aa r X i v : . [ m a t h . C T ] N ov Toposes, quantales and C ∗ algebrasin the atomic case Simon HenryNovember 6, 2018
Abstract
We start by reviewing the relation between toposes and Grothendieckquantales. We improve results of previous work on this relation by givingboth a characterisation of the map from the tensor product of two inter-nal sup-lattices to another sup-lattice and a description of the category ofinternal locales of a topos in terms of the associated Grothendieck quan-tale. We then construct a convolution product, corresponding to internalcomposition of matrices, on the set of positive lower semi-continuous func-tions on the underlying locale of the quantale attached to a topos. In goodcases, this convolution product does restrict into a well defined convolu-tion product on a subset of the set of continuous functions and defines aconvolution C ∗ algebra attached to the quantale. In the last part of thisarticle we investigate in details these attached C ∗ algebras in the spe-cial case of an atomic topos. In this situation the related Grothendieckquantale corresponds to a hypergroupoid. Relatively simple finitenessconditions on this hypergroupoid appear in order to obtain an interesting C ∗ algebra. This algebra corresponds to a hypergroupoid algebra whichcomes endowed with an arithmetic sub-algebra and a time evolution. Weconclude by showing that the existence of a hypergroupoid satisfying allthe requirements attached to a specified atomic topos is equivalent to thefact that the topos is locally decidable and locally separated. Also in thissituation the time evolution only depends on the topos and is described bya (canonical) principal Q ∗ + bundle on the topos. The BC-system and moregenerally the double cosets algebras are special cases of this situation. Contents sl ( T ) of sup-lattices . . . . . . . . . . . . . . . . . . 8 Key words. toposes, quantales, hypergroups, atomic toposes, C ∗ algebras, time evolution sl . . . . . . . . . . . . . . . . . . . . . . 113.4 Quantale Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.5 Relational representations of Grothendieck quantales . . . . . . . 213.6 Internally bi-linear maps between Q -modules . . . . . . . . . . . 233.7 Representations of modular quantales . . . . . . . . . . . . . . . 293.8 Towards a convolution C ∗ algebra attached to a quantale . . . . 30 T -groups . . . . . . . . . . . . . . . . . . . . . . 434.6 A G as a quantum dynamical system . . . . . . . . . . . . . . . . 474.7 The time evolution of an atomic locally separated topos. . . . . . 504.8 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 Both C ∗ algebras and toposes yield natural generalizations of topological spaces:locally compact topological spaces correspond to commutative C ∗ algebras ofcontinuous functions, and sober topological spaces correspond to toposes ofsheaves over them. Non commutative C ∗ algebras and general toposes thusextend the notion of topological space beyond its classical framework. It is hencenatural to ask whether these two generalizations are in any sense related to eachother, and such a relationship should be extremely interesting for both areas:toposes are closely related to geometry thanks to their relation with localicgroupoids developed in [16], [6], [15] , while non commutative C ∗ algebras arealso intimately related to groupoids ([7]) and have proven themselves extremelypowerful through their connection with the quantum formalism, the theory oftime evolution on type III factors and quantum statistical mechanics.More recently, a third family of objects appeared in this picture: Quantale. Inoperator algebra they have been introduced by C.J.Mulvey (see [22] and [13]) inan attempt to formalize the notion of “quantum topology” studied by R.Gilesand H.Kummer in [9] and C.A.Akemann in [1]. In topos theory they arise inthe description of the category of sup-lattices of a given topos studied in [16]and, because of the result [12] (see also the first part of the present article) theycompletely describe a topos in the sense that a topos endowed with a bound isessentially the same thing as a special kind of quantale, called a “Grothendieckquantale”.The type of quantale appearing in operator algebra and in topos theory haveextremely different properties: the Grothendieck quantales are quantales of re-lations on a bound of a topos, and behave like the quantale of relations on a set,in particular they are distributive and modular. On the other side, quantalesappearing in operator algebra have different properties, they are in general notmodular and correspond to particular subquantales of the quantale of projec-tions in a Hilbert space, hence deserving the name “quantum” in a more precise2anner. These differences preclude a straightforward comparison of the theo-ries of Grothendieck toposes and of operator algebras through the associatedquantales and show that the relation between the two theories is necessarilymore involved. We nevertheless use the Grothendieck quantale associated to atopos as a starting point and show in this paper that under suitable hypothe-sis a Grothendieck quantale can be used to construct a convolution C ∗ algebraattached to a topos.We would also like to stress out that modular quantales and Grothendieck quan-tales are extremely good candidates to be thought of as characteristic one opera-tor algebras. First there are several formal similitudes: the fact that sup-latticesenriched categories are a form of “characteristic one additive categories”, thepresence and the important role of the ∗ involution, and other more specificpoints like the fact that the initial and terminal support of a are given by a ∗ a and aa ∗ . Secondly, Grothendieck quantales (and conjecturally modular quan-tales) are interpreted as quantales of relations on objects of topos (see 3.4.5 and3.7.1), i.e. as characteristic one matrix algebras. Hence results of sections 3underline the fact that there is a close relation between topos theory and non-commutative geometry in characteristic one. It might be interesting to makethis relation more precise, for example by giving an interpretation of the dis-tributivity (Q3) and the modularity (Q9) conditions in term of characteristicone semirings.One of the most powerful features of topos theory is probably the internal logic:it has been shown in [4] that a topos can be considered as a “mathematicaluniverse” which differs from the classical universe of sets we are used to workwith by the fact that the law of excluded middle and the axiom of choice mightfail. This means that there is a way to interpret any mathematical (formallywritten) statement about sets as a statement about objects of a topos, and anytheorem which has a constructive proof becomes a theorem about objects inany topos. One can consult the part D of [14] for a detailed presentation of thelogical aspect of topos theory.We will make an extensive use of this internal logic, and we hope that all thetransition between the world of usual sets and the world of sheaves will be clear.Also, all the mathematics presented in this article are constructive. This is notbecause of any form of belief from the author that the law of excluded middleshould be systematically avoided, but because it was possible to dispense fromit without adding too much complexity and it opens the possibility of applyingour results internally.In section 3 we focus on the relation between “Grothendieck quantale” andGrothendieck toposes. Most results of the section 3 . . By constructive we mean which does not use the law of excluded middle or the axiom ofchoice. Q in term of a notion of Q -set completely similarto the notion of L -set when L is a frame. Section 3.5 produces a descriptionof the topos attached to a quantale as a classifying topos making the previouscorrespondence functorial.The main (and essentially only) new contribution of section 3 is in 3.6 andconsists of an extremely elegant description of the locales internal to a topos interms of “modular actions” of the corresponding quantale on classical locales,as well as more generally a description of bi-linear maps between sup-lattices interms of the corresponding Q -modules.In 3.8 we explain why we think that attaching a Grothendieck quantale to atopos is an interesting step towards the construction of C ∗ -algebras.In section 4 we focus on the case of an atomic topos; we show that in this casethe attached Grothencieck quantale corresponds to a “Hypergroupoid” and thatunder some reasonable finiteness assumptions there is indeed a “Hypergroupoid C ∗ -algebra” attached to that quantale in the way sketeched in 3.8. This C ∗ alge-bra comes in two forms: a reduced algebra and a maximal algebra; in both casesit comes with a natural and explicit time evolution attached through Tomitatheory to a “regular” representation, and with a generating Z sub-algebra withinteresting combinatorial properties. We also characterise in section 4.7 theatomic toposes for which the construction is possible as the locally decidablelocally separated toposes. We also show that in this situation the time evolutionis canonical and described by a principal Q ∗ + bundle. The main example of thissituation are the well-known double-cosets algebras. By toposes we always mean Grothendieck toposes, i.e. categories of sheaves ona Grothendieck site. Most of the basic notions of topos theory can be found in[18], for the others we will give precise references in [14]. • if C is a category (or a topos) then |C| denotes the set (or the class) ofobjects of C . The symbol C denotes the set (or class) of all maps, and wewill equivalently use the notation C ( a, b ), hom( a, b ) or hom C ( a, b ) for theset of morphisms from a to b . • The letter T always denotes a topos. • Ω T denotes its sub-object classifier. • T denotes its terminal object. • If X ∈ |T | then P ( X ) stand for the power object of X (isomorphic toΩ T X ), and Sub ( X ) for the set of sub-objects of X , i.e. the set of globalsections of P ( X ). 4 A set (or an object X ∈ |T | ) is said to be inhabited if it satisfy (internally) ∃ x ∈ X (which in constructive mathematics is stronger than the assertionthat X is not empty). For an object of a topos it is equivalent to the factthat the canonical map X → • N T , Z T and Q T denote the sheaves of natural numbers, integers andrational numbers in the topos T . As we restrict ourselves to Grothendiecktoposes, they are simply p ∗ ( N ), p ∗ ( Z ) and p ∗ ( Q ) where p is the canonicalgeometric morphism from T to the topos of sets. • R T denotes the sheaf of continuous real numbers, i.e. two sided Dedekindcuts (see [14] D4.7). It can be described externally by the following prop-erties: for any X ∈ |T | , hom( X, R T ) is the set of continuous functionsfrom the underlying locale of X ( whose frame of opens is Sub ( X )) to thespace of real numbers . • R lsc + T denotes the set of positive lower semi-continuous real numbers (pos-sibly infinite). In presence of the law of excluded middle it is the set R + ∪{∞} . In a topos it is the sheaf defined by the fact that hom( X, R lsc + T )is the set of functions from the locale Sub ( X ) to R + ∪ {∞} endowed withthe topology where the ( a, ∞ ] are a basis of open sets, i.e. it is the set oflower semi-continuous functions (possibly infinite) on the locale Sub ( X ).Internally, R lsc + T is defined as the set of P ⊂ Q T such that if q < q ∈ P , and q ∈ P ⇔ ∃ q ′ ∈ P, q < q ′ . See [14] D4.7. • A sub-quotient of an object X ∈ |T | is a quotient of a sub-object of X (orequivalently, but less naturally, a sub-object of a quotient). • A proposition (internal to a topos) is said to be decidable if it is com-plemented (i.e. equal to its double negation). An object is said to havedecidable equality, or to be decidable if its diagonal embedding X → X × X is complemented.Also an object B ∈ |T | is said to be a bound of T if any object of T can bewritten as a sub-quotient of an arbitrary co-product of copies of B (see [14]B3.1.7.). Equivalently B is a bound of T if Sub ( B ) is a generating family of T ,i.e. Sub ( B ), seen as a full subcategory of T and endowed with the restrictionof the canonical topology of T , forms a site of definition for T . This meansessentially that B is big enough to generate T : in the topos of G − Set for agroup G , an object X is a bound if and only if the map G → Aut ( X ) is injective.A topos is the topos of sheaves over a locale if and only if 1 T is a bound (see[14]Definition A4.6.1 and theorem C1.4.6). When a topos is given by a site, thesimplest way to obtain a bound is to choose an object which contains a copy ofeach representable object (for example, the direct sum of all the representableobjects, see [14]B3.1.8(b)).Existence of a bound, together with the existence of enough (co)limits charac-terize Grothendieck topos among elementary topos (see [14] C2.2.8) In a non-boolean context, the “space of or real numbers” has to be interpreted as “theformal locale of real numbers”. Here we have assumed the law of excluded middle in the topos of set in order to simplifythe notation. X (or an object of a topos) will be said to be finite if it is (internally)Kuratowski finite, i.e. if ∃ n ∈ N , x , . . . , x n ∈ X such that for all x ∈ X ∃ i, x = x i . On can consult [14] D4.5 for the theory of Kuratowski finite set.Roughly, a quotient of a finite set is finite, but the proof that a subset of afinite set is finite requires the subset to be complemented and may fail in fullgenerality. If a set X is finite and has decidable equality, then there exists n ∈ N such that X is isomorphic (internally ) to { , . . . , n } , and a subset of X is finiteif and only if it is complemented. X be any object of a topos T , we will denote by Rel ( X ) the setof relations on X , i.e. the set Sub ( X × X ) of sub-objects of X × X . Then Q = Rel ( X ) is endowed with several structures:(Q1) The inclusion of subobjects gives an order relation on Q .(Q2) Q has arbitrary supremum for this order relation.(Q3) Finite intersections distributes over arbitrary union: a ∧ W i b i = W i ( a ∧ b i ).(Q4) There is an associative composition law on Q defined internally by RP = { ( x, y ) |∃ z ∈ X, xRz and zP y } .(Q5) The composition law is order preserving and commutes to supremum ineach variable.(Q6) The diagonal subobject of X provide an element 1 ∈ Q which is a unit forthe composition law.(Q7) There is an order preserving involution: R R ∗ = { ( y, x ) | xRy } of Q .(Q8) For all x, y ∈ Q one has: ( xy ) ∗ = y ∗ x ∗ .(Q9) For all x, y, z ∈ Q one has x ∧ yz y ( y ∗ x ∧ z ).If we assume additionally that X is a bound of T , then one has additionally:(Q10) For all q ∈ Q there are two families ( v i ) i ∈ I , ( u i ) i ∈ I of elements of Q suchthat: ∀ i, u i u ∗ i , v i v ∗ i ⊤ = W i v i u ∗ i . where ⊤ denote the topelement of Q .Some of these points deserve a proof and a few comments. • ( Q
9) is called the modularity law. It is easy to prove using internal logic:Let ( a, b ) ∈ ( x ∧ yz ). One has: ( a, b ) ∈ x and there exists c ∈ X suchthat ( a, c ) ∈ y and ( c, b ) ∈ z . Hence ( a, c ) ∈ y and ( c, b ) ∈ ( y ∗ x ∧ z ) so( a, b ) ∈ y ( y ∗ x ∧ z ). As this proof uses only intuitionist logic, it is valid inany topos. As the isomorphism is not canonic, it might not lift to a global isomorphism if we areworking internally in a topos. ( Q
3) is sometimes also called the modularity law, which gives rise to aconflict of terminologies. We will prefer the term distributivity law for( Q • ( Q
10) expresses the fact that, as X is a bound of T , X × X has to be asub-quotient of a co-product of a set I of copies of X .Indeed, in this situation, there is a family ( u i , v i ) i ∈ I of partial applicationsfrom X to X × X . A partial application f from X to X can be representedby its graph: the relation R such that ( yRx ) if and only if f ( x ) is definedand y = f ( x ). A relation R on X is the graph of a partial applicationif and only if RR ∗
1. So one has two families of relations on X , alsodenoted ( u i ) and ( v i ), such that for all i , u i u ∗ i v i v ∗ i
1. Therelation W i v i u ∗ i is the union of the image of X in X × X by all the partialmaps ( v i , u i ). So the relation W i v i u ∗ i = ⊤ expresses the fact that thecorresponding map is onto.3.1.2. Definition :
A Set satisfying ( Q and ( Q is a sup-lattice. A Setsatisfying ( Q , ( Q , ( Q and ( Q is called a quantale, (unital if it alsosatisfies ( Q ). We will call a modular quantale, a quantale satisfying all theaxioms from ( Q to ( Q , and a Grothendieck quantale one satisfying all theaxioms from ( Q to ( Q . The term quantale is due to C.J. Mulvey in [22]. The name Grothendieck quan-tale has been given by H.Heymans in [11]. For the term “modular quantale”,our terminology differs slightly from previous work (like [12]), where the axiom( Q
3) is not included in the definition of a modular quantale. The main reasonsfor our choice of terminology is simply that we only want to consider quantalethat arise as relations on objects in a topos and hence satisfy the axiom ( Q Q Q Q T is a topos and B isa bound of T then T can be completely reconstructed from the Grothendieckquantale Q = Rel ( B ). And that every Grothendieck quantale can be written(essentially uniquely) in the form Rel ( B ) for a bound B of a topos T .This result (at least its first part, the second part being a little harder) canactually be proven directly using the following construction: Definition : If Q is a Grothendieck quantale, we will denote by Site ( Q ) thesite whose objects are the q ∈ Q such that q , whose morphisms are givenby: hom( q, q ′ ) = { f ∈ Q | ∧ f ∗ f = q and f f ∗ q ′ } . he composition is given by f ◦ g = f g . The identity morphism of q is q itself.And a Sieve J on an object q is covering if: _ q ′ ,f ∈ J ( q ′ ) f f ∗ = q The fact that for any Grothendieck quantale Site( Q ) is indeed a site is notstraightforward. Apparently it can be checked directly, but this proof is quitelong and is not necessary because one has a more abstract proof, using thefollowing easier proposition, and theorem 3.4.5. Proposition : If Q = Rel ( B ) , for a bound B of a topos T , then Site ( Q ) is thesite of subobjects of B . In particular, it is a site of definition for T . Proof :
We use the same kind of argument as the proof that Q satisfies ( Q ∈ Q = Sub ( B × B ) corresponds to the diagonal sub-object of B × B , anelement q ∈ Q such that q B . Let q and q ′ be two subobjects of B , and f ∈ Q = Rel ( B ) satisfying the two conditions1 ∧ f ∗ f = q and f f ∗ q ′ . The first condition asserts (internally) that all x suchthat ∃ y , yf x is an element of q , and the second condition asserts that if yf x and y ′ f x then y = y ′ and y ∈ q ′ . This is exactly the condition that characterisesthe graph of a function from q to q ′ , hence hom Site( Q ) ( q, q ′ ) is indeed isomorphicto hom T ( q, q ′ ) and as the composition of relations extends the composition offunctions this correspondence is indeed an equivalence of categories.It only remains to check that the topology of Site( Q ) is indeed the canonicaltopology of the topos, but for any collection of map f i : q i → q , the sub-object f i f ∗ i q is exactly the image of f i in q and hence the condition that: _ i f i f ∗ i = q simply asserts that the family is jointly surjective. (cid:3) One of our goal is to provide a way to reconstruct T from Q without using sites.This construction gives an alternative to sites for working with Grothendiecktopos. sl ( T ) of sup-lattices In this subsection we recall the definition and basic properties of the categoriesof sup-lattices of a topos as it is defined and studied in [16]. We will not giveany proofs, but most of them are straightforward and they all can be found in[16]. We checked it, but unfortunately, it does not seems that a proof of this kind had everbeen published.
Definition :
We will denote by sl ( T ) the category of sup-lattices of T . This means that objects of sl ( T ) are objects S of T endowed with a relation ( ) ⊂ S × S , such that ( S, ) is internally a sup-lattice (i.e. is a partial orderrelation and S admits arbitrary supremum). The morphisms are the morphisms f : S → S ′ in T which (internally) preserve supremums (and hence also theorder relation). In all of this sub-section, we will fix the topos T and work internally inside it.We will denote simply by sl the category sl ( T ) and consider objects of T asusual sets.3.2.2. Although we use the term “sup”-latices, it is a classical fact of orderedsets theory that if every subset admits a supremum then every subset alsoadmits an infimum, and hence a sup-lattice is the same thing as an inf-lattice.The term “sup” is here to denote the fact that we are considering sup-preservingmorphisms (which are different from inf-preserving morphism).This duality has a consequence on the category sl : it is endowed with an in-volutive contravariant functor, that we will denote by ( ) ∗ . Indeed if S is asup-lattice then if we define S ∗ as being S endowed with the reverse order re-lations it is again a sup-lattice, and if f is a morphism then we denote by f ∗ its right adjoint (it always exists because f commutes to supremum) and as f ∗ commutes to infimum it is a morphism of sup-lattice for the opposite orderrelations. One has f ∗∗ = f because of the reversing of the order relations, andhence ∗ is an involutive anti-equivalence of categories.This involution allows to compute colimits in the category of sup-lattices: indeedone can easily check that sl has all limits and that they are computed at the levelof the underlying set. As ( ) ∗ transforms co-limits into limits, the computationof a colimit can be brought to the computation of a limit.3.2.3. If X is a set, then P ( X ) = Ω T X (the power object of X ) is a free sup-lattice generated by X , i.e.:hom T ( X, S ) = hom sl ( P ( X ) , S )This adjunction formula turns P into a functor from sets to sl that sends a map f : X → Y to the direct image map P ( f ) : P ( X ) → P ( Y ).3.2.4. Knowing how to construct a free sup-lattice (using P ) and a quotient ofsup-lattice (using the involution ∗ ), one can construct sup-lattices by “generatorsand relations”. More precisely, if I is a set, and R is a family of couples of subsets( r , r ) of I , interpreted as relation of the form: _ x ∈ r x _ y ∈ r y then the sup-lattice presented by the set of generators I and the set of relations R identifies with: 9 V ⊂ I |∀ ( r , r ) ∈ R, ( r ⊂ V ) ⇒ ( r ⊂ V ) } S and S ′ are sup-lattices, then the set of sup-lattice morphisms be-tween S , S ′ is again a sup-lattice, for the point-wise ordering, with supremumcomputed point-wise. This sup-lattice is denoted by [ S, S ′ ].These internal hom objects come with a monoidal structure given by the uni-versal property: hom( M ⊗ N, P ) ≃ hom( M, [ N, P ])Equivalently, the morphisms from M ⊗ N to P , are the applications from M × N to P which are morphisms of sup-lattice in each variable (when fixing the othervariable). We will call such applications bilinear maps from M × N to P . Theexplicit construction of the tensor product is conducted exactly as for modulesover a ring by a construction by generator (the ( m ⊗ n ) for m, n ∈ M × N ) andrelations expressing the notion of bi-linear map.3.2.6. In addition of being a closed monoidal category endowed with an involu-tion, the category sl also satisfies the following interesting properties.Ω T ⊗ N = N [Ω T , N ] = NM ∗ = [ M, Ω T ∗ ]( M ⊗ N ) ∗ = [ M, N ∗ ]In particular, even if we will not use this concept here, this means that sl (en-dowed with all these structures) is a ∗ -autonomous category in the sense of [3],with Ω T ∗ as dualizing object.3.2.7. Let T and E be two toposes, and f a geometric morphism from T to E . Let also S be a sup-lattice in T , then f ∗ ( S ) is a sup-lattice in E : indeed(working internally in E ) if P is a subset of f ∗ ( S ) then by adjunction there is amap from f ∗ ( P ) to S , we can consider the supremum s of the image of this map.As s is a uniquely defined element, it is a global section of S , i.e. an element of f ∗ ( S ). From here one can check that s is also a supremum for P . This definesa functor f ∗ : sl ( T ) → sl ( E ). We also note that f ∗ preserves bi-linear mapsbetween sup-lattices.In the other direction, if S is a sup-lattice in E then f ∗ ( S ) is in general justa pre-order set in T , but one can construct a completion, denoted by f ( S ).In order to do so, we chose any presentation by generators and relations of S (for example, taking all elements and all relations), and then we define f ( S )by generators and relations using the pull-back of the system of generators andrelations chosen for S . At first sight, it is not clear that this definition of f ( S )10oes not depend of the presentation of S , but one can prove an adjunctionformula: hom sl ( T ) ( f ( S ) , T ) ≃ hom sl ( E ) ( S, f ∗ ( T ))which is natural in T . This implies that f ( S ) does not depend on the presenta-tion of S , that it is functorial in S and that f is a left adjoint of f ∗ . We will usethe same technique in the proof of the third point 2 of proposition 3.3.3. Thisresult can actually be seen as a special case of the first two points of proposition3.3.3 applied internally in E to the category C = sl ( T ) with B = Ω T . sl sl one can talk about sl -enrichedcategory. Precisely, a sl -enriched category C is a category such that morphismsets are endowed with an order relation which turns them into sup-lattices andcomposition into a bi-linear map.Here are the two main examples of sl -enriched category we want to consider: Proposition :
Let T be a Grothendieck topos, then sl ( T ) is a sl enrichedcategory. Proof : If S and S ′ are two objects of sl ( T ) and p denotes the structural geometricmorphism from T to the topos of sets, thenhom( S, S ′ ) = p ∗ ([ S, S ′ ])which is a sup-lattice thanks to 3.2.7. The composition is a bi-linear map becauseit is given (through an application of p ∗ ) by an internal bi-linear map:[ S, S ′ ] × [ S ′ , S ′′ ] → [ S, S ′′ ] . (cid:3) A (unital) quantale, as defined in 3.1.2, is exactly a monoid object of sl , i.e. itis a sup-lattice endowed with the structure of a (unital) monoid such that thecomposition law is bi-linear. A right (or left) module over a unital quantale Q ,is a sup-lattice S endowed with a right (or left) action of the underlying monoidof Q such that the corresponding map S × Q → S is bi-linear.The category of right modules over Q (with Q -linear morphisms) is denotedby Mod Q , this is the other important example of sl -enriched category we willconsider.If one thinks of the supremum of a family of elements as a form of addition, asup-lattice enriched category is really close to being an additive category (maybesomething we would like to call a “locally complete characteristic one additivecategory” as our addition is characterised by the properties that x + x = x ).The following two results are in this spirit.11.3.2. From the technique of computation of co-limits in sl explained in 3.2.2one can see that the co-product of a family of objects in sl is isomorphic to theproduct of the same family. This is actually a general well known result: Proposition :
Let C be a sl -enriched category, let ( A i ) i ∈ I be a family of objectsof C and A be an object of C , then the following three conditions are equivalent:1. A is the co-product of the family ( A i ) .2. A is the product of the family ( A i ) .3. There are two families of morphisms f i : A i → A and p i : A → A i suchthat sup i f i ◦ p i = Id A and for all i, j ∈ I : p j ◦ f i = sup { f : A i → A j | i = j and f = Id A i } Moreover, in this situation, the morphisms f i and p i given in . are the naturalmorphisms asserting that A is the (co)-product of the A i . Proof :
Passing from C to C op preserves property 3 . and exchanges properties 1 . and 2 . ,hence it is enough to show that 2 . and 3 . are equivalent.We will start by showing that 3 . ⇒ . .We assume 3 . holds, in particular A is already endowed with maps ( p i ) from A to A i for each i , we have to show that A and the ( p i ) are universal for thisproperty.Let X ∈ C be any object and assume we have a collection of map h i : X → A i .Let h = sup i ( f i ◦ h i ) : X → A . Then for every i : p i ◦ h = sup j p i ◦ f j ◦ h j = sup j sup { f ◦ h j | i = j and f = Id M i } = h i . We also have to show that this map is unique: let h ′ be any other map from X to A such that for every i , p i ◦ h ′ = h i . Then: h = sup f i ◦ h i = (sup f i ◦ p i ) ◦ h ′ = h ′ . Assume now that A is the product of the A i . The maps p i are the structuralmaps, the maps f i are uniquely defined morphisms (using the universal propertyof the product) by the formula given for p j ◦ f i . Hence the formula for p j ◦ f i holds by definition, and the equality sup i f i ◦ p i = Id A because of the relation p j ◦ sup i f i ◦ p i = p j (obtained by the same computation as in the first part of the proof) and theuniqueness in the universal property of the product. (cid:3) it appears, for example, under a slightly different form in [8] 2.214 and 2.223. If we assume the law of excluded middle, or more specifically that the set of indicies I has a decidable equality, then this formula reduces to the more classical: p i ◦ f i = Id A i and p j ◦ f i = 0 if i = j sl -enriched functor willautomatically preserve each product and each co-product (because 3 . is clearlypreserved by any sl -enriched functor).Additionally, one can describe the morphisms between two co-products (or prod-ucts) by something which looks like (infinite) matrix calculus. More precisely,a morphism f from ` j ∈ J A j to ` i ∈ I B i is the same thing as a morphism from ` j ∈ J A j to Q i ∈ I B i , hence it is given by the datum of a morphism f i,j : A j → B i for each i and each j .The composition with a g : ` i ∈ I B i → ` k ∈ K C k , is: g ◦ f = _ i ∈ I g k,i ◦ f i, j In the special case where all the A j and B i are isomorphic to a same object A ,then hom( A, A ) = Q is a quantale and we will denote by M I,J ( Q ) the set ofmorphisms from A ( J ) to A ( I ) , which can be identified with Q I × J .3.3.3. The next result can be thought of as a sl -enriched form of the Mitchellembedding theorem which asserts that every abelian category is a full sub-category of a category of modules over a ring, but restricted to the case wherethere are enough “projective” objects. Proposition :
Let C be a sl -enriched category, A an object of C and Q =hom C ( A, A ) .1. Q is a quantale for composition, and R A : X hom C ( A, X ) induces afunctor from C to Mod Q .2. If C has all co-limits, then R A has a left adjoint denoted T A : Y Y ⊗ Q A .3. If in addition R A commutes to co-equaliser (we will say that A is regularprojective), then T A is fully faithful.4. If in addition C ( A, ) detects isomorphisms (i.e. if f is a map in C suchthat C ( A, f ) is an isomorphism then f is an isomorphism), then R A and T A realize an equivalence of categories between C and Mod Q . Proof :
1. Clear: As C is an sl enriched category, composition are bilinear, hence Q =hom C ( A, A ) is a quantale for composition, the action of Q on hom C ( A, X )by pre-composition is also bi-linear, and for any morphism f : X → Y the induced morphism from hom C ( A, X ) to hom C ( A, Y ) is a Q -linear mor-phism of sup-lattices.2. Let X be a right Q module, then (in Mod Q ) one has a surjection p : ` x ∈ X Q ։ X . Let f , g : K ⇒ ` x ∈ X Q be the kernel pair of p . Let p : ` k ∈ K Q ։ K , and let f = f ◦ p and g = g ◦ p . X is the co-equaliser of the two Q linear maps (for the right action) f and g : ` k ∈ K Q ⇒ ` x ∈ X Q , which correspond to elements of M X,K ( Q ).13et A ( X ) = ` x ∈ X A and A ( K ) = ` k ∈ K A . Thanks to a remark done in3.3.2, maps between A ( K ) and A ( X ) can also be identified with elementsof M X,K ( Q ), hence there are two maps corresponding to f and g from A ( K ) to A ( X ) . We define T A ( X ) to be the co-equaliser of these two maps.One easily checks that for any B ∈ C , there is a canonical (functorial in B ) isomorphism Hom ( T A ( X ) , B ) ≃ Hom ( X, R A ( B )) (they are the sameonce we develop all the inductive limits involved) which implies both theadjunction between T A and R A and the functoriality of T A .3. As T A ( X ) is computed as the co-equaliser of two arrows f, g : A K ⇒ A X such that the co-equalizer of R A ( f ) , R A ( g ) is X , if R A commutes to co-equalizer then one can deduce that R A ( T A ( X )) ≃ X which (thanks to theadjunction) means that T A is fully faithful.4. We already know that X ≃ R A ◦ T A ( X ) (by the unit of the adjunction).Let c X : T A ( R A ( X )) → X be the co-unit of the adjunction then, R A ( c X ) : R A ( T A ( R A ( X ))) → R A ( X ) is a retraction (by general properties of theunit and co-unit) of the unit of the adjunction evalued in R A ( X ) (i.e. thecanonical map R A ( X ) → R A ( T A ( R A ( X )))) but this map is known to bean isomorphism, hence R A ( c X ) is an isomorphism and since R A detectsisomorphism, we proved that c X is an isomorphism. (cid:3) The following theorem can then be seen as a corollary of the previous proposi-tion.3.3.4.
Theorem :
Let T be a Grothendieck topos, and B a bound of T . Then hom T ( B, ) induces (one half of ) an equivalence of categories from sl ( T ) to Mod Q where Q is the quantale Rel ( B ) . This result is essentially the same as the theorem 5.2 of [24].
Proof :
We will prove that with C = sl ( T ) and A = P ( B ), all the hypotheses of the fourpoints of the previous proposition are verified, and Q = Rel ( B ).Note that: hom T ( B, S ) = hom sl ( T ) ( A, S ) . • sl ( T ) has all co-limits (and also all limits) because they can be computedinternally in T . • R A commutes to co-equalizer because of the following formula: R A ( X ) = hom T ( B, X ) ≃ hom T ( B, X ∗ ) ∗ ≃ hom( A, X ∗ ) ∗ ≃ hom( X, A ∗ ) ∗ And the last term clearly commutes to every inductive limit.14 Q is identified with Rel ( B ) through the isomorphism:hom sl ( T ) ( P ( B ) , P ( B )) ≃ hom T ( B, P ( B )) ≃ Sub ( B × B )Internally, this corresponds to the map which sends a morphism f to therelation y ( R f ) x := ‘ x ∈ f ( { y } ) ′ . The fact that composition of morphismscoincides with the composition of relations is checked internally: zR f R g x = ( ∃ y, x ∈ f ( { y } ) and y ∈ g ( { z } )) = xR f ◦ g z. • R A detects isomorphisms:Let f : S → S ′ such that R A ( f ) is an isomorphism. For any subobject U ⊂ B , every map t : U → S can be extended canonically to a map˜ t : B → S by the (internal) formula:˜ t ( x ) = sup { y | x ∈ U and y = t ( x ) } If t is a map from B to S , we can restrict t to U and then extend t | U into˜ t , one has then the formula˜ t ( x ) = sup { y | x ∈ U and y = t ( x ) } = t.δ U where δ U is the element of Q corresponding to the diagonal embedding of U in B × B and the product is the natural right action of Q on hom( B, S ).Finally, as δ U = δ U , hom( U, S ) is identified with hom(
B.S ) .δ U .As R A ( f ) is an isomorphism, all the maps hom( U, f ) for every subobject U of B are isomorphisms, because they are retractions of the map R A ( f ).The object B being a bound of T , the subobjects of B form a generatingfamily and so f is an isomorphism. (cid:3) In the previous section we showed that, for any Grothendieck topos T endowedwith a bound B , the quantale Q = Rel ( B ) already determined the category sl ( T ). We will now show that if we add the operation ( ) ∗ on Q , then we cangive a complete description of T in terms of Q .The theorems 3.4.3 and 3.4.5 are the classical results relating Grothendiecktopos to Grothendieck quantales they can be found explicitly in [11] and [12]and under different forms in [8] and [24]. Actually, because we know that Q is of the form Rel ( B ), the ∗ operation is fully determinedby the underlying quantale. This comes from the property (Q10) together with this lemma:the condition f = g ∗ and gg ∗ ∃ u uf = f , gu = g , gf u fg . This lemma is proved using internal logic. Lemma :
Let X be an object of T , and Y be a sub-quotient of X , then therelation on X defined by xRy = “ x and y both have an image in Y and these coincide ”is symmetric ( R ∗ = R ) and transitive ( R R ) . This induces a correspondencebetween sub-quotients of X and relations R on X such that R ∗ = R and R R . Proof :
The symmetry and transitivity of the relation are clear. Y is fully determinedby R : it is the quotient of U = { x | xRx } by R (which is an equivalence relationon U ). Conversely, let R be any symmetric and transitive relation on X . Let U = { x | xRx } , R induces an equivalence relation on U , and we have xRy ⇒ xRx .Hence, xRy ⇔ ( x ∈ U ) ∧ ( xRy ) ∧ ( y ∈ U ), i.e. R is indeed the relation inducesby the sub-quotient U//R . (cid:3) Lemma :
In the situation of the previous lemma, one actually has R = R , and the map which sends a sub-object of Y to its pullback in X identifies P ( Y ) with R ( P ( X )) (where R denotes the endomorphism of sup-lattice of P ( X ) corresponding to the relation R ). Proof :
Indeed, if ( xRy ) then ( xRx ) and ( xRy ) hence ( xR y ), this proves that R R ,and hence R = R . Let P be a subset of X , R ( P ) = { x ∈ X |∃ z ∈ P, xRz } . So P = R ( P ) if and only if P is included in U = { x | xRx } and saturated for theequivalence relation induced by R on U . These are exactly the subsets whichare pull-backs of subsets of Y . (cid:3) Theorem :
The category
Rel ( T ) whose objects are the objects of T andmorphisms from X to Y are sub-objects of Y × X (the composition being givenby the composition of relations) is equivalent to the following category Proj ( Q ) : • The objects are the couples ( I, P ) where I is a set, and P is a matrix in M I,I ( Q ) such that P = P and P ∗ = P where ( P ∗ ) i,j = ( P j,i ) ∗ . • The morphisms from ( J, P ′ ) to ( I, P ) are the matrices M ∈ M I,J ( Q ) suchthat P.M = M and M.P ′ = M (the composition being the product ofmatrices). nder this equivalence, the opposite of a relation corresponds to the “trans-conjugation” of a matrix: ( M ∗ ) i,j = ( M j,i ) ∗ . Before proving this theorem We will need one more simple lemma, which isactually the last point of the theorem:
Lemma :
Let R be a sub-object of ( ` i ∈ I B ) × ( ` j ∈ J B ) corresponding to amorphism R : P ( B ) I → P ( B ) J represented by a matrix: ( R i,j ) i ∈ I,j ∈ J , then theopposite relation corresponds to the trans-conjugate matrix ( R ∗ ) j,i = ( R i,j ) ∗ Proof :
This can be checked internally: Since R i,j corresponds to the intersection of R with the inclusion ( f i , f j ) of B × B in ( ` i ∈ I B ) × ( ` j ∈ J B ), taking the oppositerelation will reverse R i,j and exchange the indices. this concludes the proof ofthe lemma. (cid:3) We now prove theorem 3.4.3:
Proof :
In order to prove the equivalence of
Proj ( Q ) and Rel ( T ), we will consider athird category C , the full sub-category of sl ( T ) of sup-lattices which are of theform P ( X ) for X an object of T , and show that both Proj ( Q ) and Rel ( T ) areequivalent to C .The association X → P ( X ) is (one half of) an equivalence from Rel ( T ) to C .Indeed, it is essentially surjective by definition of C , and we have already men-tioned that morphisms between power objects are the same thing as relations,so it is also fully faithful.The association ( I, P ) → P ( P ( B ) I ) is (one half) of an equivalence from Proj ( Q )to C . Indeed:as B is a bound of T , any object X of T is a sub-quotient of some ` i ∈ I B ,hence there is an endomorphism F of P ( ` i ∈ I B ) = P ( B ) I such that F = F , F ∗ = F , and P ( X ) = F ( P ( B ) I ). By the result of the previous section, such anendomorphism corresponds exactly to a matrix P such that ( I, P ) is indeed anobject of our category. So this functor is full and well defined (at least on theobject). Now a morphism from P ′ ( P ( B ) J ) to P ( P ( B ) I ) is exactly the data ofa matrix M such that P.M = M and M.P ′ = M . This concludes the proof ofthe equivalences. The last point of the theorem being proved by the lemma. (cid:3) Corollary :
The topos T is equivalent to the (non full) subcategory of Proj ( Q ) , with all objects and with morphisms from ( J, P ′ ) to ( I, P ) only thematrices M which satisfy the additional condition: such that M M ∗ P and P ′ M ∗ M . roof : This two additional conditions indeed characterise functional relations amongarbitrary relations, and in a topos functional relations are in correspondencewith morphisms. (cid:3)
Theorem :
For every Grothendieck quantale Q , there exists a topos T and a bound B of T such that Q = Rel ( B ) . Of course, from the previous theorem, such a topos is unique.
Proof :
One could use the construction of Site( Q ) given in the introduction, but theproof that this is indeed a site and that it gives back Q = Rel ( B ) is long andnot really illuminating. Instead, we will use results from the theory of allegories(see [8], or [14]A.3) which is closely connected to what we are doing here:In the language of [8] a modular quantale Q is a one object locally completedistributive allegory, and Proj ( Q ) is the systemic completion of Q . The result[8]2 .
434 proves that
Proj ( Q ) is a power allegory and [8]2 .
226 proves that it is hasa unit. So in order to apply [8]2 .
414 and conclude that
Proj ( Q ) is the categoryof relations on an elementary topos, we need to proove that it is ‘tabular’.Using [8]2 . for each set X the maximal matrixof M X,X ( Q ) can be written F G ∗ for F, G ∈ M X,Y ( Q ) with F F ∗ Id Y and GG ∗ Id Y . But ( Q
10) is exactly the assertion that this is true when X is asingleton, and the general case follows easily from ( Q
10) by taking Y = X × I .The elementary topos we obtain in this way has arbitrary co-products and isbounded, hence it is a Grothendieck topos.Finally, if B is the object of Proj ( Q ) represented by the set X = {∗} and P = 1,then Rel ( B ) = Q and this concludes the proof. (cid:3) Proj ( Q ) in term of Q -Set inspired from the notion of L -sets, when L isa locale. Our aim is both to provide a formalism suitable for computation andto show that Proj ( Q ) is exactly a non-commutative generalisation of L -set. Wedo not know if this formulation has already been presented somewhere or not. Definition : • A Q -Set is a set X endowed with an application [ ≈ ] : X × X → Q suchthat: ( S ∀ x, y ∈ X, [ x ≈ y ] = [ y ≈ x ] ∗ . ( S ∀ x, y, z ∈ X, [ x ≈ y ] [ y ≈ z ] [ x ≈ z ] . The reader should note that [8] uses a reverse composition order for morphisms in category,whereas we use the standard composition order. This explains why the formula we give isdifferent from the one given in the reference. A Q -relation R from X to Y (two Q -sets) is a map: Y × X → Q ( y, x ) [ yRx ] such that: ( R
1) [ y ≈ y ′ ] [ y ′ Rx ] [ yRx ] with equality whenever y = y ′ ( R
2) [ yRx ′ ] [ x ′ ≈ x ] [ yRx ] with equality whenever x = x ′ . • A Q -function from X to Y is a Q -relation: Y × X → Q ( y, x ) [ y ≈ f ( x )] which (in addition to ( R and ( R ) satisfies: ( F
1) [ y ≈ f ( x )] [ y ′ ≈ f ( x )] ∗ [ y ≈ y ′ ]( F
2) [ x ≈ x ] W y [ y ≈ f ( x )] ∗ [ y ≈ f ( x )] • Q -relations and Q -functions can be composed by the formula: [ zRQx ] = _ y [ zRy ] [ yQx ][ z ≈ f ◦ g ( x )] = _ y [ z ≈ f ( y )] [ y ≈ g ( x )] • The opposite of a Q -relation is given by: [ xR ∗ y ] = [ yRx ] ∗ Proposition :
Consider the following modification of the axioms:(S2’) [ x ≈ y ] = W t [ x ≈ t ] [ t ≈ y ] .(R1’) W y ′ [ y ≈ y ′ ] [ y ′ Rx ] = [ yRx ] (R2’) W x ′ [ yRx ′ ] [ x ′ ≈ x ] = [ yRx ] (F2’) [ x ≈ x ′ ] W y [ y ≈ f ( x )] ∗ [ y ≈ f ( x ′ )] Then assuming ( S holds, ( S and ( S ′ ) are equivalent.And assuming X and Y are Q -sets, ( R is equivalent to ( R ′ ) , ( R is equiva-lent to ( R ′ ) and assuming additionally ( R then ( F is equivalent to ( F ′ ) .In particular Q -Sets are exactly the same as objects of Proj ( Q ) , and Q -relationsand Q -functions correspond respectively to morphisms in Proj ( Q ) , and mor-phisms which are sent to functional relations by the equivalence of 3.4.3.
19e will need the following lemma:
Lemma :
In any Q -set, one has [ x ≈ y ] [ y ≈ y ] = [ x ≈ y ][ x ≈ x ] [ x ≈ y ] = [ x ≈ y ] Proof :
Indeed (for the second equality), for any q ∈ Q element of a modular quantaleone has: q (1 .q ∧ q ) (1 ∧ qq ∗ ) q qq ∗ q So: [ x ≈ y ] [ x ≈ y ] [ y ≈ x ] [ x ≈ y ] [ x ≈ x ] [ x ≈ y ]The reverse inequality being a consequence of ( S (cid:3) We now prove the theorem:
Proof : • Clearly, ( S ′ ) implies ( S S
2) and ( S
1) hold, hence that X is a Q -set. One can apply the lemma and one has:[ x ≈ y ] _ [ x ≈ t ] [ t ≈ y ]by taking t = x or t = y . The reverse inequality follows from ( S • ( R ⇒ ( R ′ ) is clear because of the equality case. Assuming ( R ′ ) onehas immediately [ y ≈ y ′ ] [ y ′ Rx ] [ yRx ]. So we just have to prove that[ y ≈ y ] [ yRx ] = [ yRx ]. But:[ y ≈ y ] [ yRx ] = _ y ′ [ y ≈ y ] [ y ≈ y ′ ] [ y ′ Rx ] = _ y ′ [ y ≈ y ′ ] [ y ′ Rx ] = [ yRx ]The equivalence of ( R
2) and ( R ′ ) is proved the same way. • ( F
2) is a special case of ( F ′ ). Assume ( F
2) then:[ x ≈ x ′ ] = [ x ≈ x ] [ x ≈ x ′ ] W y [ y ≈ f ( x )] ∗ [ y ≈ f ( x )] [ x ≈ x ′ ] W y [ y ≈ f ( x )] ∗ [ y ≈ f ( x ′ )]20he fact that Q -Sets are exactly the same as objects of Proj ( Q ), and Q -relationsand Q -functions correspond respectively to morphisms in Proj ( Q ), and mor-phisms which are sent to functional relations by the equivalence of 3.4.3 is nowimmediate: If we replace the original axioms by these modified version, and ifwe interpret [ x ≈ y ],[ xRy ] and [ x ≈ f ( y )] as matrix coefficients then the condi-tions imposed on them are exactly those for being objects and morphisms of Proj ( Q ). (cid:3) In the previous section we constructed a topos Q -Sets from a Grothendieckquantale Q . In this section we describe the theory classified by this topos, thatis study the morphisms from an arbitrary topos T to the topos of Q -sets. Thisalso explains in which sense the equivalence between Grothendieck quantalesand Grothendieck toposes is functorial.3.5.1. Definition :
A morphism of modular quantales is an application f : Q → Q ′ between two modular quantales such that: • f commutes to arbitrary supremum (in particular it preserves the smallestelements) • f commutes to finite intersections (in particular it preserves the top ele-ment ⊤ ). • f is a morphism of unitary monoids (in particular it preserves ). • f commutes to the involution.A Relational representation of a modular quantale Q is the datum of an inhabitedset X endowed with a modular quantale morphism π from Q to Rel ( X ) . Amorphism of relational representations is a map from X to X ′ such that foreach q ∈ Q if ( x, y ) ∈ π ( q ) then ( f ( x ) , f ( y )) ∈ π ′ ( q ) . Theorem :
The topos of Q -sets classifies the relational representationsof Q , the universal representation being given by the action of Q on the bound B (which corresponds to the Q -set {∗} with [ ∗ ≈ ∗ ] = 1 ). In other words, if E is any topos then there is an equivalence of categories between the geometricmorphisms from E to Q -sets, and the relational representations of Q inside E .And this equivalence is given by f f ∗ ( B ) . This theorem is essentially the same as theorem 2.9 of [24].
Proof :
21s a bound, the object B has to be in particular inhabited, hence it is indeeda relational representation. So any geometric morphism from E to Q -sets doesyield a relational representation of Q on f ∗ ( B ) and any natural transformationgives a morphism of representation. So the functor mentioned in the theoremindeed exists.If f is a geometric morphism from E to Q -sets, then f ∗ induces a sl -enrichedfunctor from Proj ( Q ) to Rel ( E ).Because Proj ( Q ) is generated by B under co-product and splitting of projection,and since by 3.3.2 arbitrary co-products (as well as spliting of projection) areuniversal co-limits in sl -enriched category, any relational representation ( X, π )of Q in a topos E extends in a uniquely defined sl -enriched functor from Proj ( Q )to Rel ( E ): one havehas to send the couple ( I, P ) on π ( P ) ` i ∈ I X , and anymorphism in Proj ( Q ) is a matrix which is sent to the matrices “ π ( M )” defininga relation in E .Moreover if f and g are two geometric morphisms from E to Q -sets, then mor-phisms between the relational representation they induce uniquely extend tonatural transformations between f ∗ and g ∗ .So we just have to prove that if ( X, π ) is a relational representation of Q , thenthe induced functor v from Proj ( Q ) to Rel ( E ) comes from a geometric morphismfrom E to Q -sets. • As π commutes with ∗ , so does v . Hence v preserves functional relationsand induces a functor from Q -sets to E . • The terminal object of Q -sets is the quotient of B by its maximal relation,and since π preserves the maximal relation, the terminal object of Q -setsis sent to the quotient of X by its maximal relation, which is the terminalobject of E because X is inhabited. So v preserves the terminal object. • Let
P XY S fg be a pull back diagram in Q -sets, then P can be identified with the relation f ∗ g on X × Y , indeed internally f ∗ g is the relation { ( x, y | f ( x ) = g ( y ) } hence it is the fiber product X × S Y . So v preserves pull-back. As v preserves the terminal object, it preserves all limit. • In a topos, a collection of maps f i : A i → A is a covering if and onlyif 1 A W i ∈ I f i f ∗ i as v preserves all the structures involved, v preservescovering families.All these properties together imply that v is indeed the f ∗ functor of a geometricmorphism and conclude the proof. (cid:3) .6 Internally bi-linear maps between Q -modules The category sl ( T ) is endowed with a tensor product. In the case where T = Sh ( L ) is the topos of sheaves on a frame L , one can see that through theidentification of sl ( T ) with Mod L this tensor product corresponds to the naturaltensor product over L , which as in the case of commutative algebras is defined bythe universal property: the maps from M ⊗ L N to P are the bi-linear morphismsfrom M × L to P such that for all l ∈ L , f ( m, n.l ) = f ( m.l, n ) = f ( m, n ) .l .The main result of this section is that, in the general case, even if the ten-sor product of two Q -modules can be difficult to compute explicitly, the set Bil T ( M × N, P ) of internal bilinear map from M × N to P has a strikinglysimple description in term of the corresponding right Q -modules. This leadsin particular to a simple description of the category of internal locales of T interms of a Grothendieck quantale representing T . More precisely:3.6.1. Definition : If A , B and C are three right modules over a Grothendieckquantale Q , we say that a map f : A × B → C is Q -bilinear if it is a bi-linearmorphism of sup-lattices and if it satisfies the following three conditions:1. ˜ f ( aq, b ) ˜ f ( a, bq ∗ ) q ˜ f ( a, bq ) ˜ f ( aq ∗ , b ) q ˜ f ( a, b ) .q ˜ f ( aq, bq ) .We will denote by Bil Q ( A × B, C ) the set of Q -bilinear maps. Bil Q ( A × B, C ) is a sup-lattice for the pointwise ordering (with supremum com-puted pointwise), and it is an sl -enriched functor in each of the three variables(contravariant in the first variables) with the functoriality given by composition.The main result of this section (theorem 3.6.3) is that this functor is isomorphicto the functor of internal bilinear maps.3.6.2. Let M , N and P be internal sup-lattices in T , let f M , e N and e P be thecorresponding right Q -modules (i.e. ˜ M = hom T ( B, M )). Let f be a bilinearmorphism from M × N to P .Then one can define a map ˜ f from f M × e N to e P by the (internal) formula:˜ f ( m, n ) := b f ( m ( b ) , n ( b )) ∈ P With m and n elements of f M and e N , that is, maps from B to M and N , then˜ f ( m, n ) is indeed an element of e P = hom T ( B, P ). Proposition :
The map ˜ f is a Q -bilinear morphism map in the sense of 3.6.1.Moreover the construction f → ˜ f defines a morphism of sl -enriched functors: µ ( M, N, P ) :
Bil T ( M × N, P ) → Bil Q ( f M × e N , e P )23 roof : The (sup-lattice) bilinearity is immediate: supremum in f M , e N and e P corre-sponds to pointwise internal supremum hence the bilinearity of ˜ f simply comesfrom the internal bilinearity of f .Recall that by definition one has internally for any m ∈ f M and q ∈ Q = Rel ( B ): m.q ( b ) = _ ( b ′ ,b ) ∈ q m ( b ′ ).All three properties defining Q -bilinearity are then easily checked internally:1. ˜ f ( mq, n )( b ) = _ ( b ′ ,b ) ∈ q f ( m ( b ′ ) , n ( b ))Whereas:˜ f ( m, nq ∗ ) q ( b ) = _ ( b ′ ,b ) ∈ q ˜ f ( m, nq ∗ )( b ′ ) = _ ( b ′ ,b ) ∈ q, ( b ′ ,b ′′ ) inq f ( m ( b ′ ) , n ( b ′′ ))So the first term corresponds to the restriction of the union to b = b ′′ ofthe second and is indeed smaller.2. Same proof.3. h ˜ f ( m, n ) q i ( b ) = _ ( b ′ ,b ) ∈ q f ( m ( b ′ ) , n ( b ′ ))Whereas˜ f ( mq, nq )( b ) = f ( mq ( b ) , nq ( b )) = _ ( b ′ ,b ) ∈ q, ( b ′′ ,b ) ∈ q f ( m ( b ′′ ) , n ( b ′ ))So the first term corresponds to the restriction of the union to b ′ = b ′′ ofthe second and is indeed smaller.Also f ˜ f commutes to supremum, because if one takes f i a (external) familyof internal bilinear maps, m ∈ f M and n ∈ e N then (internally) for any b ∈ B : ] _ i f i ! ( m, n )( b ) = _ i f i ( m ( b ) , n ( b )) = _ i ˜ f i ( m, n ) ! ( b )And the functoriality is immediate: ˜ f ( m, g ( n )) := b f ( m ( b ) , g ( n ( b )) is in-deed the map attached to f ( , g ( )) and ˜ g ( ˜ f ( m, n ) := b g ( f ( m ( b ) , n ( b )) = ] g ◦ f ( m, n ). (cid:3) Theorem :
The construction f ˜ f from 3.6.2, defines an isomorphismof sl enriched functors: µ : Bil T ( M × N, P ) ≃ Bil Q ( f M × e N , e P ) . The functoriality of the association and the fact that it commutes to supremumhave already been mentioned, so it only remains to prove that it is a bijection.The proof of this theorem will be completed in 3.6.8 after proving a few lemmas.3.6.4.
Lemma :
The association f ˜ f of 3.6.2 is injective. Proof :
Let f and g be two internal bi-linear maps from M × N to P such that ˜ f = ˜ g .This means that for each map ( m, n ) : B → M × N one has internally: ∀ b ∈ B, f ( m ( b ) , n ( b )) = g ( m ( b ) , n ( b ))i.e.: f ◦ ( m, n ) = g ◦ ( m, n ) . But we already explained in the last part of the proof of 3.3.4 that any mapfrom a sub-object U of B to a sup-lattice can be extended (canonically) to amap on all of B . As B is a bound, maps from sub-objects of B can cover M × N and by the extension arguments, maps from B cover M × N , so we can concludefrom the previous formula that f = g . (cid:3) Lemma :
Let h : Q × Q → ˜ P ∈ Bil Q ( Q × Q, ˜ P ) where Q is endowedwith its right action on itself. Then: • Let c ∈ ˜ P and a, b ∈ Q such that a.a ∗ , b.b ∗ . If one has c h ( a, b ) then for all x, y ∈ Q : c. ( a ∗ .x ∧ b ∗ .y ) h ( x, y ) • For all x, y one has: h ( x, y ) = _ aa ∗ ,bb ∗ h ( a, b )( a ∗ x ∧ b ∗ y )25 roof : For the first point:Let t = ( a ∗ x ∧ b ∗ .y ) ∈ Q . Then one has: a.t ( aa ∗ x ∧ ab ∗ y ) aa ∗ x xb.t ( ba ∗ x ∧ bb ∗ y ) bb ∗ y y So: c.t h ( a, b ) .t h ( a.t, b.t ) h ( x, y )For the second point:Let h ′ ( x, y ) = W aa ∗ ,bb ∗ h ( a, b )( a ∗ x ∧ b ∗ y ).Clearly, h ′ is also in Bil Q ( Q × Q, ˜ P ).The first point shows that h ′ h . For the reverse inequality we will proceed inseveral steps: • If ( xx ∗ , ( yy ∗ D ( x ) = 1 ∧ x ∗ x and D ( y ) = 1 ∧ y ∗ y . Wenote that for elements smaller than 1, the involution is the identity andcomposition and intersection coincide (these can be proved by applyingthe modularity law, or by using theorem 3.4.5 as a black box and checkingit internally in the corresponding topos). So x.D ( x ) = x (1 ∧ x ∗ x ) > ( x ∧ x ) = x hence x.D ( x ) = x and y.D ( y ) = y also, x ∗ x ∧ y ∗ y > D ( x ) ∧ D ( y ) = D ( x ) .D ( y )Aslo for any e h ( xe, y ) h ( x, ye ) e h ( x, y ) e h ( xe, ye ) h ( xe, y )hence h ( xe, y ) = h ( x, ye ) = h ( x, y ) e Finally: h ( x, y )( x ∗ x ∧ y ∗ y ) > h ( x, y ) D ( x ) D ( y ) = h ( xD ( x ) , yD ( y )) = h ( x, y )So h ′ ( x, y ) > h ( x, y ) • We will now assume that x is arbitrary and y is simple. As Q is a Gro-thendieck quantale, x can be written as a supremum of elements of theform uv ∗ with u and v simple. So, by bi-linearity of h , it is enough toprove that h ( x, y ) h ′ ( x, y ) when x is of the form uv ∗ . In this case: h ( uv ∗ , y ) h ( u, yv ) v ∗ = h ′ ( u, yv ) v ∗ h ′ ( uv ∗ , yvv ∗ ) h ′ ( uv ∗ , y )26 If both x and y are now arbitrary, then the same technique allows ones toconclude. (cid:3) Corollary :
Theorem 3.6.3 holds whenever M = N = P B (ie. µ ( P ( B ) , P ( B ) , ˜ T ) is an isomorphism) Proof :
Injectivity is known by lemma 3.6.4. So we just have to prove surjectivity. Let h be any bi-linear map from Q × Q = ˜ M × ˜ N → ˜ P satisfying our three conditions.Then, by lemma 3.6.5, one has that: h ( x, y ) = _ aa ∗ ,bb ∗ h ( a, b )( a ∗ x ∧ b ∗ y ) . One can see that maps of the form ( x, y ) t ( ux ∧ vy ) with u, v ∈ Q and t ∈ ˜ T correspond to the internal bi-linear map which sends ( p, q ) ∈ P ( B ) × P ( B ) to t ( u ( p ) ∧ v ( q ) where u and v are seen as endomorphisms of P ( B ) and t as amorphism from P ( B ) to T . Indeed if g ( p, q ) = t ( u ( p ) ∧ v ( q )) then˜ g ( x, y ) = t ( ux ( b ) ∧ vy ( b )) = _ ( b ′ ,b ) ∈ ( ux ∧ vy ) t ( b ′ ) = t ( ux ∧ vy )Hence h can be written as a supremum of maps coming from internal bi-linearmaps, but as f ˜ f commutes to arbitrary supremum, this shows that h doesalso come from a bilinear map. (cid:3) Proposition :
Assume that µ ( P ( B ) , M, P ) is an isomorphism for some M and P in sl ( T ) . Then µ ( N, M, P ) is an isomorphism for any N in sl ( T ) . Proof :
We already know that µ is injective. Hence it remains to show the surjectivity.Let M and P such that µ ( P ( B ) , M, P ) is an isomorphism. the internal sup-lattice [ M, P ] then corresponds to the right Q -module:hom T ( B, [ M, P ]) = hom sl ( T ) ( P ( B ) , [ M, P ]) =
Bil T ( P ( B ) × M, P )= Bil Q ( Q × ˜ M , ˜ P )where the action of Q on the last term is given by the left action of Q on itself.Now let g ∈ Bil Q ( ˜ N × ˜ M , ˜ P ). For any n ∈ ˜ N , the map:27 n : ( q, m ) g ( nq, m )is an element of Bil Q ( Q × ˜ M , ˜ P ). The map ( n g n ) is a morphism of right Q -modules and hence internally corresponds to a map f : N → [ M, P ] which inturn corresponds to a map f ∈ Bil T ( N × M, P ). Finally ˜ f = g because for any n ∈ ˜ N , ( q, m ) ˜ f ( nq, m ) is by construction of ˜ f the map g n ∈ Bil Q ( Q × ˜ M , ˜ P )and hence ˜ f agrees with g . This concludes the proof. (cid:3) T ∈ sl ( T ), µ ( P ( B ) , P ( B ) , T ) is an isomorphism. Hence by 3.6.7, µ ( N, P ( B ) , T )is an isomorphism for each N and T , as one can freely exchange the first twovariables, µ ( P ( B ) , N, T ) is also an isomorphism, and a second application of3.6.7 allows one to conclude.3.6.9. Corollary :
The category of internal locales of T (equivalently, thecategory of toposes which are localic over T ) is equivalent to the category oflocales L endowed with a right action of Q such that: • As a sup-lattice L is a right Q -module. • One has the modularity condition: ∀ m, n ∈ L , ∀ q ∈ Q, m ∧ nq ( mq ∗ ∧ n ) q We will call such an action a modular action. (the morphisms of this categorybeing the Q -equivariant morphisms of locale). Of course, the equivalence is given by the usual functor
L 7→ hom T ( B, L ). Proof :
Let L be a locale in T then ˜ L is indeed endowed with an operation ˜ ∧ which is Q -bilinear. Also: ˜ ∧ ( m, n )( b ) = m ( b ) ∧ n ( b )is the intersection of hom T ( B, L ), hence L is indeed a locale and intersection isindeed Q -bilinear.Conversely, if L is a locale endowed with a modular action of Q , then theoperation ∧ is Q -bilinear (the second axiom comes from the symmetry, andthe third axiom because multiplication by q is order preserving). Hence L corresponds to an internal sup-lattice L equipped with a bi-linear map m comingfrom ∧ . This bi-linear map has to be the intersection map of L because both m and ∧ L induce the same map when we externalise it by watching the morphismsfrom B to L × L , and the proof of the injectivity of the externalisation processdone in 3.6.4 works without assuming that the map f is bilinear. (cid:3) Bil T ( M × N, P ) it is possible to obtain anexplicit description of both the tensor product and the internal hom objects interms of the corresponding Q -modules: for the hom-object it has been done inthe proof of 3.6.7 and for the tensor product, the description of Bil T ( M × N, P )translates into a presentation by generators and relations of ^ M ⊗ N . In orderto completely handle the monoidal structure of sl ( T ) in terms of the categoryof Q -modules it remains to understand what g M ∗ is.An element of g M ∗ is an application from B to M ∗ , hence it is the same thing asan application from B to M (i.e. an element of f M ) but with the reverse orderrelation, hence as a sup-lattice g M ∗ = f M ∗ . A simple computation shows that q ∈ Q acts on f M ∗ by the adjoint of the action of q ∗ on M .We also note that the notion of Q -bilinear map makes sense when Q is only amodular quantale, that one can define a “tensor product” M ⊗ Q N universalfor Q -bilinear maps from M × N . But in general this tensor product fails to beassociative. Rel ( X ) with X anobject of a topos.3.7.2. Let Q be any modular quantale, we can try to consider the classifyingtopos of the theory of relational representations of Q and hope that the quantaleof relations on the universal representation is isomorphic to Q . Unfortunatelythis is not true in general as the following example shows:Let n be any integer >
2, and G = S n be the permutation group and considerthe set X n = { , . . . , n } endowed with its natural action of S n as a S n -set. Theonly relations on X n (in S n -sets) are: ∅ , , ⊤ , ∆ where ∆ is the complementaryrelation of 1.As n > = ⊤ . Let Q be the quantale of relations on X in S n -sets. Q does not depend on n . A relational representation of Q (in a topos)is just an object S which has a decidable equality (the diagonal sub-object iscomplemented) and at least three distinct elements. The universal model U ofthis theory has more relations on it than just the four elements of Q :Indeed, let P = { ( x, y ) ∈ U |∃ x , . . . , x ∈ U pairwise disjoint in U } . then the pull-back of P in the representation X is ∅ whereas the pull-back of P in the representation X n for n > ⊤ hence P cannot be any of the objectsof Q in the universal representation. 29 .8 Towards a convolution C ∗ algebra attached to a quan-tale In the special case where the topos T is an ´etendue, ie the topos of equivariantsheaves over an ´etale (localic) groupoid G = ( G , G ), the quantale associatedto the bound B such that T B = G is the set of open subsets of G , and thecomposition law is given by the direct image of open subsets in the compositionlaw of G .With this fact and the usual construction of a C ∗ -algebra from a groupoid (see[23] ) in mind it is natural to try to construct a C ∗ -algebra from a quantale bydefining a convolution product over a subset of continuous functions “over Q ”(or over open subspaces of Q ). Indeed we can view Q as a locale by forgettingits composition law and involution, then use the involution to get an involutionon continuous functions and hope that the composition law on Q will allowus to construct a convolution product on continuous functions.In order to perform this construction the general idea is the following: a contin-uous function on Q is the same thing as a function from B × B to the sheaf ofreal or complex numbers. Hence it can be thought of, internally, as an infinitematrix whose rows and columns are indexed by B , and we can use the multipli-cation of matrices to define the convolution product. If the coefficients are allpositive and we allow infinite coefficients the product should be always defined.There are two difficulties that arise when we try to define the matrix productinternally: • Matrix multiplication requires a summation indexed by the elements of B . It can not be done if we don’t assume that B has a decidable equality.The reason for this is that without this assumption, when we look atpartial sums f ( b ) + · · · + f ( b n ), for ( b , · · · b n ) elements of B we cannotsay whether the elements b i are distinct, hence we cannot assert that wehave not counted some value of f twice. • The sum of an infinite number of terms will in full generality be definedas the supremum of all the possible finite sums. In general the object R T of continuous real numbers (i.e. two sided Dedekind cut) does notalways have supremum. In order to define a supremum we need to replacethe usual “continuous” real numbers by the lower semi-continuous realnumbers (one-sided Dedekind cut), so the result of the convolution will ingeneral be a lower semi-continuous function.3.8.1. We now move to the precise definition: Proposition :
Let B be an object of a topos T with a decidable equality. Let(internally) f and g be functions from B × B to R lsc + T . Then we can define afunction ( f ∗ g ) from B × B to R lsc + T such that for all q ∈ Q T , one has q < ( f ∗ g )( b, b ′ ) This will not be the case in full generality. f and only if: ∃ n ∈ N T , b , · · · , b n ∈ B such that: ∀ i = j, B i = B j and q < n X i =1 f ( b, b i ) g ( b i , b ′ ) where q < n X i =1 f ( b, b i ) g ( b i , b ′ ) naturally means: ∃ u , · · · , u n , v , · · · v n ∈ Q T , such that u i < f ( b, n ) , v i < g ( b i , b ′ ) and q < P ni =1 u i v i . Proof :
All we have to do is check that the set X of q such that: ∃ n ∈ N T , b , · · · , b n ∈ B, such that ∀ i = j, B i = B j and q < n X i =1 f ( b, b i ) g ( b i , b ′ )is indeed a positive one-sided Dedekind cut.It is positive because, by taking n = 0 all negative q are in X . If q ′ < q and q ∈ X then clearly q ′ ∈ X . If q ∈ X , then q < P f ( b, b i ) g ( b i , b ′ ) so there exists q ′ > q such that q ′ < P f ( b, b i ) g ( b i , b ′ ) hence q ′ ∈ X . This concludes the proof. (cid:3) Proposition :
The convolution product defined is associative and thecharacteristic function of the unit of Q is a unit. Proof :
This an immediate consequence of the fact that internally, the composition ofmatrices is associative and that the identity matrix is a unit for the compositionof matrices. All we need is a constructive version of Fubini’s theorem for sums(indexed by decidable sets) of positive lower semi-continuous real numbers. Theusual proof can easily been made constructive, or we can apply the generalFubini’s theorem proved in [26]. (cid:3)
This is interpreted externally as the construction of a convolution product onthe set of lower semi-continuous functions on the underlying space of Q . Thiscorresponds exactly to the construction of the convolution algebra of an ´etalegroupoid (see for example [23] for this construction).It should be possible to obtain something similar to the more general construc-tion of the convolution algebra of a locally compact groupoid endowed with aHaar system, by replacing the bound B by an internal space endowed with an In this reference, ´etale groupoids are called r-discrete groupoids. Q is discrete. In this section we focus on a really special case of the theory explained in theprevious section: when the underlying space of the quantale Q (that is, thespace obtained by forgetting the product on Q and seeing it as a locale) is adiscrete topological space. This corresponds to the study of atomic toposes (see[14] C.3.5 for the theory of atomic toposes).In this situation, the convolution product constructed at the end of the previoussection is easier to understand, and among other things we will explain a simplenecessary and sufficient condition for this convolution product to be defined andinteresting on continuous functions. In this special case two additional featuresappear: a canonical ‘time evolution’ on the C ∗ algebra obtained this way, andwe will observe that when the convolution product is well defined it restrictsinto a product on finitely supported integer valued functions, which gives riseto an algebra over Z with possibly interesting arithmetic and combinatorialproperties. This algebra can be interpreted as a subalgebra of the algebra ofendomorphisms of the free Z T modules of base B .All the constructions will be made constructively, but at some point we willneed to take the assumption that the underlying space of Q is decidable. An object X of a topos is said to be an atom if Sub ( X ) ≃ Ω . A direct imageof an atom by a morphism is again an atom. A topos is said to be atomic if itsatisfies one of the following equivalent properties: • The atoms form a generating family. • For every object X ∈ |T | , Sub ( X ) is an atomic locale (i.e. of the form P ( S ) form some S . • T is the topos of sheaves over an atomic site (i.e. a site such that thecovering sieves are exactly the inhabited sieves). Assuming classical logic in the base topos, this means that X is non-empty and has nonon trivial sub-object. .4.2.1. Proposition :
Let T be a topos and B be a bound of T , then thefollowing conditions are equivalent:1. T is atomic over sets.2. Q = Rel ( B ) is (as a locale) atomic, i.e. its underlying poset is of the form P ( X ) for some set X .3. Z ( Q ) = Sub ( B ) = { q ∈ Q | } is atomic. Proof : . ⇒ . is clear, because in an atomic topos all the lattices Sub ( X ) are atomic.The implication 2 . ⇒ . is also clear. We will prove 3 . ⇒ . : Every object of T can be covered by subobjects of B , so if every subobject of B can be coveredby atoms (which is the assumption in 3 . ) every object of T can be covered byatoms and hence T is atomic. This concludes the proof. (cid:3) Of course, if T is an atomic topos and X is any object of T , then Rel ( X ) is anatomic modular quantale.4.2.2. The notion of atomic quantale will be closely related to the notion ofHypergroupoid, which is a natural generalisation of the notion of canonicalhypergroup which can be found in [19] or [17]. Definition :
A hypergroupoid G is the data of: • A set E ( G ) of objects. • For each e, e ′ ∈ E ( G ) a set G ( e, e ′ ) of “arrows” from e to e ′ . • For each g ∈ G ( e , e ) and h ∈ G ( e , e ) an inhabited set hg ⊂ G ( e , e ) of “possible compositions”. • And for each x ∈ G ( e, e ′ ) an element x ∗ ∈ G ( e ′ , e ) called the inverse of x .With the following axioms:(HG1) ∀ e ∈ E ( G ) , ∃ e ∈ G ( e, e ) , such that ∀ x ∈ G ( e ′ , e ) , e x = { x } and ∀ x ∈ G ( e, e ′ ) , x e = { x } . Such a e is unique and is also denoted by e .(HG2) for all x, y, z three arrows such that the composition xy and yz are defined,one has ( xy ) z = x ( yz ) where the product of an element x with a set S isdefined by xS = W s ∈ S xs . This reference studies atomic geometric morphisms, but a geometric morphism f : E → T is atomic if and only if E is atomic as a T -topos. HG3) if x ∈ yz then z ∈ y ∗ x and y ∈ xz ∗ . We will adopt the convention that if g and g ′ are two non-composable ar-rows of a hypergroupoid then gg ′ is defined to be the empty set. Or con-structively that for two general arrow g and g ′ of a hypergroupoid gg ′ = { u | g and g ′ are composable and u ∈ gg ′ } .4.2.3. Proposition :
Let G be a hypergroupoid, and let X be the set of allarrows: X = _ e,e ′ ∈ E ( G ) G ( e, e ′ ) Then P ( X ) endowed with the following structure: U ∗ = { x ∗ , x ∈ U } U V = { t |∃ u ∈ U, v ∈ V, u, v are composable and t ∈ uv } = _ u ∈ Uv ∈ V uv is a (atomic) modular quantale. Proof : P ( X ) is by definition an atomic locale. We check the remaining axioms: • The associativity of the product: if g ∈ U ( V W ) then ∃ u ∈ U , f ∈ ( V W )such that g ∈ uf but b ∈ V W means that there exists v ∈ V , w ∈ W such that f ∈ v.w . hence g ∈ u ( vw ) = ( uv ) w . So there exists an h ∈ ( uv )(in particular h ∈ U V ) such that g ∈ hw and, g ∈ ( U V ) W . The reverseinclusion is exactly the same. • The composition is clearly bi-linear because it is defined so that:
U V = _ u ∈ Uv ∈ V { u } . { v }• The Set 1 = { e , e ∈ E ( G ) } is also clearly a unit for Q , because for any u ∈ G ( e, e ′ ), { u } . { e } = { u } and { u }{ e ′′ } = ∅ for any other e ′′ , hence { u } { u } , one obtains the general result by bi-linearity (and symmetryfor the fact that 1 is also a left unit). • ( U V ) ∗ = V ∗ U ∗ : Let x ∈ V ∗ U ∗ i.e. x ∈ v ∗ u ∗ for u and v respectively in U and V . Then v ∗ ∈ x.u , u ∈ x ∗ v ∗ , and finally x ∗ ∈ uv , ie x ∈ ( U V ) ∗ . Thisreasoning can be conducted backwards to obtain the reverse inequality. • The modularity law:Let x ∈ U ∧ V W then x ∈ U and x ∈ vw with v ∈ V , w ∈ W . We have v ∈ xw ∗ hence v ∈ ( U W ∗ ∧ V ) and x ∈ ( U W ∗ ∧ V ) W . (cid:3) Proposition :
Conversely, any atomic modular quantale is of the form P ( X ) where X is the set of arrows of a hypergroupoid. Proof : Q be an atomic modular quantale. So Q = P ( X ) for some set X .Let E = { x ∈ X | x ∈ Q } . In order to simplify notations we will identify anelement of X with the corresponding singleton element in Q .For any q ∈ X , as 1 Q q = q there exists e ∈ E such that ex = x . Such an e isunique because if e ′ x = x then e ′ ex = x . But as e ′ and e are subobjects of 1 ina modular quantale, one has ee ′ = e ∧ e ′ and in particular x = ( e ′ ∧ e ) x . Hence e ′ ∧ e is inhabited and finally e = e ′ . Similarly, for each x ∈ Q there is a unique e ′ ∈ E such that xe ′ = x .Let: G ( e, e ′ ) = { x ∈ X | xe = x and e ′ x = x } We just show that X is the disjoint union of all the G ( e, e ′ ). If x ∈ G ( e, e ′ )then x ∗ ∈ G ( e ′ , e ). Also, if a ∈ G ( e, e ′ ) and b ∈ G ( e ′ , e ′′ ) then ba ⊂ G ( e, e ′′ ),indeed, if c ∈ ba then there exists a unit f ∈ E such that f c = c . In particular c ∈ f ba = f e ′′ ba = ( f ∧ e ′′ ) ba . Hence e ′′ = f .We will prove that this is indeed a hypergroupoid structure: • Let a ∈ G ( e, e ′ ) and b ∈ G ( e ′ , e ′′ ). One has e ′ a = a and be ′ = b . hence e ′ ∈ aa ∗ and b ∈ baa ∗ so ba is inhabited. • if a ∈ G ( e ′ , e ) then ea = a and ae ′ = a by definition. • the associativity of the product comes from the associativity of the productof the quantale and the fact that if there exists a ∈ uv for u, v ∈ X then u and v are composable elements: this assert that the product of thehypergroupoid is exactly the product of the quantale, and its associativityfollows by restriction to composable pairs of morphisms. • If we assume that x ∈ yz then x = x ∧ yz ( xz ∗ ∧ y ) z hence y ∧ xz ∗ isinhabited so y ∈ xz ∗ .Finally as we have proved already that multiplication in Q and in X are essen-tially the same it is a routine check to prove that Q will be isomorphic to P ( X )as a modular quantale. (cid:3) G to G ′ to be an application f from E ( G ) to E ( G ′ ) and a collection of maps (allcalled f ) f : G ( e, e ′ ) → G ′ ( f ( e ) , f ( e ′ )) such that f (1 e ) = 1 f ( e ) and f ( xy ) ⊂ f ( x ) f ( y ) , then one has: Theorem :
There is an anti-equivalence of category between the categories ofatomic modular quantale (with weakly unital morphisms) and the categories ofhypergroupoid. Of course f ( xy ) denote the direct image by f of the set xy . f (1). Proof :
Let Q and Q ′ be two atomic modular quantales and g : Q ′ → Q a morphismof modular quantales. By 4.2.4, Q and Q ′ can be written Q ′ = P ( X ′ ) and Q = P ( X ) where X ′ and X are the sets of all arrows of two hypergroupoids G and G ′ . As g is in particular a morphism of locale, it induces an application f : X → X ′ characterised by the fact that for all x ∈ X , f ( x ) is the uniqueelement of X ′ such that x ∈ g ( f ( x )).In particular, let c ∈ ab then ab ⊂ g ( f ( a )) g ( f ( b )) hence c ∈ g ( f ( a ) f ( b )) ie f ( c ) ∈ f ( a ) f ( b ). This proves that f ( ab ) ⊂ f ( a ) f ( b ). As 1 g (1), and 1 ∈ G corresponds to E ( G ) ⊂ X , the application f acts on the unit set and preservesthe identity element. One also has x ∗ ∈ g ( f ( x )) ∗ = g ( f ( x ) ∗ ) hence f ( x ) ∗ = f ( x ∗ ) and finally, if g ∈ G ( e, e ′ ) then e ∈ g ∗ g , e ′ ∈ gg ∗ and f ( e ) ∈ f ( g ) ∗ f ( g ), f ( e ′ ) ∈ f ( g ) f ( g ) ∗ which proves that f ( g ) is an element of G ( f ( e ) , f ( e ′ )) whichconcludes the proof that f is a morphism of hypergroupoid.Conversely, if f is a morphism of hypergroupoids, then as f (1 e ) ∈ f ( g ) f ( g ∗ )one can conclude that f ( g ∗ ) ∈ f ( g ) ∗ f (1 e ) = f ( g ) ∗ hence f ( g ∗ ) = f ( g ) ∗ . Onecan then define g = f − which is a frame homomorphism and compatible withmultiplication and involution, and 1 g (1) because each unit is sent by f toa unit, hence it is a morphism of modular quantales. These two constructionsare clearly compatible with compositions and inverse from each other, hence,together with propositions 4.2.3 and 4.2.4, this concludes the proof of the the-orem. (cid:3) Proposition :
Let Q be a modular quantale, and G be the corresponding hy-pergroup. The following conditions are equivalent:1. Q is a Grothendieck quantale2. for every arrow f in G there exists two arrows u and v such that f = uv ∗ with uu ∗ and vv ∗ units. Proof : If Q is a Grothendieck quantale, then (by ( Q Q can bewritten as a supremum of elements of the form uv ∗ with uu ∗ vv ∗ Q is atomic we can write these u and v as union of atoms and hence anyelements of Q can be written as a supremum of element of the form uv ∗ where uu ∗ and vv ∗ are units. In particular, any f ∈ G is an atom of Q , and henceshould be of the form uv ∗ .Conversely, if G satisfies condition (2) then any element of Q can be written asa union of its atoms, which are all of the form uv ∗ with uu ∗ and vv ∗ units (andhence Definition :
An element f of a hypergroupoid such that f f ∗ is a unit is called asimple element. An element which can be written in the form f g ∗ with f and g simple is called a semi-simple element. A hypergroupoid satisfying the conditionof the proposition, i.e. such that every element is semi-simple, will be called asemi-simple hypergroupoid. Hence an atomic Grothendieck quantale is essentially the same thing as a semi-simple hypergroupoid.
In this section we consider an atomic topos T , an arbitrary object X , thequantale Q of relations on X and the corresponding hypergroupoid G . Weassume that G (the set of all arrow of G ) is decidable. This implies that X is adecidable object, indeed: Lemma :
Let G be a decidable hypergroupoid, then its set of units is comple-mented. And if G is associated to an object X of an atomic topos then X isdecidable. Proof :
Let ∆ c = { g ∈ G |∀ e ∈ E ( G ) , e = x } . We will prove that ∆ c is a complement of E ( G ). They are disjoint and forevery g ∈ G there exists a unique e such that eg = g . As G is decidable, either g = e or g = e . If g = e then g ∈ E ( G ). If g = e then for all e ′ , one has e ′ = g ⇒ e ′ = e because of the uniqueness of e , and hence e ′ = g yields acontradiction, so g ∈ ∆ c .In particular, as P ( G ) is isomorphic to Sub ( X × X ), and as the diagonal sub-object of X corresponds to the set of units of G , this proves that X is decidable. (cid:3) In this situation, the convolution product defined in section 3.8 gives a convolu-tion on functions on G with value in R lsc + T . We will give necessary and sufficientconditions in order that the convolution product induces an interesting multi-plication on some algebra.As, in this situation, the convolution product depends on T and not only on G ,these conditions will be expressed in terms of the logic of T . In the next sectionwe will focus on the case of a semi-simple hypergroupoid, in this situation thetopos T will be canonically determined and it will be possible to reformulatethe definition given here in more explicit terms.37.3.1. We will need a few generalities about cardinals of sets in a constructivesetting in order to be able to give an internally valid proof of the mains resultsof this section. Definition :
Let X be a decidable set, then the cardinal of X is defined by: | X | = X x ∈ X ! ∈ R lsc + T We remind the reader that R lsc + T contains an element + ∞ . The following lemmagives two properties that completely characterise the cardinal of a set. Lemma : • For n ∈ N one has n | X | if and only if there exists x , · · · x n pairwisedisjoint elements of X . • For q ∈ Q , q < | X | if and only if there exists an n ∈ N such that q < n | X | . Proof :
Let q ∈ Q , such that q < | X | . By definition, there exists x , . . . , x n ∈ X pairwisedistinct such that ∃ q , . . . , q n < q < P q i . This can be rewritten as ∃ x , . . . , x n ∈ X such that q < n . This proves the second point of the lemmaassuming the first.If there are n distinct elements in X , any q < n is also smaller than | X | hence n | X | . Conversely, if n | X | then ( n − ) < | X | , so there is an integer m with ( n − ) < m and x , . . . x m distinct element in X . As n m one also has n distinct elements, this concludes the proof of the first point of the lemma. (cid:3) Proposition : If X is a decidable set, the following conditions are equivalent:1. X is finite.2. | X | is an integer.3. | X | is a (finite) continuous real number (ie an element of R T ). For an example of a set X with | X | < ∞ but not satisfying these properties,one can take any non-complemented sub-set of a finite (decidable) set. Proof : . ⇒ . is clear because a finite decidable set is isomorphic to { , . . . , n } andhence as cardinal n .2 . ⇒ . is also clear.Assume 3 . , then there exist q, q ′ such that | q − q ′ | < and q < | X | < q ′ . Thereexists an integer n such that q < n | X | < q ′ , and x , . . . , x n pairwise distinctelements of X . Let x ∈ X then there are two possible cases: either x = x i for38ome i , or x is distinct from all the other x i (this is proved recursively on n usingthe decidability of X ). But if x is distinct from all the x i then ( n + 1) | X | and q n < ( n + 1) q which yields a contradiction. So x = x i for some i ,and X is indeed finite. (cid:3) We also note that the same argument yields the following result: If X is decid-able, and we have a function p: X → N > such that P x ∈ X p ( x ) is an integer,then X is finite.4.3.2. Let g and g ′ be two arrows in G , and [ g ],[ g ′ ] the characteristic functionsof the singletons { g } and { g ′ } then:([ g ] ∗ [ g ′ ])( x, y ) = X z [ xgz ∧ zg ′ y ] = |{ z | xgz and zg ′ y }| Definition :
Let (cid:18) g, g ′ a (cid:19) denote the evaluation in a of the function [ g ] ∗ [ g ′ ] .We also define for g ∈ G ( e, e ′ ) , | g | l = (cid:18) g ∗ , ge (cid:19) | g | r = (cid:18) g, g ∗ e ′ (cid:19) Proposition : (cid:18) g, g ′ a (cid:19) , | g | l and | g | r can be computed internally using theformulas:For any u ∈ e the source unit of g : | g | l = |{ z | zgu }| . For any v ∈ e ′ the target unit of g : | g | r = |{ z | vgz }| For any ( x, y ) ∈ a : (cid:18) g, g ′ a (cid:19) = |{ t | xgt and tg ′ y }| Proof :
The formula for (cid:18) g, g ′ a (cid:19) is essentially its definition. The two other formulasfollow easily. The fact that each time the value internally does not depend onany choice of (internal) elements is clear because the various possible choices allbelong to a same atom. (cid:3) One also mentions the two easy (but important) relations: | g ∗ | l = | g | r , (cid:18) g, g ′ a (cid:19) = (cid:18) g ′∗ , g ∗ a ∗ (cid:19) Theorem :
For all pair g, g ′ of composable arrows and all a ∈ gg ′ on has:1. (cid:18) g, g ′ a (cid:19) | a | l = (cid:18) g ∗ , ag ′ (cid:19) | g ′ | l (cid:18) g, g ′ a (cid:19) | a | r = (cid:18) a, g ′∗ g (cid:19) | g | r | g | l | g ′ | l = X a ∈ gg ′ (cid:18) g, g ′ a (cid:19) | a | l Proof : let e , e and e be the units such that g ′ ∈ G ( e , e ) and g ∈ G ( e , e ).Let e ∈ e be arbitrary then let: X a = { ( u, v ) | ugv and vg ′ e and uae } X = { ( u, v ) | ugv and vg ′ e } = G a ∈ gg ′ X a . The cardinality of X a can be computed in two different ways, on one side: | X a | = X u s.t. uae |{ v | ugv and vg ′ e } = X u s.t. uae (cid:18) g, g ′ a (cid:19) = | a | l (cid:18) g, g ′ a (cid:19) . On the other side: | X a | = X v s.t. vg ′ e |{ u | vg ∗ u and uae }| = X v s.t. vg ′ e (cid:18) g ∗ , ag ′ (cid:19) = | g ′ | l (cid:18) g ∗ , ag ′ (cid:19) . The equality of the two results gives (1 . ). the result (2 . ) is the dual (one canuse a similar proof or apply ∗ every where).Similarly, | X | = X v s.t. vg ′ e |{ u | ugv }| = X v,vg ′ e | g | l = | g | l | g | r Hence 3 . comes from the fact that X is the disjoint union of the X a . (cid:3) Theorem :
Let G be a decidable hypergroupoid represented in a toposi.e. corresponding to the modular quantale Rel ( X ) for some object X of a topos.The following propositions are equivalent: • for all g ∈ G, g : e ′ → e the value of [ g ] ∗ [ g ∗ ] at e is a finite continuousreal number. • for all g, g ′ ∈ G the set gg ′ is a finite set and one has a formula of theform: [ g ] ∗ [ g ′ ] = X a ∈ gg ′ (cid:18) g, g ′ a (cid:19) [ a ] with (cid:18) g, g ′ a (cid:19) (finite) positive integers.In the case where this conditions are verified we say that ( G, X ) is locally finite. Proof :
The second condition clearly implies the first. We assume the first condition.This means that for all g ∈ G one has internally ∀ y ∈ X the set { x | xgy } is a finite set (its cardinal is (cid:18) g, g ∗ e (cid:19) and is continuous by assumption). Inparticular { x | xgy } as a finite subset of a decidable set is complemented in X ,hence { z | xgz and zg ′ y } is finite as a complemented subset of a finite set.This proves that the evaluation of [ g ] ∗ [ g ′ ] at every point is indeed a positiveinteger. All that remains to do is check that gg ′ is finite but this comes from thefact that all coefficients appearing in the sum of lemma 4.3.3 (3 . ) are strictlypositive integers, hence the sum has to be indexed by a finite set (see the remarksat the end of 4.3.1). (cid:3) The situation described in the second condition is basically the best we canhope: we get a Z algebra generated (as a group) by the symbol [ g ] for g ∈ G .We will call this algebra A G .But conversely, if we want to have any interesting convolution structure comingfrom the construction done in 3.8 on a set of functions on G with value incontinuous numbers, we need to have the first condition. This proves thatthe “locally finite” hypothesis is exactly the good hypothesis for getting aninteresting convolution product. In this section we assume that X is a bound of T . The hypergroupoid G is nowsemi-simple, and T is fully determined by G , so we should be able to expressthe value of (cid:18) g, g ′ a (cid:19) in terms of the structure of G .41.4.1. Proposition :
In a semi-simple decidable hypergroupoid one has: (cid:18) g, g ′ a (cid:19) = sup a = xy ∗ x,y simples | g ∗ x ∧ g ′ y | The sup in the proposition is taken in R lsc + T , this means in particular that thecoefficient is an integer if and only if the supremum is reached. Proof :
Let q < (cid:18) g, g ′ a (cid:19) i.e. q < n (cid:18) g, g ′ a (cid:19) . This means that for every ( x, y ) ∈ a , thereexists ( v , . . . v n ) pairwise distinct in B such that for all i , ( xgv i ) and ( v i g ′ y ).This means that there is a surjection ( x, y ) : t ։ a and a collection of n maps v , . . . v n from t to B pairwise distinct , such that for all i , ( x, v i ) has value in g and ( v i , y ) has value in g ′ .If we choose any atom on B which maps to t , the composite is still a sur-jection on a . Hence we can freely assume that t is a unit of G , and that x, y, v , . . . v n are arrows in G . And the two relations x ( t ) gv i ( t ) and v i ( t ) g ′ y ( t )become: v , . . . v n ∈ g ′ y ∧ g ∗ x and x ( t ) ay ( t ) became a = xy ∗ , so q < n sup a = xy ∗ | g ′ y ∧ g ∗ x | . Conversely, if q < sup a = xy ∗ | g ′ y ∧ g ∗ x | , then for some x, y simple such that a = xy ∗ , q < n | g ′ y ∧ g ∗ x | . So there exist v , . . . , v n pairwise distinct in g ′ y ∧ g ∗ x . If n >
0, this impliesthat g, g ′ is composable (if n = 0, then q is smaller than any cardinal). Let e be the target of g and the source of g ′ .In this situation, for any u ∈ e , v ( u ) . . . v n ( u ) are n pairwise distinct elementsin { z | x ( u ) gz and zg ′ y ( u ) } and hence q < n (cid:18) g, g ′ a (cid:19) .And this concludes the proof. (cid:3) Proposition :
An element g ∈ G is simple if and only if | g | l = 1 . Proof :
Let g ∈ G ( e, e ′ ). Internally one has by the proposition 4.3.2: ∀ x ∈ e, | g | l = |{ z | zgx }| Hence | g | l = 1 exactly means that for all x there exists a unique z such that zgx which means that g is a partial function, i.e. a simple element. (cid:3) As we will soon assume that t is an atom the precise meaning of ‘distinct’ is not important. Proposition :
Let G be a semi-simple decidable hypergroupoid, then g ∈ G is left finite if and only if there exists a simple arrow u such that ( g, u ) is composable , gu is finite and contains only simple elements. Moreover in thiscase | g | l is the cardinal of the set ( gu ) . Proof :
Translating into an external result the internal formulation of the left finitenessof g would actually give us exactly the statement of this theorem, but thistranslation require some work (essentially done in the proof 4.4.1) which can beavoided by the use of the combinatorial identities we already proved.Assume first that gu is finite and contains only simple elements { x , . . . , x n } .then by the formula 3 . of 4.3.3 and replacing by 1 the left cardinal of simpleelements, one gets that: | g | l = n X i =1 (cid:18) g, ux i (cid:19) But one can see on the formula given in the proposition 4.3.2 that (cid:18) g, ux i (cid:19) | u | l = 1, hence (cid:18) g, ux i (cid:19) = 1 and | g | l = n which implies that g is left finite.Conversely, assume that | g | l = n for some n . By 4.4.1 one has: | g | l = sup u simple | gu | In particular as ( n − / < | g | l , there exists u simple such that | gu | = n . onehas then: | g | l | u | l = X x ∈ gu (cid:18) g, ux (cid:19) | x | l n = n X i =1 (cid:18) g, ux i (cid:19) | x i | l This implies first that all (cid:18) g, ux i (cid:19) | x i | l have an opposite, hence they are all con-tinuous numbers, and hence integers. Moreover as all the (cid:18) g, ux i (cid:19) | x i | l are > x i are simple and this concludes theproof. (cid:3) T -groups In this section, we will show that in the locally finite case, the algebra A G weobtained can be seen as a particular subalgebra of endomorphisms of the freegroup Z B generated by B in the logic of T . This gives an abstract interpretationof the algebra A G . 43e will also show that in the semi-simple case the category of groups of T embeds as a full subcategory of the category of A G -modules, and that thisembedding induces an equivalence between Q T -vector spaces in T and full A G ⊗ Q -modules.4.5.1. Let E = Z B be the free group generated by B in T ( E is a group object of T ). In particular E ⊗ E = Z ( B × B ) is the free group generated by B × B , henceas B is assumed to be decidable one can define a bi-linear map ∆ : E × E → Z T which sends ( b, b ′ ) to 1 if b = b ′ and to 0 if b = b ′ .Let f be an endomorphism of E . One can associate to E the function of “matrixelements” of f , ρ ( f ) : ( b, b ′ ) ∆( b, f ( b ′ )) from B × B to Z . The map ρ : f ρ ( f ) is injective, and one has: f ( b ) = X b ′ ρ ( f )( b ′ , b ) b ′ Also, ρ ( f ◦ f ′ ) = ρ ( f ) ∗ ρ ( f ′ ) for the convolution of functions on B × B (herethe sum involved in the convolution product will be finite, hence it is definedfor integer valued functions). So we just have to understand the image of ρ : Proposition :
A function f from G to Z belongs to the image of ρ if and onlyif it verifies the following two properties:1. If f ( g ) = 0 then g is left finite.2. For each unit e of G , there is at most a finite number of arrows g ∈ G pointing to e such that f ( g ) is non zero. In particular, property (1 . ) tells us that if we are not in the locally finite case,then the algebra of group homomorphisms of E is in some sense ‘too small’.Also, in the locally finite case, the algebra A Q is identified with the sub-algebraof endomorphisms of E such that f ( g ) is non zero only for a finite number of g ∈ G . Proof :
Internally, a function f : X × X → Z T corresponds to a group homomorphismif and only if for all x ∈ X there is only a finite set of x ′ such that f ( x ′ , x )is non zero. Indeed, the corresponding group homomorphism has to send x to P x ′ f ( x ′ , x ) .x ′ .The cardinality of the set of x ′ such that f ( x ′ , x ) is non zero defines a function τ on X , whose value at any atom e of X (ie at any unit of G ) is given by: τ ( e ) = X g ∈ G ( e ′ ,e ) ,f ( g ) =0 | g | l Indeed, for any x ∈ X the set of x ′ such that f ( x ′ , x ) is non zero is partitioned bythe various g ∈ G ( e ′ , e ) such that ( x ′ , x ) ∈ g and each of this set has cardinality | g | l (because the value of f ( x ′ , x ) only depends on the atom that contains it).So τ ( e ) is an integer if and only if each of the | g | l are integers (this is condition1 . ) and if they arise in finite number (this is condition 2 . ).44 In the rest of this section (and also in the rest of this paper), we assume thatthe representation of G in T is locally finite.4.5.2. Let F be any group object of T . Then hom( E, F ) is a right hom(
E, E )module (where the hom denotes the internal group homomorphisms). The unitsof G act as family of disjoint projections on E , and hence on hom( E, F ). Let:˜ F = M e ∈ E ( G ) Hom ( E, F ) .e equivalently, ˜ F is the subset of hom( E, F ) of elements x such that there existsa finite set I of units such that x.e I = x where e I = P e ∈ I e . Proposition :
For all F , ˜ F is a full right A Q -module.This gives a functor from group objects of T to full right A Q -modules.Also ˜ E is A Q seen as a right A Q -module. Proof :
Clear from the observation that ˜ F is the subset of hom( E, F ) of elements x suchthat there exists a finite set I of units such that x.e I = x where e I = P e ∈ I e ∈ A Q . And ˜ E identify with A Q thanks to 4.5.1 (cid:3) Actually, ˜ F = hom( E, F ) .A Q .4.5.3. Assume that G is semi-simple, so B is a bound of T and the categoryof atoms of B and morphisms between them (i.e. the category of units of G and simple arrows between them) endowed with the atomic topology is a siteof definition of T .If F is a group object of T then for each e atom of B , F ( e ) = ˜ F .e and theaction of a simple arrow f from e to e ′ is given be the action of [ f ] on ˜ F . Inparticular, the sheaf corresponding to F is fully determined by ˜ F and any A Q -linear morphism from ˜ F to ˜ F ′ gives rise to a morphism of sheaves and one canconclude that: Lemma :
When G is semi-simple (and locally finite) the functor from T -groupsto A G -modules defined in 4.5.2 is fully faithful. Unfortunately, if we start from a general A G -module we only get a pre-sheaf overthe site of units. In the general case, we have not found a characterisation of the A G -modules corresponding to T -group simpler than the definition of a sheaf.But in the case where we assume that all the coefficients | g | l are invertible, then A module M over a non unitary ring A is said to be full if the map A × M → M issurjective. g ] for non simple g will automatically turn our pre-sheaf intoa sheaf. Proposition : (Still under the assumption that G is semi-simple and locallyfinite) Let M be an A G module such that for every g ∈ G the integer | g | l acts(by multiplication) as a bijection on M . Then M comes from a T -group.In particular there is an equivalence of categories between Q T -vector spaces andfull right A G ⊗ Q -modules. Proof :
We will check that under this assumption the pre-sheaf of ˜
M .e is actually asheaf for the atomic topology.Let e be a unit of G , let f be any simple arrow starting at e . And let m ∈ M.e such that for any to simple arrows g, h targeting e such that f g = f h one has mg = mh . We need to prove that there exists a unique n such that nf = m .The uniqueness is easy: if m = n.f then m.f ∗ = n.f f ∗ = | f | r n so as | f | r isinvertible on has n = | f | r m.f ∗ .Conversely, we will prove that n = | f | r m.f ∗ provides a solution. n.f = 1 | f | r m.f ∗ f = 1 | f | r X a ∈ f ∗ f (cid:18) f ∗ , fa (cid:19) m.a let a ∈ f ∗ f . a can be written in the form a = gh ∗ with g and h simple. gh ∗ ∈ f ∗ f ⇒ f ∈ f gh ∗ ⇒ f g ∈ f h Hence one has f g = f h and by the assumption on m , m.h = m.g .the relation m.g = m.h implies, m. [ g ][ h ∗ ] = m. [ h ][ h ∗ ][ h ][ h ∗ ] = | h | r . [ e ]By 4.3.3 one has: [ g ][ h ∗ ] = | h | r | a | r [ a ]so m. [ g ][ h ∗ ] = m. [ h ][ h ∗ ] becomes: m.a = | a | r .m and we can conclude that: n.f = 1 | f | r X a ∈ f ∗ f (cid:18) f ∗ , fa (cid:19) | a | r m = | f ∗ | r m = m again by 4.3.3 and the fact that f is simple hence | f ∗ | r = | f | l = 1. (cid:3) .6 A G as a quantum dynamical system In this section we construct the regular representation of A G . We show that the C ∗ algebra generated by A G comes with a canonical action of R . There is alsoa regular representation of A G , attached to a KM S state and defining a C ∗ algebra C ∗ red ( G ) by completion.4.6.1. Let H be a real Hilbert space in T . Then˜ H = M e ∈ E ( G ) hom( e, H )is an A G vector space. The internal scalar product on H gives rise to a scalarproduct on each of the hom( e, H ) and turns ˜ H into a pre-Hilbert space. Proposition :
In the action of A G on ˜ H one has: [ g ] ∗ = | g | l | g | r [ g ∗ ] And [ g ] has norm smaller than | g | l . Proof :
Let g ∈ G , v ∈ hom( e, H ) and v ′ ∈ hom( e ′ , H ).If g is not an arrow from e to e ′ , then both h v, v ′ [ g ] i and h v [ g ∗ ] , v ′ i are zero(hence equal). If g is an arrow from e to e ′ , then for any x ∈ e one has: h v, v ′ [ g ] i ( x ) = * v ( x ) , X y,ygx v ′ ( y ) + = X y,ygx h v ( x ) , v ′ ( y ) i But g is an atom of B × B , and h v ( x ) , v ′ ( y ) i is a function on B × B so its valuedoes not depend on x , y as long as they belong to g . Hence, for any ( x, y ) ∈ g one has: h v, v ′ [ g ] i = | g | l h v ( x ) , v ′ ( y ) i Similarly: h v [ g ∗ ] , v ′ i = | g | r h v ( x ) , v ′ ( y ) i Finally: h v, v ′ [ g ] i = | g | l | g | r h v [ g ∗ ] , v ′ i And the first result follows.The second result follows from h v, v ′ [ g ] i = | g | l . h v ( x ) , v ′ ( y ) i | g | l k v kk v ′ k . As k v ( x ) k = k v k . (cid:3) Definition :
For g ∈ G , We will denote: χ ( g ) = | g | l | g | r One has: χ ( g ∗ ) = χ ( g ) − [ g ] ∗ = χ ( g )[ g ∗ ]and also the more surprising result: Proposition :
For any three arrows a, g, g ′ of G such that a ∈ gg ′ : χ ( a ) = χ ( g ) χ ( g ′ ) Proof :
We will just need several applications of the first two points of theorem 4.3.3. χ ( a ) = | a | l | a | r = | a | l (cid:18) g, g ′ a (cid:19) | a | r (cid:18) g, g ′ a (cid:19) = | g ′ | l | g | r (cid:18) g ∗ , ag ′ (cid:19)(cid:18) a, g ′∗ g (cid:19) but: | g ′ | r | g | l (cid:18) g ∗ , ag ′ (cid:19)(cid:18) a, g ′∗ g (cid:19) = | g | l (cid:18) g ′ , a ∗ g ∗ (cid:19) | g ′ | r (cid:18) a ∗ , gg ′∗ (cid:19) = | g | l (cid:18) a, g ′∗ g (cid:19) | g ′∗ | l (cid:18) a ∗ , gg ′∗ (cid:19) = 1So we can conclude that: χ ( a ) = | g ′ | l | g | r (cid:18) g ∗ , ag ′ (cid:19)(cid:18) a, g ′∗ g (cid:19) = | g ′ | l | g | r | g | l | g ′ | r = χ ( g ) χ ( g ′ ) (cid:3) Definition :
Let A G, Q be the algebra A G ⊗ Q . It is endowed with theinvolution ∗ defined by 4.6.1, i.e.: [ g ] ∗ = χ ( g )[ g ∗ ] We also define the elements: e g = 1 | g | l [ g ] ∈ A G, Q which are additive generator such that ( e g ) ∗ = e g ∗ . Proposition : If G is semi-simple and locally finite, the functor which sendsa T -Hilbert space H to the completion of ˜ H is (one half of ) an equivalence ofcategories between the category of internal Hilbert spaces of T , and the full right Hilbert ∗ -representations of A G, Q . Proof :
The proof is really similar to the case of Q T -vector spaces done in 4.5.2 and4.5.3. If we start from a T -vector space, we already proved that A G, Q acts on˜ H by bounded morphisms and in a way compatible with the involution. So itextends to a full Hilbert ∗ representation of A G, Q on the completion of H .In the other direction, the sheaf of complex numbers on the site of units of G is the constant sheaf. Hence if H is a Hilbert ∗ representation of A G, Q then e → H [ e ] defines a pre-sheaf of C T -modules that we will denote by H . Thepre-sheaf H is a sheaf by 4.5.3.For every simple arrow f : e → e ′ one has:[ f ] . [ f ] ∗ = [ f ][ f ∗ ] 1 | f | r = [ e ′ ] | f | r | f | r = [ e ′ ]and the induced map H [ e ′ ] → H [ e ] (i.e. the structural map of H ) is an isometricinjection. This proves that the scalar product H [ e ] × H [ e ] → C is in fact amorphism of sheaves H × H → C T and hence this endows H with an internalscalar product.It remains to show that H is internally complete. Let ˜ H be its completion, let h ∈ ˜ H ( e ).Then, by (internal) density of H in ˜ H , for every n ∈ N , there exists f : e ′ → e ,and h ′ ∈ H ( e ′ ) such that k h ′ − h.f k < /n . But one can write: k h ′ [ f ] ∗ − h k < /n and h ′ [ f ] ∗ ∈ H ( e ), hence h can be approximated by elements of H ( e ). As H ( e )is complete, this proves that h ∈ H ( e ) and hence that H is internally complete.Finally this is an equivalence, because if we start from a full Hilbert ∗ -representation H of A G, Q then the construction we just made corresponds to that of 4.5.3 ap-plied to H.A G, Q hence as we applied a completion at the end, we will get H back because H.A G, Q is dense in H by assumption. (cid:3) Here full, mean that H .A G, Q is dense in H . A G, C = A G ⊗ C ina specific representation: the one corresponding to the internal Hilbert space l ( B ) of square summable functions on B , defining a C ∗ algebra C ∗ red ( G ), or wecan take the universal C ∗ algebra generated by A G, Q that we will denote by C ∗ max ( G ).Both these algebras come with a time evolution ( σ t ) t ∈ R given by: σ t ([ g ]) = χ ( g ) it [ g ]This is a morphism of algebras because of 4.6.2.Let e be an atom of B . Then one has a map e ֒ → B which gives rise to a map e → l ( B ) and hence to a vector of the corresponding representation of C ∗ red ( G )that we will simply denote l . An easy computation shows that the state on C ∗ red induced by this vector is: η e ([ q ]) = (cid:26) q = e G is finite then the (renormalized) sum of all the η e is astate, in general we can define it without renormalization as a semi-finite weight(it is finite on the algebra A G ), we denote it by η . Proposition :
The GNS representation induced by η is the l representation,and η verifies the KMS condition at temperature one. Proof :
The first part is clear: the GNS representation induced by η is included in l and contains all the vectors corresponding to the e ∈ G (indeed, [ e ] gives rise tothis vector through the GNS construction). If it were a strict sub-representationthen it would correspond internally to a sub Hilbert space of l ( B ) containingall the basis vectors, which is impossible.For the second part: η ([ q ] σ i ([ q ′ ])) = χ ( q ′ ) − η ([ q ][ q ′ ])If q ′ = q ∗ then both η ([ q ][ q ′ ]) and η ([ q ′ ][ q ]) are zero (because e ∈ qq ′ ⇒ q ′ = q ∗ ).If q ′ = q ∗ then η ([ q ] σ i ([ q ′ ]) = χ ( q ) η ([ q ][ q ∗ ]) = χ ( q ) (cid:18) q, q ∗ e (cid:19) = χ ( q ) | q | r = | q | l = η ([ q ′ ][ q ]) (cid:3) In this subsection, we will first show that for a decidable bound B of an atomictopos T , the hypergroupoid of atoms of B × B is locally finite if and only ifthe slice topos T /B is separated (or Hausdorff) in the sense of [20]. Then wewill show that an atomic topos admits such a bound if and only if it is locally50ecidable and “locally separated”, that is if there exists an inhabited object X of T such that T /X is separated. And finally, that in this case the time evolutionconstructed in 4.6 is completely canonical when seen as a family of functors onthe category of Hilbert space of T and is described by a canonical principal Q ∗ + bundle χ T : T → B Q ∗ + attached to every locally separated (locally decidable)atomic topos T .4.7.1. We recall that a geometric morphism f : T → E is said to be properif f ∗ (Ω T ) is a compact locale internally in E , and is said to be separated ifthe diagonal map T → T × E T is proper. A topos is said to be compact (resp.separated) if the geometric morphism from T to Set is proper (resp. separated).These notions have been studied in [20] and in [14]C3.2 (see also C5.1).We will say that a topos is locally separated if there exists an inhabited object X of T such that the slice topos T /X is separated.4.7.2. We start by the following proposition which relates finiteness conditionsto the separation property. Proposition :
An atomic locally decidable topos T is separated if and only ifevery atom of T is internally finite. Also, the “only if” part holds without assuming that T is locally decidable. Proof :
We start by assuming that T is separated, and that a ∈ |T | is an atom. Thenthe topos T /a is hyperconnected and hence proper. Proposition II.2.1(iv) of[20] asserts that when one has a commutative diagrame: T /a T∗ gh f with h proper and f separated then g is proper. But the map T /a → T is properif and only if the discrete space a (internally in T ) is compact if and only if a isfinite.Conversely let T be an atomic topos whose atoms are internally finite.The commutative diagram: T T × TT ∆ Id π This mean that the locale p ∗ (Ω T /a ) whose open are subobjects of a is trivial, which isthe definition of the fact that a is an atom. T × T internally in T . As T × T is thepullback of T by the canonical geometric morphism from T to the point, it willstill be an atomic topos internally in T , and it will still have a generating familyof finite objects and hence all its atoms will be finite internally in T .Hence our problem is equivalent to prove (constructively) that if T is atomicwith a point p and that all the atoms of T are finite then p is proper. But anatomic topos with a point is equivalent to BG for G the localic group of auto-morphisms of the point, and the fact that the atoms are finite means that all the G -transitive sets are finite, and as G has been taken to be set of automorphismsof the point, this implies that G is compact:Indeed, the localic monoid of endomorphisms M ( G ) = lim G/U constructed in[21] is compact by (localic) Tychonoff’s theorem, and separated because thanksto the locale decidability one can restrict to the U such that G/U is decidable,hence separated. In particular the point 1 is closed, and as G can be identifiedwith the subspace of M ( G ) × M ( G ) of f, g such that f g = 1 which is a closedsubspace in a separated compact space, G is also compact.Finally, the map 1 → BG is proper because its pull-back along itself is the map G → ∗ which we just showed to be proper, and the map ∗ → BG is always anopen surjection (for exemple by [14]C3.5.6(i)) hence the fact that proper mapdescend along open surjection (see [14]C5.1.7) allows us to conclude. (cid:3) e and e ′ are two decidable atoms and f : e → e ′ then the internal(semi-continuous) number | f − ( y ) | does not depend on y because e ′ is an atomand hence gives an (externally) well defined number called the degree of f . Proposition :
Let T be an atomic topos, and B be a decidable bound of T thenthe associated semi-simple hypergroupoid is locally finite if and only if for anycouple e , e ′ of atoms of B , any map f : e → e ′ has finite degree. Proof :
If Γ f denotes the graph of f in B × B then it is an atom of B × B and the degreeof f is equal to | Γ f | r . Hence if the hypergroupoid is locally finite, then everysuch map has a finite degree. Conversely, if any such map has a finite degreethen every simple element is right (and left) finite and as any element can bewritten g = uv ∗ with u and v simples, one has: | g | l = |{ z |∃ t, z = u ( t ) and v ( t ) = y }| and this set is finite because it is a quotient of v − ( y ). (cid:3) Proposition :
Let T be an atomic topos, B a decidable bound of T then the associated hypergroupoid is locally finite if and only if T /B is separated.Moreover, for an arbitrary atomic topos, such a bound exists if and only if it islocally decidable and locally separated. roof : Let T be an atomic topos, and B an arbitrary object of T . One has: T /B = a a atom of B T /a Hence T /B is separated if and only if for every atom a of B the topos T /a isseparated. Also, by 4.7.2, T /B is separated if and only if for every atom v of T ,every map v → B has finite fiber.In particular, if B is decidable and T /B is separated, then for any atom e of T and any atom e ′ of B , any map f : e ′ → e has finite degree, and hence theassociated hypergroupoid is locally finite.Conversely, let B be a decidable bound satisfying the condition of 4.7.3, then inorder to show that T /B is separated we need to show that for any map f : e ′ → e where e is an atom of B and e ′ an arbitrary atom of T , f has finite degree. Butin this situation, as B is a bound there exists an atom e ′′ of B and a map e ′′ → e ′ → e . As the map e ′′ → e ′ is surjective because they are atoms, anyfiber of the map e ′ → e is covered by a fiber of the map e ′′ → e , which are finiteby assumption. This concludes the proof of the first part of the theorem.If T admits a bound satisfying the locale finiteness assumption, then it is inparticular locally separated. Conversely assume that T is atomic, locally de-cidable and locally separated. Let X be an inhabited object such that T /X isseparated, B an arbitrary bound and B be a decidable cover of B × X . Then B is decidable, it is a bound because it is a cover of B and T /B is separated be-cause B can be seen as a decidable object of T /X , hence the geometric morphism T /B → T /X is separated (see [20] II.1.3(1) ) and by composition of separatedmorphisms T /B is separated (see [20]II.2.1(ii) ). (cid:3) H → H t for t ∈ R acting on the category of Hilbert spaces of T , correspondingto the functor which send a representation of C ∗ max ( Q ) to the representationtwisted by σ t .By the previous theorem we know that any atomic locally decidable locallyseparated topos has such a time evolution. We will show that this time evolutionis canonical by giving a construction of it which does not depend on the choiceof the bound B .To be more precise, let T be a locally decidable locally separated topos. Wewill construct a Q T principal bundle χ T in T the following way. The decidableatoms a of T such that T /a is separated form a generating family. Hence todefine an object of T it is enough to define a sheaf for the atomic topology onthe full subcategory of these atoms.We define : hom( a, χ T ) = Q ∗ + and if f : a → a ′ is any map then it acts on Q ∗ + by multiplication by its degree.All these maps are bijective, hence it defines a sheaf. Also Q ∗ + acts on χ T bymultiplication turning it into a principal Q ∗ + bundle.53e note in particular that if T is itself separated, then the terminal object of T is among the atoms of the site we consider and hence hom(1 , χ T ) is inhabitedand hence χ T is the trivial bundle. But saying that χ T is trivial only meansthat it is possible to construct a global section d of it, which will be a mapassociating to any decidable atom a such that T /a is separated a finite rationalnumber d ( a ) such that if f : a → a ′ is a map between two of these atoms then d ( a ) = d ( a ′ ) deg ( f ).4.7.6. Finally, the time evolution given by any bound can be described in termsof this invariant χ T . Indeed if H is an arbitrary Hilbert space on T , and wechoose an ’admissible’ bound B , then the effect of the time evolution on H canbe described by the fact that H t is the same sheaf as H on the site of atoms of B but with the action of a map f : e → e ′ twisted by χ ( f ) it = ( degf ) − it .Hence if we see χ T as a morphism from the topos T to the classifying space B Q ∗ + of principal Q ∗ + bundles (i.e. the topos of Q ∗ + sets), and if we call E t the onedimensional Hilbert space in B Q ∗ + defined by C with its usual Hilbert spacestructure and endowed with action q.z = q − it ( z ) then the previous formula for H t can be rephrased as: H t = H ⊗ χ ∗T ( E t )where the tensor product is just the internal tensor product of Hilbert spacesin T . G is a discrete group, T the topos of G -sets and B is G endowedwith its (left) action on itself, then the corresponding quantale is P ( G ), thehypergroupoid is the group G , the integral algebra is Z [ G ], and the reduced andmaximal C ∗ -algebras are the usual reduced and maximal group C ∗ algebras. Inthis situation the time evolution is trivial.4.8.2. The best-known example of this situation is the case of double cosetsalgebras. Let G be a discrete group, and ( K i ) a family of subgroups of G (onecan generalize to G a localic group and K i open subgroups). Let X i be the G -set G/K i . The topos of G -set is atomic and Rel ( X i , X j ) can be identifiedwith the subset of G stable by the action of K i on the left and K j one the right,hence the atom of Rel ( X i , X j ) are exactly the ( K i , K j ) double-cosets. Underthis identification, the composition of a ( K i , K j ) cosets with a ( K j , K t ) cosetsis the set of ( K i , K t ) cosets included in the product of their elements, and thecoefficients (cid:18) g, g ′ a (cid:19) are exactly the usual coefficient involved in the definitionof the double cosets modules and double cosets algebras (hence we are in thelocally finite case if and only if one has the usual almost normality condition).54.8.3. The previous example in particular gives back the BC-system constructedin [5] with both its time evolution and its integral sub-algebra by consideringthe topos of continuous actions of the group G = A f Q ⋊ Q ∗ + , where A f Q denotethe additive group of finite adele of Q , i.e. the restricted product of all p -adiccompletions of Q , with the bound B = G/ ˆ Z .Unfortunately, trying to replace Q by another number field in this constructiondoes not seem to give the “good” BC -system for number field constructed in[10], and certainly not the good arithmetic subalgebra. Actually the variant ofthe BC algebra associated to a number field K constructed in [2] corresponds tothe topos of continuous G K = ( A fK ) ⋉ K ∗ -sets with the bound B = G K / d O K .4.8.4. As a spatial atomic topos is automatically a disjoint sum of toposes of theform BG for G a localic group, the case of double cosets algebras (attached todiscrete, topological or localic groups with possibly several subgroups involved)exactly corresponds to the case of spatial atomic toposes. We actually do notknow if in the classical set theoretical case there exists examples of non-spatiallocally separated atomic toposes, but these examples definitely exist internallyto other toposes, and we will exploit them in a forthcoming article. References [1] Charles A Akemann. The general stone-weierstrass problem.
Journal ofFunctional Analysis , 4(2):277–294, 1969.[2] Jane Arledge, Marcelo Laca, and Iain Raeburn. Semigroup crossed productsand hecke algebras arising from number fields.
Doc. Math , 2:115–138, 1997.[3] M. Barr and P.H. Chu. *-Autonomous categories . Springer-Verlag, 1979.[4] A Boileau. et joyal, a.-la logique des topos.
J. Symbolic Logic , 46(1):616,1981.[5] Jean-Benoit Bost and Alain Connes. Hecke algebras, type iii factors andphase transitions with spontaneous symmetry breaking in number theory.
Selecta Mathematica , 1(3):411–457, 1995.[6] Marta Bunge. An application of descent to a classification theorem fortoposes. In
Mathematical Proceedings of the Cambridge Philosophical So-ciety , volume 107, pages 59–79. Cambridge Univ Press, 1990.[7] Alain Connes.
Noncommutative geometry . Academic press, 1995.[8] Peter J Freyd and Andre Scedrov.
Categories, allegories , volume 39. AccessOnline via Elsevier, 1990.[9] Robin Giles and Hans Kummer. A non-commutative generalization oftopology.
Indiana Univ. Math. J , 21(1):91–102, 1971.[10] Eugene Ha and Fr´ed´eric Paugam. Bost-connes-marcolli systems for shimuravarieties. part i. definitions and formal analytic properties.
InternationalMathematics Research Papers , 2005(5):237–286, 2005.5511] Hans Heymans and Bob Adviser-Lowen. Sheaves on quantales as general-ized metric spaces. 2010.[12] Hans Heymans and Isar Stubbe. Grothendieck quantaloids for allegoriesof enriched categories.
Bulletin of the Belgian Mathematical Society-SimonStevin , 19(5):859–888, 2012.[13] Christopher J Mulvey and Joan Wick Pelletier. On the quantisation ofpoints.
Journal of Pure and Applied Algebra , 159(2):231–295, 2001.[14] P.T. Johnstone.
Sketches of an elephant: a topos theory compendium .Clarendon Press, 2002.[15] A Joyal and I Moerdijk. Toposes as homotopy groupoids.
Advances inMathematics , 80(1):22–38, 1990.[16] A. Joyal and M. Tierney.
An extension of the Galois theory of Grothendieck .American Mathematical Society, 1984.[17] Marc Krasner. Approximation des corps values complets de caractristique p = 0 par ceux de caract´eristique o. In Colloque d’Algebre Superleure(Bruxelles , 1956.[18] S. MacLane and I. Moerdijk. Sheaves in geometry and logic: A first intro-duction to topos theory (universitext). 1992.[19] Jean Mittas. Hypergroupes canoniques.
Mathematica Balkanica , 2:165–179,1972.[20] Ieke Moerdijk and Jacob Johan Caspar Vermeulen.
Proper maps of toposes ,volume 705. AMS Bookstore, 2000.[21] Izak Moerdijk.
Morita equivalence for continuous groups . Cambridge UnivPress, 1987.[22] Christopher J Mulvey. &, suppl.
Rend. Circ. Mat. Palermo II , 12:99–104,1986.[23] Alan LT Paterson.
Groupoids, inverse semigroups, and their operator al-gebras , volume 170. Springer, 1999.[24] Andrew M Pitts. Applications of sup-lattice enriched category theory tosheaf theory.
Proceedings of the London Mathematical Society , 3(3):433–480, 1988.[25] Steven Vickers. A localic theory of lower and upper integrals.
MathematicalLogic Quarterly , 54(1):109–123, 2008.[26] Steven Vickers. A monad of valuation locales. , 2011.
Simon Henry, Paris, France.
E-mail adress: [email protected]@math.jussieu.fr