aa r X i v : . [ m a t h . C T ] N ov Measure theory over boolean toposes
Simon HenryNovember 7, 2014
Abstract
In this paper we develop a notion of measure theory over booleantoposes which is analogous to noncommutative measure theory, i.e. tothe theory of von Neumann algebras. This is part of a larger project tostudy relations between topos theory and noncommutative geometry. Themain result is a topos theoretic version of the modular time evolution ofvon Neumann algebra which take the form of a canonical R > -principalbundle over any integrable locally separated boolean topos. Contents
This paper is part of a larger project to understands the relation between non-commutative geometry and topos theory, and is more precisely focused on themeasure theoretic aspect of this relation.An extremely efficient way to relate a topos T to objects from noncommutativegeometry is the study of internal Hilbert spaces of T . Let us explain a littlemore this point. In the same way that an abelian category is a category whichbehave like the category of abelian groups, a topos is a category that behave Keywords.
Boolean topos, Measure theory, Modular theory, time evolution. T , to transport, for example, the definition of an Hilbertspace into a definition of an internal Hilbert space in T . An Hilbert space of T (or a T -Hilbert space) is an object of T endowed with a series of operationsatisfying the axiom for an Hilbert spaces in the internal logic of T . It appearsthat the category of T -Hilbert spaces and bounded operators between themis an “external” C ∗ -category (in the sense of for example [1, Definition 1.1])naturally attached to the topos T and related to its geometry.When the internal logic of a topos T satisfies the law of excluded middle ( T isthen said to be a boolean topos) then one can, using internal logic, construct thesupremum of a bounded directed net of positive operators on an internal Hilbertspace simply by constructing their supremum internally (the law of excludedmiddle is all we need to construct supremum of such families, and one easilysee that the internal supremum will also be an external supremum), this turnthe C ∗ -category of T -Hilbert space into a “Monotone complete C ∗ -category”,in the sense that all the C ∗ -algebras of endomorphisms are monotone complete C ∗ -algebras (see [7, Definition III.3.13]).We will show that this monotone complete C ∗ -category is a W ∗ -category if andonly if T satisfies a condition of existence of measures called “integrability” (seedefinition 3.5 and theorem 3.6).For these reasons, it seem natural to think of boolean toposes as the topostheoretic analogue of monotone complete C ∗ -algebras and of integrable booleantopos as the analogue of von Neumann algebras. This is encouraged by thefact that the “commutative case” agrees: the category of boolean locales iswell known to be anti-equivalent to the category of commutative monotonecomplete C ∗ -algebras and normal morphisms between them; and, under thiscorrespondence, commutative von Neumann algebra correspond exactly to theintegrable boolean locale.This being said we acknowledge the fact that there do exist non-boolean toposwhose category of internal Hilbert space is a W ∗ -category, but it seems to usthat these are examples of toposes whose geometry is not reflected by their C ∗ -category of internal Hilbert spaces, and that they should be discarded of thepresent work.For these reasons, and as we are concerned with measure theory, we will onlyfocus on boolean toposes. All the topos mentioned in this paper are booleanGrothendieck toposes.
Von Neumann algebras (and also W ∗ -categories) are well known to have acanonical “modular” time evolution. The main result of this paper is to pro- We might need T to be a Grothendieck topos and not just an elementary topos for this. see for example [8, Chapter VIII] for von Neumann algebras and [1, section 3] for the caseof W ∗ -categories. T in terms of a certain(canonical) principal R > T -bundle over T analogous to the bundle of positive lo-cally finite well supported measures over a boolean locale (which is a principalbundle exactly because of the Radon-Nikodym theorem).This gives a classification in types I, II and III of boolean integrable locallyseparated toposes analogous (but not totally equivalent) to the classification intype of von Neumann algebras: Type I corresponds to separated toposes, typeII to toposes which are not separated but which have a trivial modular bundle,and type III to toposes which have a non-trivial modular bundle.Of course in full generality, one obtains that every boolean integrable locallyseparated topos decomposes in a disjoint sum of a topos of each of the threetypes by applying this disjunction internally in its localic reflection.In the last section of this paper, we consider T a boolean integrable locallyseparated toposes, X an object of T such that T /X is separated and l ( X ) theHilbert space of square summable sequences. In this situation, we show thatthe modular time evolution of the von Neumann algebra A of globally boundedendomorphisms of l ( X ) is indeed the time evolution on A described by χ , andthat an invariant measure on T induces a trace on A .The following table sum-up the dictionary between topos theory (in the leftcolumn) and operator algebra (in the right column) that arise in this paper. Thisis of course just a vague analogy that we have observed while developing thistheory, we do not claim that there is any sort of rigorous correspondence here.In particular we think that this dictionary is meant to be made more precise inthe future. For example a work in progress (mentioned in the introduction ofthe author’s thesis under the name “Non abelian monoidal Gelfand duality”)highly suggest that it can be made a lot more precise if we take into account onthe right hand side the monoidal structures that arise from the internal tensorproduct of Hilbert space and the compatibility to these structures.Boolean topos (locally separated) Monotone complete C ∗ -algebra (upto morita equivalence)Boolean integrable topos (locallyseparated) W ∗ -algebra (up to moritaequivalence)Localic reflection CenterMeasure on an object Semi-finite normal weightMeasure on an object of mass 1 Normal stateInvariant measure Normal (semi-finite) traceThe modular bundle The ∆ operator of theTomita-Takesaki constructionThe family of line bundles ( F t ) t ∈ R The modular time evolution3his paper essentially corresponds to the second chapter of the author thesis([2]). Although the presentation of the present paper has been revised, and theproof technique improved in comparison to the version present in the author’sthesis. T will denotes a boolean (Grothendieck) topos. Its terminalobject is denoted 1 T , if X is an object of T we denote by T /X the slice toposwhose objects are couples ( Y, f ) where f is a morphism from Y to X . It corre-sponds geometrically to the etale space of X . A family ( X i ) of object of a toposis called a generating family or a family of generator if any object of T can becovered by epimorphic image of the X i (in which case on can construct a sitefor T on the full sub-category of T whose objects are the X i ).2.2. R T denote the object of real number of the topos T . As T is boolean, allreasonable definitions of the real numbers agree and give rise to an internalordered fields satisfying the property that every bounded set has a supremum.2.3. If X is an object of T we denote by | X | the internal cardinal of X , whichis a function from the terminal object of T to N ∪ ∞ (i.e. a function to theobject N ∪ ∞ of T or equivalently a map from the locale associated to the frameof sub-object of 1 T to the discrete locale whose points are N ∪ ∞ ).2.4. An object of a topos will be called finite if it is internally finite (eitherKuratowski finite or cardinal finite which are equivalent in boolean topos). Amap f : X → Y between object of a topos is said to be finite if internallyone has ∀ y ∈ Y, f − ( { y } ) is finite. It is said to be n -to-1 if internally onehas ∀ y ∈ Y, f − ( { y } ) has cardinal n , one can eventually take n = ∞ in thisdefinition.2.5. Let B be a complete boolean algebra with 0 and 1 the bottom and topelemnts. A valuation on B is a non-decreasing function µ from B to the set R ∞ + of possibly infinite non-negative real numbers, such that µ ( U ) + µ ( V ) = µ ( U ∪ V ) + µ ( U ∩ V ), µ (0) = 0 and µ preserve arbitrary directed supremum.A valuation is said to be finite if µ (1) < ∞ , locally finite if 1 is a union ofelement of finite valuation and well supported if the only element of valuation0 is 0. except the definition by Cauchy sequences which require the axiom of depend choice tobe equivalent to the others. X is a boolean locale a measure on X is valuation on the completeboolean algebra O ( X ). If f : X → C is a continuous map and µ a valuation on X one can integrate f against µ . (see for example [9] for a general treatment ofthe integration over locale in constructive mathematics). This integral satisfy allthe properties of the usual notion of integral: it is linear, additive and preservedirected supremum when applied to positive functions. For general function,we need to restrict to function for which the integral of | f | is finite and thenit is linear and satisfy the various form of Lebesgue’s dominated convergencetheorem .2.7. Boolean locales satisfy the Radon-Nikodym theorem: If µ and ν are twolocally finite well supported measures on a boolean locale X , then there existsa (unique) continuous map f from X to R > such that µ = f.ν . This lastequality mean that for all open sublocales U ⊂ X one has µ ( U ) = R f I U dν , seefor example [3, section 3.4] for more details. T -Hilbert space an object of T endowed with operations making it into an internalHilbert space. A morphism of T -Hilbert space, is a map f which is internally abounded operator, and such that the norm is externally bounded, i.e. such thatthere exists an external constant K such that internally k f k K (we also callthem “globally bounded operator”). We denote k h k ∞ the smaller such constant K , while k h k will denote the internal norm (which is a function on the terminalobject).Endowed with this norm and the internal adjunction, the category of T -Hilbertspace is a C ∗ -category . Moreover, as T is boolean, it is monotone complete asmentioned in the introduction.3.2. If X is an object of a topos, then l ( X ) denote the T -Hilbert space a squaresumable sequences of complex numbers indexed by X . Equivalently, l ( X ) is theHilbert space generated by elements e x for each x ∈ X such that h e x , e x ′ i = 1if x = x ′ and 0 otherwise.3.3. Let T be a boolean topos, and X be an object of T , and Sub ( X ) thecomplete boolean algebra of sub-object of X . We call measure on X a valuationon the complete boolean algebra Sub ( X ) as in 2.5.If f : X → R + T is a morphism from X to the object of positive real numbersof T , then f corresponds externally to a continuous function from the locale This last fact can be deduced from the constructivity of Vickers’ theory: it allows todeduce directly lower semi-continuity of integrals with parameters, and using to the domina-tion hypothesis one obtains the lower semi-continuity of − R and hence the continuity of theintegral. See for example [1, Definition 1.1] for the definition of C ∗ -category. Sub ( X ) to the locale of real numbers. In particular, if µ is a measure on X one can define: Z X f dµ Or Z x ∈ X f ( x ) dµ as the integral of this corresponding function with respect to µ (as in 2.6).3.4. This external integration of internal functions allows to relates measureson objects to states of the C ∗ category of the topos: Lemma :
Let X ∈ T be an object of T , let H be an internal Hilbert space of T , µ a measure on X and v : X → H a function in T such that: Z x ∈ X k v ( x ) k dµ = 1 then, for f : H → H a globally bounded operator, the formula: η ( h ) := Z x ∈ X h v ( x ) , hv ( x ) i dµ defines a normal state η on the monotone complete C ∗ -algebra B ( H ) of globallybounded operators on H . Proof : η is well defined and of norm smaller than one because: Z x ∈ X | h v ( x ) , hv ( x ) i | dµ k h k ∞ Z x ∈ X k v ( x ) k = k h k ∞ the linearity and the positivity are immediate by linearity and positivity of thescalar product and the integral. The value of of η (1) follows from the conditionrelating v and µ .The normality of η follow from the fact that the supremums are computedinternally and that the integral defined in [9] commute to directed supremums. (cid:3) Definition :
A boolean topos T is said to be integrable if for all non-zeroobject X ∈ T there exists a non zero finite measure on X . Theorem :
Let T be boolean topos, then the C ∗ -category of T -Hilbertspaces is a W ∗ -category if and only if T is integrable. Proof :
Assume first that T is integrable. As the category of T -Hilbert spaces has bi-product, it is enough to show that the algebra of endomorphism of any internalHilbert space is a W ∗ -algebra (because of [1, Proposition 2.6]), and as it isalready know that these are monotone complete, we only have to prove thatthey admit enough normal state (see [7, Theorem III.3.16]).Let h : H → H be a non zero positive self adjoint operator, let X ⊂ H the set of x such that h x, h ( x ) i > k x k
1. If X = 0 then h = 0 hence, X is a nonzero subset of H which hence admit a positive finite measure. After rescaling,lemma 3.4 gives a state η such that η ( h ) > W ∗ -category, let X ∈T a non-zero object of T , then there is a state η on the algebra of endomorphismsof l ( X ). For any subobject S ⊂ X one can define P S to be the endomorphismof l ( X ) defined by P S ( e x ) = e x if x ∈ S and 0 otherwise. Then µ ( S ) = η ( P S )defines a non zero finite measure on X . (cid:3) Corollary :
Let T be an integrable boolean topos, then there exists avon Neumann algebra A (uniquely determined up to Morita equivalence of vonNeumann algebras) such that the category of T -Hilbert spaces is equivalent tothe category of self-dual Hermitian A -modules (in the sense of [1, example 1.4]). Proof :
As the category of T -Hilbert spaces has arbitrary orthogonal sums and splittingof projections (they are computed internally) it is enough, by proposition 7 . . G a set of generators of T . Let E be the family of all isomorphisms classof triples ( H, g, f ) where H is a T -Hilbert space, g is an element of G and f is amap g → H such that internally the image of this map spans a dense subspaceof H . To an element ( H , g, f ) of E one can associate a continuous function on g × g defined by ( x, y ) → h f ( x ) , f ( y ) i , and if ( H, g, f ) and ( H ′ , g, f ′ ) define thesame function on g × g then they are isomorphic. In particular the isomorphismclass of elements of E form a set.Now, for any f : A → B an operator between two T -Hilbert spaces, there exists g ∈ G and a map λ : g → A such that f ◦ λ = 0, the adherence of the spanof λ gives an element H of E and a map i from H to A such that f ◦ i = 0.This proves that elements of E form a family of generators of H ( T ), i.e. theirorthogonal sum is a generator of this W ∗ -category. This concludes the proof. (cid:3) Even if this von Neumann algebra is naturally attached to the topos T anduniquely determined by it up to Morita equivalence, it is in most case “too big”for example, if T is the topos of G -sets for some discrete group G , then H ( T ) is7he category of unitary representations of G and the associated von Neumannalgebra is the enveloping von Neumann algebra of the maximal C ∗ -algebra ofthe group, which is an enormous algebra.It seems that a more reasonable algebra to consider in practice is the algebraof operators on a space l ( X ) for X a separating bound of T . In the case of G -sets, this gives the Von Neumann algebra of the group, eventually up to Moritaequivalence depending on the choice of the object X . The results of the lastsection suggest this algebra can be controlled by the geometry of T , whereasin general an algebra of operators on an arbitrary Hilbert space over a topos T can have nothing in common with the geometry of T . For example, any vonNeumann algebra arises as the algebra of globally bounded endomorphisms ofsome representation of a discrete group. Definition :
Let T be a boolean topos. An invariant measure on T is afunction which to every object X of T associates a real number µ ( X ) ∈ [0 , ∞ ] such that:(IM1) For each X ∈ T , the restriction of µ to sub-objects of X defines a locallyfinite valuation on Sub ( X ) .(IM2) There exists a generating family of T of objects X on which the valuationinduces on Sub ( X ) is well supported.(IM3) If f : Y → X is a n -to- map in T and if µ ( Y ) < ∞ then: µ ( X ) = µ ( Y ) n Note that the third condition implies that µ ( X ) = µ ( Y ) when X and Y areisomorphic. In the case n = ∞ , the axiom ( IM
3) has to be interpreted as µ ( X ) = 0 if µ ( Y ) < ∞ .4.2. This definition, which might seems ad-hoc, essentially come from the studyof explicit example of topos where the associated Von Neumann algebra havean explicite time evolution and found is justification in the results of section 6.Some detail on its origin can be found in the author’s thesis [2] in chapters oneand two. It should probably be more correct to call this a “well supported locally finite invariantmeasure”. IM Proposition :
Let T be a boolean topos endowed with an invariant measure µ .Let f : Y → X be a finite map and h be a complex valued function on Y (i.e. afunction from Y to the object of complex number of T ).Assume that h is positive or that: Z y ∈ Y | h ( y ) | dµ < ∞ Then one has: Z y ∈ Y h ( y ) dµ = Z x ∈ X X y ∈ f − ( y ) h ( y ) dµ Proof :
We will proceed in a series of steps:1. Assume first that h is the constant equal to one function, that µ ( Y ) < ∞ and that f is a n -to-1 map for some (external) integer n . Then the resultsis exactly ( IM
3) applied to f : Y → X .2. Still assume that h = 1 and µ ( Y ) < ∞ , but f is an arbitrary finite map.Let: X n = { x ∈ X | x has exactly n antecedent by f } Y n = f − ( X n ) , and let f n be the restriction of f as a map Y n to X n . One has X = ` X n and Y = ` Y n . For all n > f n to get that µ ( Y n ) = nµ ( X n ). This formula also holds for n = 0 as Y n is empty by definition. Using the decomposition of X and Y into the X n and Y n one obtain that the left hand side of the formula is P µ ( Y n ) andthe right hand side is P nµ ( X n ) which proves they are equals.3. Assume now that Y is arbitrary and that h is the characteristic functionof S ⊂ Y with µ ( Y ) < ∞ . Then the function P f ( y )= x h ( y ) is supportedon f ( S ) hence the result follows from the previous point applied to f : S → f ( S ).4. By linearity, this automatically imply the result for any linear combina-tion of such characteristic functions. And as every positive function is adirected supremum of such linear combination and that both the integraland the sum commute to directed supremum of positive function it implythat the result holds for any positive function h .5. if h is an arbitrary complex function such that R | h | < ∞ then we candecompose it into a − b + ic − id where a, b, c, d are four positive functions,and the result follows by linearity of the integral and of the sum. (cid:3) Corollary :
Let T be a topos endowed with an invariant measure and let f : Y ։ X be an arbitrary epimorphism, with µ ( Y ) < ∞ then: µ ( X ) = Z y ∈ Y | f − ( f ( y )) | dµ Proof : If f is a finite map then the result follow directly from 4.3 applied to h = | f − ( f ( y )) | . In the general case, we decompose X and Y into two components:one where f is finite and the other where f is ∞ -to-1. on the first component theprevious result apply, one the second component µ ( X ) = 0 because of ( IM h = 0. (cid:3) Proposition :
Let C be a class of object of T such that if C ∈ C and f : X → C is any map in T then X ∈ C . Also assume that every object of T can be covered by objects in C .Let µ be a function which associate to every object in C a non-negative possiblyinfinite real number, and which satisfy all the axiom of the definition of aninvariant measure when restricted to object and map in C . Then µ extend to aunique invariant measure on T . We insist on the fact that any subobject of an object in C is also in C , henceaxiom (IM1) do mean something when we restrict to C . Proof :
First observe that the results of 4.3 and 4.4 still holds for our function µ when X, Y ∈ C . Let C f the subclass of C ∈ C such that µ ( C ) < ∞ .Let X be an object of T and assume that there is two epimorphisms f : C ։ X and f ′ : C ′ ։ X with C, C ′ ∈ C f . Then one has: Z c ∈ C | f − f ( c ) | dµ = Z c ∈ C ′ | f ′− f ′ ( c ) | dµ Indeed, to see that, let P = C × X C ′ and g and g ′ the two maps from P to C and C ′ , let h be the function on P defined by h ( c, c ′ ) = | f ′− f ′ ( c ′ ) | ∗ | f − f ( c ) | .Now, (internally) for any c ∈ C one has: X x ∈ Pg ( x )= c h ( x ) = X c ′∈ C ′ f ′ ( c ′ )= f ( c ) | f ′− f ′ ( c ′ ) | | f − f ( c ) | = 1 | f − f ( c ) | And the same computation exchanging C and C ′ together with proposition 4.3,yields that: Z c ∈ C | f − f ( c ) | dµ = Z P h = Z c ′ ∈ C ′ | f ′− f ′ ( c ′ ) | dµ In particular, for any object X of topos which is a surjective image of C ∈ C f there is a uniquely defined extension of µ to X as:10 ( X ) = Z c ∈ C | f − f ( c ) | dµ If S ⊂ X then S is the surjective image of f − ( S ) hence one also have adefinition of µ ( S ) and this defines a valuation on X . If X and X ′ are twosubobjects of Y begin epimorphic images of objects in C f , then the measuredefined on them agree on their intersection by the uniqueness property, hencethis measure extend as a measure on their union, and as every object is coveredby surjective image of object of C f by hypothesis this defines a unique measureon each object of T . We already proved that the extension satisfy (IM1), andit clearly satisfy (IM2) because C already contains a generating family of objecton which the measure is well supported.In order to conclude, we just need to prove that this extension satisfy (IM3).Let f : Y → X be a n -to-1 map. Let C ∈ C f and v : C → X . Let C ′ := C × X Y .The map π : C ′ → C is again n -to-1 hence: µ ( v ( C )) = Z c ∈ C | v − v ( c ) | dµ = 1 n Z c ′ ∈ C ′ | v − v ( π ( c ′ )) | dµ. On the other hand: µ ( π ( C ′ )) = Z c ′ ∈ C ′ | π − ( π ( c ′ ) | dµ But if c ′ = ( c, y ) then π − ( π ( c ′ )) is the set of ( a, y ) with a ∈ v − v ( c ) hence, | π − ( π ( c ′ )) | = | v − v ( π ( c ′ )) | so that: µ ( v ( C )) = 1 n µ ( π ( C ′ )) = 1 n µ ( f − ( v ( c )))As the measure on X is defined out of the measure on such v ( C ) this showsthat µ ( X ) = n µ ( Y ) and concludes the proof. (cid:3) T be a topos, if X is an object of T we denote by M ( X ) the set ofinvariant measures on the slice topos T /X . If f : Y → X is a map in topos,and µ ∈ M ( X ) then we define f ∗ µ ∈ M ( Y ) by the formula f ∗ ( µ )( V ) = µ ( f ! V ).Where f ! is the “composition with f ” functor from T /Y to T /X . This defines acontravariant functor from T to Set. Theorem :
Let T be a boolean topos, then the contravairant functor M ofinvariant measure is representable by an object denoted χ . Proof : As T is a Grothendieck topos we just need to prove that M is a sheaf for thecanonical topology of T . More precisely: let f : X ։ Y be an epimorphism, P = X × Y X and π , π : P ⇒ X the two projections, we need to prove thatfor any µ ∈ M ( X ) such that π ∗ µ = π ∗ µ there exists a ν ∈ M ( Y ) such that µ = f ∗ ν . 11y working with T /Y instead of T one can freely assume that Y is the terminalobject of T and P = X × X .Let C be the class of object V of T such that there exists a map from V to X . Let V ∈ C and let f, g : V ⇒ X two maps from V to X , then ( V, f, g ) isan object of T /P one has by hypothesis π ∗ µ ( V, f, g ) = π ∗ µ ( V, f, g ), but one theother hand, π ∗ µ ( V, f, g ) = µ ( V, f ) and π ∗ µ ( V, f, g ) = µ ( V, g ). Hence µ ( V, f )does not depends on f : V → X and we define, for V ∈ C , ν ( V ) = µ ( V, f ) forany f : V → X . It is easy to check that this satisfy the axiom of an invariantmeasure restricted to C : for each axiom, one can chose a coherent family offunction to X and apply the corresponding axiom for the measure µ in T /X .Hence by proposition 4.5 this extend to a unique measure ν on T which bydefinition satisfy f ∗ ν = µ (cid:3) χ comes with several structures (due to structureson M ). The most important is the multiplicative action of R > T on χ whichcorresponds to the fact that if µ is an invariant measure on T /X and f a functionfrom X to R > T then one can define, exactly as in the case of classical measuretheory, f.µ by f.µ ( Y, p ) = R f ( p ( y )) dµ . One can also define an addition on χ ,but there is no “zero” element because it is ruled out by (IM2). In this section we introduce the notion of separated and locally separated toposesdue to I.Moerdjik and C.J.Vermeulen in [6], and we prove a new characterizationseparation for boolean toposes (see Theorem 5.2).5.1. The following definitions are due to Moerdijk and Vermeulen (see [6])
Definition : • A topos is said to be compact if its localic reflection is compact. • A geometric morphism f : E → T is said to be proper if internally in T ,the T -topos E is compact. • A geometric morphism f : E → T is said to be separated if its diagonalmap ∆ :
E → E × T E is a proper map. • A topos is said to be separated if its canonical geometric morphism to thebase topos is separated.
Being separated is a strong property, for example the topos of G -sets for adiscrete group G is separated if and only if G is finite.12.2. The main result of this section is the following characterization of separa-tion in the case of boolean toposes: Theorem :
Let T be a boolean topos. Then T is separated if and only if T hasa generating family of internally finite object. The proof of this theorem will be dived in various lemma, one implication isproved in corollary 5.4 and the other by corollary 5.6.5.3.
Proposition :
Let T be a hyperconnected separated topos. Then T isatomic and all its atoms are (internally) finite. Proof :
Let T be a hyperconnected topos. Let B be a non zero boolean locale endowedwith a geometric morphism to T . In the logic of B , the pullback of the topos T is hyperconnected separated and has a point. Applied internally in B , thetheorem II.3.1 of [6] shows that the pullback of T is equivalent to the topos of G -sets for G a compact localic groups in B . In particular the pullback of T isatomic in the logic of B and as the map from B to the base topos is an opensurjection (because the base topos is boolean) it implies that T is atomic in thebase topos (by [4, C5.1.7]).Let now a be an atom of a separated topos T , then T /a → ∗ is a proper map(because as a is an atom it is hyperconnected), and T → ∗ is separated byhypothesis. Hence, by Proposition II.2.1(iv) of [6] this proves that the map T /a → T is proper, i.e. that a is internally finite. (cid:3) Corollary :
Let T be a boolean separated topos, then T admit a generatingfamily of internally finite objects. Proof :
Let L be the localic reflection of T . The geometric morphism from T to L ishyperconnected by definition and separated by [6, II.2.3 or II.2.5], hence we canapply proposition 5.3 internally in L (which is boolean) and this proves that themap from T to L is atomic and internally with finite atoms.We need to externalise this result: Let X be any object of T . Internally in L ,the object X still corresponds to an object of T (now seen as a topos internallyin L ). Because of the first observation, it is true internally in L that X is aunion of finite subobject (even finite atoms). Externally this mean that X canbe covered by subobjects S defined over an open sublocale U ⊂ L with the map S → U finite (such a S corresponds to a section over U of the object of finitesubobject of X ). But as U is complemented, S is finite in T : internally in T if This means that its localic reflection is the point, i.e. that 1 T has no non-trivial subob-jects, see [4, A4.6]. then S is finite but if ¬ U then S is empty hence also finite. This proves thatany object of X can be covered by finite objects and this concludes the proof. (cid:3) Lemma :
Let T be a (possibly non-boolean) topos which admit a generatingfamily of of objects X which are finite in the sense that ∃ n, X ≃ { , . . . , n } holdsinternally. Let p and q be two points of T . Then the locale I of isomorphismsfrom p to q , defined as the pullback: I ∗∗ T pq is compact.Moreover, this lemma holds in intuitionist mathematics. Proof :
This locale I classifies the theory of isomorphisms from p to q . By hypothesis,one can construct a site C for T whose representable objects corresponds tointernally finite objects. As point, p and q can then be seen as flat continuousfunctor from C to sets (see [5, VII.10]), and a morphism from p to q can thenbe describe as a collection of map from p ( c ) to q ( c ) for c ∈ C satisfying thenatural transformation condition. Because of the assumption, all the p ( c ) and q ( c ) are finite sets, hence the theory of collection of map from p ( c ) to q ( c ) iscompact by the localic Tychonof theorem and the condition of being a naturaltransformation can be written as a series of equality which, because equalityis decidable in finite objects with this notion of finiteness, is going to be anintersection of closed sublocale hence is again closed and is hence compact.An isomorphism is given by a couple of morphisms which are inverses of eachother hence the classifying locale for isomorphisms is a closed sublocale of the lo-cale hom( p, q ) × hom( q, p ) and hence is also compact, which concludes the proof. (cid:3) Corollary :
Let T be a topos admitting a generating family of object whichare internally finite (in the sense that internally ∃ n, X ≃ { , . . . , n } ) then T isseparated. Moreover this corollary holds in constructive mathematics.
14s a finite object in a boolean topos automatically satisfy this strong from offiniteness (because under the law of excluded middle all these possible definitionsof finite sets are equivalent) this automatically imply the last part of the theorem
Proof :
The key observation, is that the hypothesis of being generated by a familly offinite object is pullback stable, in the sense that if T satisfy it and E is anothertopos then T × E satisfies it internally in E because a pullback of a finite objectis finite and a site for T × E internally in E is given by pulling back to E asite for T in sets. Hence as the lemma 5.5 has been proven constructively itsconclusion will holds internally in E for all the the pullback of T . The trick isthen as usual to apply this to the universal case: that is internally in E = T × T in which (the pullback of) T has two canonical point, namely π and π and thelocale of isomorphismes between them is exactly the diagonal embedings of T in T × T , hence one get that this map is proper and hence that T is separated. (cid:3) Note that even when T is boolean, T × T is in general not (essentially unless T is atomic) hence we really needed lemma 5.5 to be proved in intuitionistmathematics in order to obtain corollary 5.6 even in classical mathematics.5.7. We conclude this section with the notion of locally separated topos. Definition :
An object X of T is said to be separating if the slice topos T /X isseparated. A topos is said to be locally separated if it has a generating family ofseparating objects. A slice of a separated topos (by a decidable object) is again separated, hence ina boolean topos as soon as one has an arrow X → Y with Y separating, X isagain separating. As moreover a coproduct of separating object is separating, aboolean topos (or more generally a locally decidable topos) is locally separatedif an only if the terminal object an be covered by separating object.5.8. In addition, the class of separating objects also enjoy another stabilityproperty: Lemma :
Let f : X ։ Y be a surjection in T . Assume that f is a finite map,then Y is separating. Proof :
The fact that f is a finite surjection means that the induced map T /X → T /Y is a proper surjection (indeed, internally in T /Y this means that the object X → Y is inhabited and finite). The result then follows immediately from [6,Prop II.2.1.(iii)], asserting that if f ◦ g is separated with g a proper surjectionthen f is separated. (cid:3) Main result and examples
Lemma :
Let T be a boolean topos endowed with an invariant measurewhich is well supported on the terminal object, then T is separated. Proof :
By (IM1), T is generated by objects with µ ( X ) < ∞ . For such an object X ,let P be the suboject of 1 T corresponding to the proposition “ X is infinite”.The map from X × P to P is ∞ -to-1, hence by ( IM
3) one has µ ( P ) = 0. As µ is supposed to be well supported on 1 T this imply that P = 0 hence X isinternally finite, and hence T is generated by finite objects and hence separatedby theorem 5.2. (cid:3) Note that a local application of this result automatically show that if X is anobject of a topos T endowed with an invariant measure which is well supportedon X then X is a separating object. In particular T is automatically locallyseparated. We will improve this with theorem 6.3.6.2. Proposition :
Let T be a boolean separated topos, and µ a well supportedlocally finite valuation on Sub (1 T ) , then there is an invariant measure ˜ µ one T defined by: ˜ µ ( X ) = Z T | X | dµ, Moreover µ is the restriction of ˜ µ to subobjects of T and every invariant mea-sure on T is of the form ˜ µ for some locally finite well supported valuation on Sub (1 T ) . Proof :
We will first show that ˜ µ satisfies the three points ( IM − ( IM
3) of the defi-nition 4.1.(IM1) One easily sees that ˜ µ is a valuation on Sub ( X ) for any X ∈ |T | essentiallybecause the cardinal is internally a valuation on X and that the integral ofan internal valuation is again a valuation because the notion of integral weare using is linear and preserve directed supremums of positive functions.It remains to prove that this valuation is locally finite. If X is a finiteobject of topos, then | X | is everywhere finite hence we can cover 1 T bysubobjects U such that R U | X | < ∞ , and for such U , the object X U = X × U satisfy ˜ µ ( X U ) = R U | X | < ∞ and as the X U form a covering of X and that every object of T can be covered by finite objects this concludesthe proof.(IM2) Let X such that ˜ µ ( X ) = 0, then as µ is well supported this imply that | X | = 0 hence X is the zero object. This proves that ˜ µ is well supportedon every object. 16IM3) if f : X → Y is a n -to-1 map then internally | X | = n | Y | . hence if R | X | < ∞ , as µ is well supported, then | X | < ∞ everywhere, and henceone indeed get R | Y | = R | X | /n .Now, if U ∈ Sub (1 T ) then | U | is just the characteristic function of U hence˜ µ ( U ) = µ ( U ).Conversely, if µ is an invariant measure on T and if X is any object such that µ ( X ) < ∞ then by proposition 4.3 applied to f : X → T (the terminal object)and h = 1 one has: µ ( X ) = Z X dµ = Z T X x ∈ X ! dµ = Z T | X | dµ Hence µ is of the announced form. We just have to check that the measureinduced by µ on Sub (1 T ) is well supported, but if v ⊂ T is of zero measure anyobject defined over V has measure zero because of the previous formula, henceaxiom ( IM
2) imply that V is empty. (cid:3) Theorem :
Let T be a boolean topos then χ is well supported (i.e. inter-nally inhabited) if and only if T is integrable and locally separated. Moreover, if T is integrable and locally separated then χ is a principal bundle for the actionof R T . Proof :
First let T be a boolean topos such that χ is well supported. We will prove that T is integrable and locally separated.Let X → χ be any map, then by definition T /X is endowed with an invariantmeasure. In particular T /X admit a generating family of object Y on whichthe measure is well supported, and hence by lemma 6.1 a generating family ofseparating object of T /X . Such object are also separating in T /X , and as χ isassumed to be well they form a generating family of T . This proves that T is locally separated. Moreover, as we have constructed a generating family ofobjects of T which admit well supported measures this also proves that T isintegrable.Conversely assume that T is integrable and locally separated, we will prove that χ is well supported and is a principal R T -bundle.Let X be any separating object of T . As T is integrable one can cover X bysubobject S ⊂ X such that S is endowed with a well supported measure. Suchan object S is separating and endowed with a well supported measure, hencethere is an invariant measure on T /S by proposition 6.2, and hence a map from S to χ . As such objects from a generating family of T , they in particular forma covering of 1 T and hence χ is inhabited.Moreover, still by proposition 6.2 any two functions from such an object S to χ correspond to two measures on S , and hence (as Sub ( S ) is boolean) theRadon-Nikodym theorem for complete boolean algebras imply that they differ17nly by multiplication by a function from S to R T . As objects of this form aregenerating this exactly proves that χ is a R T -principal bundle. (cid:3) T is a topos of sheaves over a boolean locale L then T is alwaysseparated and χ is the sheaf of locally finite well supported measure on L . Itis inhbited if and only if L is integrable and in this case it always has a section(assuming the axiom of choice, every inhabited sheaves over a boolean localehas a section). Moreover, it is a principal bundle exactly because of the Radon-Nikodym theorem.b) If T is the topos of G -sets for G a discrete group. Let G be endowed withits left action on itself then T /G is isomorphic to the topos of sets. Hence G isseparating, and χ is as a set R > (indeed, element of χ are morphism from G to χ , i.e. invariant measure on T /G ≃ sets. On then easily check that the G actionon χ is trivial, and that T do have an invariant measure given by: µ ( X ) = X x ∈ ( X/G ) | Stab( x ) | Finally, T is separated if and only if G is finite, in which case the previousformula can be rewritten as | X | / | G | .c) More generally, If T is the topos of G equivariant sheaves over a boolean locale L endowed with an action of a discrete group G , then there is an object X suchthat T /X is isomorphic to the topos of sheaves over L . In particular morphismfrom X to χ are exactly measure on L , and a short calculation involving thisobject X allows to deduce from this that χ is the sheaves of measure on L endowed with the natural action of G on it.In particular, an invariant measure on T is exactly a well suported locally finitemeasure on L invariant by the action of G (this is where the name “invariantmeasure” comes from). Also note that the formula giving the invariant measureon T in terms of the G -invariant measure on L might be complicated in themore general case. This can be generalized easily to any boolean etendu.d) Consider now the case where G is a etale complete localic group and T isthe topos of continuous G -sets. On can then see that T is always integrable, itis separated if and only if G is compact and locally separated if and only if G islocally compact. A short computation yields that χ is simply R > endowed withthe action of G trough its modular character. In particular, T has an invariantmeasure if and only if G is unimodular and in this case the invariant measure isgiven by the same formula as for discrete group but by replacing | Stab( x ) | by µ (Stab( x )) for a bi-invariant Haar measure µ . A reader unfamiliar with this terminology can replace “etale complete localic group” bypro-discrete topological group. The reader can observe that as x is an element of X/G and not of X this only make senseif µ is bi-invariant.
18) In the general case where T is the topos of equivariant sheaves over an etalecomplete groupoid with a boolean space of objects, then one can think of χ as the sheave of right invariant left Haar systems endowed with the left actionof the groupoid, and hence an invariant measure on T can be thought of as abi-invariant Haar system. But this cannot be made into a precise statementwithout developing first a theory of Haar systems well adapted to the localicframework. χ one can define the time evolution ofHilbert spaces in the following way: χ is R > T bundle, and for each t , as x x it define a continuous morphism from R > to the circle group, one can define for each t a principal bundle χ it for thecircle group χ it . A principal bundle for the circle group is the same as a internalone dimensional Hilbert space, hence for each t one has a internal one dimensionHilbert space F t in T . One can then easily see that because χ it ⊗ χ it ′ = χ i ( t + t ′ ) one has F t ⊗ F t ′ = F t + t ′ . Hence σ t : H 7→ H ⊗ F t defines a one parameter familyof endofunctor of the W ∗ -category of T -Hilbert space.7.2. One can give a more explicit formula for these constructions: For t ∈ R , F t is the sheaf defined as: F t = { f : χ → C |∀ α ∈ χ, ∀ r ∈ R > , f ( r.α ) = r − it f ( α ) } The addition, multiplication and scalar product are defined pointwise, and theinternal choice of a v ∈ χ induce an internal isomorphism between F t and C preserving all these structures.The isomorphism between F t ⊗ F ′ t and F t + t ′ is given by sending u ⊗ v to thepointwise product of u and v .7.3. The time evolution is then defined by: σ t ( H ) = F t ⊗ H for H a T -Hilbertspace. Equivalently, one can define: σ t H = { f : χ → H|∀ α ∈ χ, ∀ r ∈ R > , f ( r.α ) = r − it f ( α ) } The σ t are endofunctors of the category of T -Hilbert spaces. this does not mean that F t is trivial, but only that it is “locally trivial of rank one”. χ give rise to trace. It is actually not true that this family of functors σ t is the modular time evolution of the full W ∗ -categories of T -Hilbert spaces.This category is too big and there are Hilbert spaces in it which have nothingto do with the geometry of T . In order to get a result in this spirit we need tofocus on a smaller, more reasonable full subcategory, essentially correspondingto the notion of “square-integrable” representations of a group. More precisely,we will focus our attention the subcategory which is generated (in the sense ofthe generators of [1, Proposition 7.3]) by T -Hilbert spaces of the form l ( X ) for X a separating object, and hence it is enough to study the Hilbert spaces of theform l ( X ) for X a separating object.7.5. For now on, X denotes a separating object of a boolean integrable locallyseparated topos T . We also choose µ a locally finite well supported measureon X , hence an invariant measure on T /X , which corresponds to a morphism λ from X to χ .We denote by l ( X ) the Hilbert space internally defined as the set of squaresummable X -indexed sequences, with (internally) its generator ( e x ) ∈ l ( X ) foreach x ∈ X . We also denote by A = B ( l ( X )) the (external) von Neumannalgebra of globally bounded operators on l ( X ).7.6. µ can be used to construct a normal locally finite weight on A , also denoted µ defined by: ∀ f ∈ A + , µ ( f ) = Z x ∈ X h x, f x i dµµ is locally finite because if V is a subset of finite measure of X , and if P V ∈ A denotes the orthogonal projection on l ( V ) ⊂ l ( X ) then P V f P V has measuresmaller than the measure of V times the norm of f and (assuming f is positive)when V vary among all finite measure subsets of X , this constitute a directednet of operator whose supremum if f .7.7. λ can be used to construct an isomorphism φ t : l ( X ) ≃ σ t l ( X ).Indeed, one can define (internally): φ t ( e x ) := α (cid:18) αλ ( x ) (cid:19) − it e x ! ∈ σ t l ( X )where α/λ ( x ) denotes the unique element r of R > such that α = r.λ ( x ). Definedthis way φ t ( e x ) indeed satisfies the relation φ t ( e x )( r.α ) = r − it φ t ( e x ), showingthat it is an element of σ t ( l ( X )). Moreover the φ t ( e x ) are of norm one and pair-wise orthogonal, hence φ t indeed defines an isometric map l ( X ) → σ t ( l ( X )).It satisfies in particular, φ = Id and σ t ( φ t ′ ) ◦ φ t = φ t + t ′ hence σ t φ − t constitutesan inverse for φ t showing that it is an isomorphism.20.8. Finally, as l ( X ) is “fixed” by the time evolution (as attested by the iso-morphism φ t ) one obtains an action of R directly on A , via: ∀ a ∈ A, θ t ( a ) = φ − t σ t ( a ) φ t One easily checks that it is an action of R (either directly or from the followingproposition). This time evolution on A can be more explicitly described on thematrix elements by: Proposition :
For a an element of A and x, y internal elements of X : h e y , θ t ( a ) e x i = (cid:18) λ ( y ) λ ( x ) (cid:19) − it h e y , ae x i With λ ( y ) λ ( x ) denoting the unique element r ( x, y ) of R > such that r ( x, y ) λ ( x ) = λ ( y ). Proof :
One has by definition of θ and as the φ t are isometric: h e y , θ t ( a ) e x i = h φ t ( e y ) , σ t ( a ) φ t ( e x ) i But: φ t ( e x ) = α (cid:18) αλ ( x ) (cid:19) − it e x And similarly for y , hence: h φ t ( e y ) , σ t ( a ) φ t ( e x ) i = *(cid:18) αλ ( y ) (cid:19) − it e y , (cid:18) αλ ( x ) (cid:19) − it ae x + = (cid:18) λ ( y ) λ ( x ) (cid:19) − it h e y , ae x i (cid:3) Proposition :
For each u ∈ A + such that µ ( u ) is finite one has µ ( θ t ( u )) = µ ( u ) .Let u, v ∈ A such that µ ( u ∗ u ) , µ ( uu ∗ ) , µ ( v ∗ v ) and µ ( vv ∗ ) are all finite. Thenthere exists a complex function F u,v defined on { z ∈ C | Im ( z ) ∈ [ − , } andholomorphic on its interior such that for all real numbers t : F u,v ( t ) = µ ( θ t ( u ) v ) F u,v ( t − i ) = µ ( vθ t ( u ))This proves that θ t is indeed the modular group of automorphisms of the al-gebra A , associated to the semi finite normal weight µ . See [8, Chapter VIII].21oreover, if T is endowed with an invariant measure, then χ as a global sec-tion, λ can be chosen to be constant equal to this global section. In this case,the formula 7.8 shows that θ t is the identity for all t and hence this last resultshows that µ is a normal semi-finite trace on A constructed out of the invariantmeasure. Proof :
From the formula given in 7.8 one can see that θ t left unchanged the diagonalcoefficients of u , and as µ is defined as the integral of the diagonal coefficientsone immediately has that µ ( θ t ( u )) = µ ( u ).Let, for any a ∈ A , the function a yx of matrix coefficients be defined internallyby a yx = h e y , ae x i (it is a function on X × X ). A general formal computationgives that for a, b ∈ A : µ ( ab ) = Z x ∈ X h e x , abe x i dµ = Z x ∈ X * e x , X y ∈ X ab yx e y + dµ = Z x ∈ X X y ∈ X b yx a xy dµ = Z ( x,y ) ∈ X b yx a xy dπ ∗ µ The last equality, corresponds to proposition 4.3, and holds only if b yx a xy is apositive function, or if the integral is finite when we replace b yx a xy by | b yx a xy | .In particular, it holds when a, b are ( u ∗ , u ),( u, u ∗ ),( v ∗ , v ) or ( v, v ∗ ), and this to-gether with the finiteness hypothesis on u and v shows that all the four integralsof | a yx | and | b yx | with respect to both dπ ∗ µ and dπ ∗ µ on X × X are finite. Alsonote that under the correspondence between measures on X × X and functionsfrom X × X to χ , the measure π ∗ µ corresponds to the function ( x, y ) λ ( x )and π ∗ µ to the function ( x, y ) λ ( y ). Hence one has: π ∗ µ = (cid:18) λ ( y ) λ ( x ) (cid:19) π ∗ µ (1)For any complex number z such that Im ( z ) ∈ [ − ,
0] one has: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18) λ ( y ) λ ( x ) (cid:19) iz u xy v yx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:18) λ ( y ) λ ( x ) (cid:19) − Im ( z ) | u xy || v yx | | u xy | + | v yx | + (cid:18) λ ( y ) λ ( x ) (cid:19) | u xy | + (cid:18) λ ( y ) λ ( x ) (cid:19) | v yx | (2)And the four terms on the right have a finite integral on X × X with respect to π ∗ µ (because of (1) for the last two), hence one can define the following function F u,v which is finite and continuous for Im ( z ) ∈ [ − ,
0] and the previous formalcomputation can be applied to it. F u,v ( z ) = Z ( x,y ) ∈ X × X (cid:18) λ ( y ) λ ( x ) (cid:19) iz u xy v yx dπ ∗ µ µ ( ab ) and the expression givenin 7.8 for the matrix coefficients of θ t ( a ), one has for t real F u,v ( t ) = µ ( θ t ( u ) v ),and using equation (1) one gets that: F u,v ( t − i ) = R ( x,y ) ∈ X × X (cid:16) λ ( y ) λ ( x ) (cid:17) it u xy v yx λ ( y ) λ ( x ) dπ ∗ µ = R ( x,y ) ∈ X × X (cid:16) λ ( y ) λ ( x ) (cid:17) it u xy v yx dπ ∗ µ = R ( y,x ) ∈ X × X (cid:16) λ ( y ) λ ( x ) (cid:17) it u xy v yx dπ ∗ µ = µ ( vθ t ( u ))It remains to be proven that F u,v is holomorphic. Let V n be the subobject of X × X on which the function λ ( y ) λ ( x ) is between (1 /n ) and n . one has S V n = X × X and consider: F nu,v = Z ( x,y ) ∈ V n (cid:18) λ ( y ) λ ( x ) (cid:19) iz u xy v yx dπ ∗ µ The functions F nu,v are holomorphic, and the inequality (2) shows that F nu,v converges to F u,v uniformly in z on all its domain of definition, showing that F u,v is holomorphic on the interior of its domain. (cid:3) References [1] Ghez, P and Lima, Ricardo and Roberts, John E. W ∗ -categories. Pacific J.Math , 120(1):79–109, 1985.[2] Henry, Simon.
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