = (0,, 0, 0), ψ 3 = (0, 0,, 0), ψ 4 = (0, 0, 0, ). Wewouldliketo define the proposition S z [ 3, ] S z [, 3], where S z represents the value of thespininthez direction of a two particle system. Total spin in the z direction can only have values 2, 0, 2 since the self-adjoint operator representing S z is Ŝ z = Thus the only value that S z can take in the interval [ 3, ] is 2 while the only value it can take in the interval S z [, 3] is 2. In this setting the proposition Ŝ z [, 3] is represented by the projection operator Pˆ = diag(, 0, 0, 0) (see Sect. 0.3). On the other hand the proposition Ŝ z [ 3, ] is represented by the projection operator Pˆ 4 = diag(0, 0, 0, ) (see Sec. 2.3). Therefore all that remains to compute is δ( Pˆ ) δ( Pˆ 4 ). Let us compute this for each context V V(H). For the maximal algebra V and for VP ˆ, Pˆ 4 we have δ( Pˆ ) V δ( Pˆ 4 ) P ˆ, Pˆ V = δ( Pˆ ) ˆ 4 P, Pˆ V δ( Pˆ 4 ) V ={λ } {λ 4 }=. 4 (B.29) For V ˆ P we have δ( Pˆ ) V δ( Pˆ 4 ) P ˆ V ={λ } {λ 23 }=. ˆ P (B.30) For V ˆ P 4 we have δ( Pˆ ) V δ( Pˆ 4 ) P ˆ V ={λ 23 } {λ 4 }=. ˆ 4 P 4 (B.3) For V ˆ P 2, ˆ P 3 we have δ( Pˆ ) V δ( Pˆ 4 ) P ˆ 2, Pˆ V ={λ 4 } {λ 4 }={λ 4 }. ˆ 3 P 2, Pˆ 3 (B.32) For V ˆ P, ˆ P j, j {2, 3} we have δ( Pˆ ) V δ( Pˆ 4 ) P ˆ, Pˆ V ={λ } {λ 4i }=. ˆ j P, Pˆ j (B.33) For V ˆ P i, ˆ P 4, i {2, 3} we have δ( Pˆ ) V δ( Pˆ 4 ) P ˆ i, Pˆ V ={λ j } {λ 4 }=. ˆ 4 P i, Pˆ 4 (B.34)

38 434 B Worked out Examples It is interesting to compare such a proposition with the proposition S z ([ 3.] [, 3]) which is represented by δ o ( Pˆ Pˆ 4 ). This is clearly equal to ˆ0 for all contexts, hence δ o ( Pˆ Pˆ 4 ) δ o ( Pˆ ) δ o Pˆ 4 ). Example B.8 Given the same setting as above we would like to give the topos analogue of the proposition S z [ 3, 3]. This is represented by the projection operators ˆ. Therefore, for each context V V(H),wehavethat which implies that δ o (ˆ) V = ˆ S δ o (ˆ) V = Σ V. (B.35) (B.36) Example B.9 Consider the algebra of bounded operators B(H) on a Hilbert space H. We then define the category V(B) of abelian sub-algebras of B(H). This can be easily seen to be a category under sub-algebra inclusion. We now would like to define a covariant functor F : V(B) V(H). first guess would be F : V(B) V(H) B F(B):= B (B.37) where B represents the double commutant of B. Is this a functor? First we need to show that, given two sub-algebras i : B i B, the following diagram commutes: B i F F(B i ) i F(i) B F F(B) The fact that the above diagram commutes follows trivially from the fact that if B i B then B i B. Given the commutativity of the above diagram it follows at once that F(i j)= F(i) F(j)for j : B j B i. Moreover F(id B ) := ( F(B) F(B) ) = id F(B). (B.38) Example B.20 Consider a set of classical observables O which you want to quantized. Such a set forms a Lie algebra with respect to an appropriately defined commutator. For example the Poisson algebra of a set of functions on phase space such that two elements,b O are considered to be non-commuting when {,B}=0.