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A topos-theoretic approach to branching space-time
Leonid V. Il'ichov Novosibirsk State University, Institute for Automation and Electrometry Novosibirsk, Russia E-mail: leonid@iae.nsk.su

1

Introduction

Isham and Dè oring [1] put forward a fundamental conception of how any physical theory should be constructed. They proposed to look for topos representation of the formal language used in the theory. The known and chronologically preceding example of such an approach is the work [2] by Fotini Markopoulou. She considered from a novel view-point the properties of the causal set C which is the discrete analog of space-time from general relativity. The interest to causal sets is stipulated by hypothetical discreteness of space-time at sub-Planck scale. The causal set C is also the central notion of the present work. It is a partly ordered set of all events. Some ordered event pairs e, e in C are causally related: e e . That means that e is a consequence of e or, equivalently, e is a cause of e . The causal relation is reflexive (for any e in C one has e e), transitive (from e e and e e follows e e ) and antisymmetric (if e e and e e, then e = e ). The last condition guarantees the absence of closed causal loops. We also assume the absence of the last event ef in which is the consequence of any other event in C . This is important for the sub ject of the present work being non-trivial. The partial order in C let one consider it as a category with the events as ob jects and causal relations between them as arrows (morphisms). So the set M orC (e, e ) of morphisms from e to e consists of at most of one element just in the case when e is the cause of e . Using the notions of category theory, Markopoulou introduced the (covariant) functor P ast from the category C to the category of sets S et. To any event e from C this functor associates the set P aste = {e C : e e} (the set of all causes with respect to e), and to any causal arrow e1 e2 í the map P aste1 e2 : P aste1 P aste2 which is the inclusion of sets. The functor P ast from [2] has simple physical meaning: P aste is the
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memory content for an observer localized in e. Note that the complete causal structure of space-time is reflected in P ast. The approach of the present work to C is also realized as a view-point of localized observers. But some other properties of the causal set are in focus. We are going to address the branching properties of space-time. The very notion of "branching" will get the strict sense. The motivation beyond the approach is to elaborate a logical framework for considering multivariant future by a local observer. So the subtitle of the present work (in the style of [2] subtitle) can be the phrase "What should the branching Universe be thought of from the inside?". We are going to build the topos representation of Nuel Belnap's branching space-time theory [3]. The aim of the branching space-time conception is the reconciliation and unification of indeterminism and relativism. In the following section the central notion of the branching space-time theory, the so-called Belnap "world" (Belnaps calls it "history"), is introduced along with basic notions of categories. Because the ideas of topos mathematics has not yet become widely known, the present work is aimed to be in its considerable part an introduction to basic notions of topoi. In the third section we give the main elements of topos approach in application to the model of branching space-time and show the origin of its natural non-classical logic. The application of universal topos construction of local semantic values to propositions of the natural logic is made in the fourth section.

2

Basic notions

First we provide and discuss the required formal and methodological tools. Let us digress for a while on the main notions of category theory. It is known [4], that any category C is specified by its ob jects and morphisms (arrows) between them. For some arrows the following composition law is defined. Let M orC (c, c ) be the set of arrows (morphisms) beginning and ending at the ob jects c and c , respectively. There is a rule by which to any ordered pair of morphisms from the direct product M orC (c1 , c2 ) ½ M orC (c2 , c3 ) an element (morphism) from M orC (c1 , c3 ) is assigned, their composit. The composition law is associative and in every M orC (c, c) there is an identity morphism 1c which acts as a left unit in compositions with elements from M orC (c, c ) and as a right unit in compositions with elements from M orC (c , c). The most important category is S et with sets as ob jects and maps between sets as morphisms. Covariant functor F from category C1 to category C2 is a rule of "pro2


jecting" the category structure of C1 onto C2 . To every ob ject c from C1 an ob ject Fc from C2 is assigned. Similarly, the functor assignes to every morphism f from M orC1 (c, c ) some morphism Ff from M orC2 (Fc , Fc ). The identity morphisms as well as composits are respected by the functor. The specific of contravariant functor is in inversion of morphism directions: to f from M orC1 (c, c ) there assigned the morphism Ff from M orC2 (Fc , Fc ). The functors from a category C1 to a category C2 can be considered C as ob jects of a new category C2 1 . Morphisms between such functors are called natural transformations. The examples of natural transformations will appear later in the text. We need not the strict definition of topos as a special type of category [5]. In some sense all topos are like the classical topos S et í the category of sets. C It is important for us that if C2 is a topos, then C2 1 is a topos also. Below the topos S etC is used. Considering C as a branching space-time, it is convenient to introduce subsets from C without branching and, hence, without indeterminism at all. The richer is the collection of such branchless subsets, more branching is the causal set C . These subsets will be called Belnap worlds and, following [3], are defined as maximal upward (i.e. towards future) directed subsets of C . The demand of being directed is natural and motivated by evidently indispensable property of any branchless world í for any two events e1 e2 in a world w, there should be their common consequence e in w: e1 e and e2 e. The maximality does not allow a world to be a proper sub-world in a wider world. Now the condition of absence of the last event ef in ("Big flap") in C is clear. In other case C is the single maximal directed set and the sub ject of the present work becomes trivial. The standard application of Kuratovski-Zorn's lemma proves any event pertaining to some Belnap world: given any growing chain of directed sets which contain the event, one can note that their union is an upper bound of the chain. Thus a maximal directed subset in C exists and contains the event. There may happen no Belnap world for a pair of events e1 and e2 to live in. Such events are called incompatible. The alternative outcomes of a quantum measurement is an example of incompatible events. One should not mix the notion of incompatible events with causally unrelated events.

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3

Main ob ject and subob jects

We are going to deal with some contravariant functors, called presheaves, op from C to S et, i.e. the ob jects of the topos S etC , where the category C op can be formed, given by C , by reversing the direction of all causal arrows. In this point our approach differs with that of [2], where covariant functors are used. Let W be the set of all Belnap worlds in C . There is a simple but important Theorem 3.1. Assigning to an event e the set of Belnap worlds Lo ce = {w W : e w}, one defines the functor Lo c from C op to S et. Proof: The sets Lo ce define the function of ob jects of the functor. We have to clarify the nature of function of causal arrows e1 e2 from C . This should be a map Lo ce1 e2 : Lo ce2 Lo ce1 , (1) The map is set inclusion. To prove the fact one should involve the maximality of Belnap worlds, which lead to the closeness of any world with respect to causes [3]: if e is a cause of e w, i.e. e e , there is in w a common consequence of e and e , where e is any event in w (due to transitivity of causal relation this common cause can be chosen among common causes of e and e which exist because of the directed nature of the world). Hence, the world w can be extended by inclusion of e. But w is maximal and can not be extended í the event e has been already included in w. It follows that every world containing e2 , contains e1 as well and, so, Lo ce1 e2 (1) is inclusion. The functor Lo c is in some sense analogous to the functor P ast from [2] and is of importance also. Lo ce is the set of worlds an observer localized in e considers as "her worlds". It is worth to introduce the presheaf Glob (the counterpart of the functor W orld from [2]): Globe = W, (2) and for e
1

e

2

Glob

e1 e

2

= id

W

: Globe2 Globe1 ,

(3)

There can be introduced morphisms between functors from C op to S et, op considered as ob jects of the category S etC . The morphisms are called natural transformations [4]. In particular, the natural transformation [
Lo c

] : Lo c Glob
4

(4)


is the set {[ Loc ]e : e C } of maps [ following diagrams commutative: Lo c
e1 e2

Lo c ]e

: Loce Globe ,which make the

e

2

Lo c

- - Globe2 - -2 Glob - - Globe - -1
[
Lo c ]e 1

[

Lo c ]e

e1 e2

=idW

(5)

Lo c

e

1

for any causal arrow e1 e2 , i.e. idW [ Loc ]e2 = [ Loc ]e1 Lo ce1 e2 . In this simple case the components [ Loc ]e , of the natural transformation (4) are set inclusions. This let the functor Lo c be considered as a sub-functor of Glob. op If one calls Glob the ob ject in S etC , then Lo c is the subobject. The set of subob jects of Glob is important in the topos approach to the branching space-time. The set of subob jects in any topos is known to be endowed with the structure of Heyting algebra [5]. It is a model of a natural language of the considered system (the branching space-time C ) [1]. The logic of the language is not classical one based on Boolean algebra. Logical operations are realized as algebraic operations on subob jects of Glob. The conjunction F G, models the logicaoperation "and". For subob jects F and G one has (F G)e =df Fe Ge . (6) Similarly, the disjunction F G models the operation "or" and is defined as follows: (F G)e =df Fe Ge . (7) For any causal arrow e1 e2 the maps Fe1 e2 and Ge1 e2 are set inclusions (this follows from the corresponding (5)-type commutative diagrams, where the horizontal arrows are inclusions). Therefor, the maps (F G)e and (F G)e
1 e2 1 e2

: (F G)e2 (F G)e1 : (F G)e2 (F G)e1 .

(8) (9)

are set inclusions as well. The binary operation F G models logical implication: (F G)e =df {w W : e e (w Fe ) (w Ge )}. (10)

Here in lhs means the binary operation on the functors F and G, the same symbol in rhs stands for the ordinary logical connection "if..., then...". It is easy to see that for any causal arrow e1 e2 the set (F G)e2 is a
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subset in (F G)e1 . Therefore, the functorial image of the causal arrow is the set inclusion: (F G)e
1 e2

: (F G)e2 (F G)e1 .

(11)

The implication operation let one define the unary operation of negation ì in the set of subfunctors of Glob. To this end one need the zero subfunctor . It assigns the empty set to any event. Let ìF =df (F ). From this definition and (10) follows (ìF)e = {w W : e e (w Fe )}. / (13) (12)

The intersection Fe (ìF)e is evidently the empty set. So F (ìF) = . In the general case F (ìF) = Glob. (15) If it were the place of exact equality in the last expression, one would be free to identify F and ììF. But there is only a weaker statement that F is a subfunctor of ììF: Fe (ììF)e . (16) Really, due to (13), we have (ììF)e = {w W : e e e e (w Fe )}. (17) (14)

Because of Fe Fe , taking place as soon as e e, the expression (16) follows. The reverse inclusion (ììF)e Fe is generally incorrect, then Fe (ììF)e . according to (14), (ìF)e (ììF)e = . Consequently, (ìF)e (F)e is a proper subset of (ìF)e (ììF)e which, in its turn, is a subset of W . Therefore, (ìF) F is a proper subfunctor in Glob. Expression (15) points the principe of excluded middle to be not fulfilled in the logic of subob jects in Glob (the presheaf Glob plays the role of identically true proposition). This is a generic property of topoi, other then S et [5]. The logic is intuitionistic. As it is pointed in [2], some important properties of the causal set C are reflected in the structure of ìP ast, viz ìP ast = provided C is a lattice (here the functor along with P ast, is the ob ject of category S etC and does not coincide with contravariant counterpart). This follows from the definition (ìP ast)e = {e C : e
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e (e P aste )}. /

(18)


We see that if any pair of events has an upper bound, the set (ìP ast)e is empty. Reformulating this observation within our approach aimed at branching properties of C , we arrive at the following Prop osition 3.1. The subject of the present work (the branching of spacetime) is non-trivial iff the functor ìP ast from [2] is non-empty. . As it has been pointed out, in our case Lo c is an analog of P ast. Consequently, one may expect the structure of Lo c reflects some important properties of C and the set of worlds. Due to (13) we get: (ìLo c)e = {w W : e It is easy to verify that (ìLo c = ) (e C w W P aste w = ). So, we have Prop osition 3.2. The emptiness of ìLo c is equivalent to non-empty intersection of any Belnap world with the past of any event. . Particularly, this is the case of C containing the initial event e Bang"), such that ein e for any event e.
in

e (e w)}. /

(19)

(20)

("Big

4

Subob ject classifier
op

In S etC , as well as in each topos, there is a subob ject classifier for any ob ject. Intuitively, the subob ject classifier delivers generalized truth values for a set of propositions. This can be illustrated by application of the general construction [5] in our setting. Let us consider the set M orC (§, e) of causal arrows ending in e. Special subsets called sieves on e are of importance. Any sieve S M orC (§, e) is closed in the following sense: if (e e) S , and there is a causal arrow e e , then (e e) S . The maximal sieve on e is the very set M orC (§, e). Its empty subset is the minimal sieve. It is worth to introduce the set e =
df

{S : S í sieve on e}.
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(21)


For any causal arrow e

1

e2 there is a map e
1 e2

: e2 e

1

(22)

by the following rule: given the sieve S on e2 , one should consider
e1 e
2

(S ) =

df

{(e

e1 ) M orC (§, e1 ) : (e

e2 ) S }

(23)

as its image in e1 . Expressions (21) í (23) let us define the contravariant functor from C to S et. For any subob ject F from Glob with the corresponding inclusion [F ] : F Glob consider the following maps [F ]e : W e , which let assign a sieve on e to any Belnap world: [F ]e (w) =
df

(24)

(25)

{(e

e) M orC (§, e) : w Fe }.

(26)

It is easy to see that this is really a sieve: if a causal pair e e belongs to rhs of (26) and there is an arrow e e , then w Fe because Fe Fe and, consequently, e e belongs to rhs of (26) as well. The fact is of importance that the maps (25) (defined for all events) make the diagram W - -2 e -
idW [F ]
e

[F ]

e

2 1 e2

e

(27)

W - -1 e1 , - commutative for any arrow e1 e2 . Therefore, [F ]e can be considered as components of the natural transformation [F ] : Glob . (28)

It follows from (26) that if w Fe , then [F ]e (w) = M orC (§, e). Oppositely, if the last equality is fulfilled then from e e M orC (§, e) one has w Fe . We see that [F ]e maps all the worlds from Fe and only these worlds to the maximal sieve on e. This makes the following diagram commutative F - Glob -
[!F ] [F ]

[F ]

(29)

1 - -
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Here the functor 1 assigns the fixed one-element set {0} to any event from C and the natural transformation components [!F ]e are the only possible maps from Fe to {0}. The natural transformation (truth) has the components e : {0} e such that e (0) = M orC (§, e) is the maximal sieve on e. F along with the natural transformations [!F ] and [F ] are the pull-back of the diagram Glob - - 1 [5]. As has been said, the presheaf Lo c is the most simple and important subob ject of Glob. To its inclusion in Glob (4), the following natural transformation corresponds [Loc ] : Glob , (30) so that [
Lo c ]e [F ]

(w) =

df

{(e

e) M orC (§, e) : e w}.

(31)

The sieve on e defined by rhs of this expression, is the generalized truth value of an e-localized observer being living or having been lived in the world w. If the event e pertains to the world w, then this is the maximal sieve on e í the maximal truth value. Oppositely, if P aste w = , the rhs of (31) is the minimal (empty) sieve í the complete false. Intermediate truth values correspond to situations when only part of events from P aste pertains to w. The truth value depends evidently on e, i.e. on the place and moment of assertion. In a similar manner for any subfunctor F from Glob the sieve defined by the map (26), is the local (from the view-point of an observer in e) truth value of the proposition w is an element of F . Note that we can look at assigning of truth values from a slightly another point. We can locally assign a sieve on e to any subfunctor F from Glob: [SF ]e =df {(e e) M orC (§, e) : Fe = }. (32)

In this setting the local truth value (31) is identical to [SwLoc ]e . Here the functor w appears such that we = {w} and all causal arrows are mapped onto identity of this one-element set. The usefulness of the last notion let one assign a local truth-value sieve to assertions concerning space-time events without any reference to particular Belnap world. For example, from the view-point of an observer in e the assertion that the event e0 takes place should intuitively be associated with the following sieve: {(e e) M orC (§, e) : Lo ce Lo ce0 = }. (33)

One can note that this sieve is yielded by (32) for the presheaf Lo c Lo ce0 , where Lo ce0 is the constant presheaf, so that [Lo ce0 ]e = Lo ce0 for any e. It
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is easy to see that the sieve (33) is maximal on e provided e0 is compatible with e. On the contrary, (33) is the empty sieve if any event from P aste is incompatible with e0 . The truth value in e is intermediate if the considered assertion is absolutely true in only part (not all) of events from P aste . In particular, this assertion is not completely false in any event if ìLo c = , since in this case any world from Lo ce0 has non-empty intersection with the past of any other event. It is worth to note that the truth value of similar assertion from [2] is given by a co-sieve on e í by the set of all causal arrows from e to all common consequences of e and e0 . There last assertion states that some consequence of the event e0 will sooner or later be fixed in the memory of an e-localized observer . Meanwhile, this statement may be not absolutely true even if the sieve (33) is maximal.

5

Conclusion

Resuming, we see that the central ob ject of our approach, the presheaf Glob, is made of Belnap's worlds í the main elements of branching spacetime C considered as a partly-ordered set. The model of Heyting-value logic is realized on the set of subob jects of Glob. The known construction of subobject classifier let one assign to propositions the generalized truth values from the view-point of a local observer. The similarities can be traced between our approach and that of the work [2] which also uses the basis of topos theory. Nevertheless, there are significant differences. Not only events of space-time are considered but also their special collections, called Belnap's worlds. Because of this the emptiness of the presheaf ìLo c, as has been show, is equivalent to non-trivial relation between this two types of entities. As a further development of the present approach we are going to present the categorial construction of local orthologics. This let us to make a step towards introducing quantum-like structures in the framework of branching space-time and, in perspective, towards a new line of reconciliation of classical and quantum conceptions.

References
[1] A. Dè oring and C.J. Isham, "A topos foundation for theories of physics", arXiv: quant-ph/0703060, 0703062, 0703064, 0703066 (2007).

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[2] F. Markopoulou, "The internal description of a causal set: What the univers looks like from the inside", arXiv: gr-qc/9811053 (1998). [3] N. Belnap, "Branching space-time", Synthese, 92 (1992), 385. [4] S. MacLane, "Categories for the working mathematician", Springer (1998). [5] R. Goldblatt, "Topoi: The categorial analisis of logic", Nort-Holland (1979).

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