aa r X i v : . [ m a t h . C T ] A ug INTERACTION DECOMPOSITION FOR PRESHEAVES
GRÉGOIRE SERGEANT-PERTHUIS
Contents
1. Introduction 12. What is an interaction decomposition? 23. Necessary and sufficient condition for the interaction decomposition ofprojectors from a finite poset to
Vect
64. Interaction Decomposition for presheaves in
Mod
Abstract.
Consider a collection of vector subspaces of a given vector spaceand a collection of projectors on these vector spaces, can we decompose thevector space into a product of vector subspaces such that the projectors areisomorphic to projections? We provide an answer to this question by extendingthe relation between the intersection property and the interaction decomposi-tion ([11]) to the projective case. This enables us to classify the decompositionsof interactions for factor spaces. We then extend these results for presheavesfrom a poset to the category of modules by adding the data of a section functorwhen it exists. Introduction
Motivation.
Let us note Gr V the poset of ( k -)vector subspaces of a given( k -)vector space V and let A be a poset. Gr V is called the Grassmannian of V ; the set of poset morphisms, i.e. increasing function, from A to Gr V shall benoted hom ( A , Gr V ); let U ∈ hom ( A , Gr V ), assume that V is a finite dimen-sional Hilbert space and that there is a collection ( e a , a ∈ A ) of elements of V suchthat for any a ∈ A , U( a ) = < e b , b ≤ a > , where < e b , b ≤ a > is the vector sub-space of V generated by the collection ( e b , b ≤ a ). One can ask when one can find( u b , b ∈ A ) an orthogonal basis of V such that for any a ∈ A , U( a ) = L b ≤ a < u b > .To answer this question one needs to know how to decompose each U( a ) in a com-patible manner with the poset structure A and compatible with the orthogonalprojection π a : V → U( a ). The study of when one can find a decomposition of U compatible with A is the subject of [11] (injective case). In Appendix B [3],in the context of factor spaces ( A = { , } n see Definition 2.1) and for a specificscalar product one can show that such decomposition exists (Proposition B.4 [3])(a similar result holds is stated in [1], [7]). However the proof heavily relies onthe explicit expression of the Mobius function of the boolean poset { , } n andseemed to be restricted to the case of factor spaces and to specific scalar products.Given A a finite poset, a collection of projectors ( π a , a ∈ A ) from V to V and U = (Im π a , a ∈ A ) ∈ hom ( A , Gr V ), by using tools and methods very differentin spirit from the ones found in the litterature on interaction models ([3]), we shallgive a necessary and sufficient condition for a decomposition of U compatible with( π a , a ∈ A ) to exist. This will enable us to classify the interaction decompositionsfor factor spaces. We shall then extend this result to posets that are not necessarilyfinite ( A ∈ ˆ P f Notation 4.2) and couples of functor, presheaf from a A to thecategory of modules, Mod .1.2.
Structure of this document.
In Section 2 we give some general propertiesfor a collection of endomorphisms of a vector space, ( π a , a ∈ A ), over a finiteposet and define (Definition 2.4) what a compatible decomposition with respectto ( π a , a ∈ A ) is and how it relates to the interaction decomposition of factorspaces (see Question). We state the intersection condition (I) (Proposition 2.6) for( π a , a ∈ A ) decomposable and set the problematic of this document.In Section 3 we show that the intersection property (Definition 3.7) implies that( π a , a ∈ A ) is decomposable (Theorem3.2, Corolary 3.2); as a consequence we showthat conditional expectations to factor spaces for a given measure are decomposableif and only if the measure is a product measure (Corollary 3.3); this classifies all thepossible interaction decompositions of factor spaces. We believe this application isin itself an improvement with respect to what is know on the decomposition ininteraction spaces. We then extend (Section 4) these results for presheaves in thecategory of ( R -)modules (Definition 4.3, Definition 4.7, Theorem 4.1, Corollary4.4). We conclude by remarking that these results can be applied to extend theGram-Schmidt process to more general posets that N .2. What is an interaction decomposition?
As the interaction decomposition in [3] is stated for collection of random vari-ables, let us first give some reasons for why solving this question can be interestingin probability; let I be a finite set, it indexes random variables, let for all i ∈ I , E i be a finite set with the discrete σ -algebra, in which the i -th random variabletakes its values, E = Q i ∈ I E i is the configuration space; for ω ∈ E , one has that pr i ( ω ) = ω ( i ), and for a ⊆ I non empty, we will note ω | a as x a . We will call E a = Q i ∈ a E i and, p a : E → E a x x a Let • be a given singleton. Then there is only one application of domain E to • that we call π ∅ ; we pose ω ∅ = p ∅ ( ω ). The σ -algebra one considers implicitlyon Q i ∈ I E i will be the Borel algebra with respect to the product topology, i.e. thesmallest algebra that makes the projections, for any a ⊆ I , π a measurable, here asthe E i are finite and I is finite is coincides with the dicrete σ -algebra on E. Let usdenote F a the smallest σ -algebra that makes p a measureable, i.e generated by thecylindric events { ω a } := { ω ∈ E : ω a = ω a } . NTERACTION DECOMPOSITION FOR PRESHEAVES 3
Notation . For any measurable space ( X, F ) let us denote M ( E, F ) the set ofmeasurable function and M b ( E, F ) the set of bounded measurable functions, i.e. L ∞ ( E, F ). Let P ( E ) be the set of probability measures of E . Notation . We shall note the image measure p a ∗ P , i.e. the marginalisation of P over E a , as P a . Definition 2.1.
For any a ∈ P ( I ), let V ( a ) be the vector subspace of V constitutedof functions f that can be factorised by p a , in other words there is ˜ f such that f = ˜ f ◦ p a . V ( a ) is called the a -factor space, V ( a ) = M b ( E, F a ).As E is finite one can define for a probability P its support,Supp P = { ω ∈ E : P ( ω ) = 0 } Let P be the probability measure on E associated to the collection of randomvariables I , for any a ⊆ I , let π a : V → V ( a ) be such that π a = E a [ | F a ] theconditional expectation with respect to the a factor space; we take as conventionthat for any cylinder events if P ( { ω a } ) = 0, E a [ | F a ]( ω ) = 0; therefore for any a ∈ A , f ∈ M b ( E ), E [ f | F a ] = 1[ ∈ Supp P a ] P ω ′ : ω ′ a = ω a P ( ω ′ ) P a ( ω a ) f ( ω ′ ). Notation . We shall note the set of endomorphisms of V as L ( V ). For π =( π i ∈ L ( V ) , i ∈ I ) the collection (Im π i , i ∈ I ) will be called the image of π anddenoted as Im π . Proposition 2.1.
Let π be a collection of projectors of V over A . Im π ∈ hom ( A , Gr V ) if and only if for any a, b ∈ A such that b ≤ a , π a π b = π b :Proof. Assume that Im π b ⊆ Im π a for b ≤ a then π a π b = π b ; suppose π a π b = π b then Im π b ⊆ Im π a . (cid:3) Remark . When Im π ∈ hom ( A , Gr V ), let for b ≤ a , G ( a ) = Im π a and G ba = i Im π b Im π a ; G is a functor from A to Gr V , where Gr V has as morphisms inclusions;we shall call G ( π ) the canonical functor associated to π . Notation . For V a vector space, whe shall note sub ( V ) the subcategory ofsubvector spaces of V . Definition 2.2.
Let A be a poset, π = ( π a , a ∈ A ) a collection of endomorphismsof V . If for any a, b ∈ A such that b ≤ a there is f ab ∈ L ( V ) such that,(1) π b = f ab ◦ π a we will say that π is presheafable in sub ( V ). Definition 2.3.
For any poset A , we shall call the nerve of A , denoted as N ( A ),any strictly increasing sequence of A ; in particular we shall call N ( A ) n any strictlyincreasing sequence of n -elements. GRÉGOIRE SERGEANT-PERTHUIS
Example . N ( A ) = { ( a, b ) ∈ A : a < b } . Proposition 2.2.
Let π be presheafable in sub ( V ) and ( f ba , ( a, b ) ∈ N ( A ) ) acollection that satisfies Equation (1); let F aa = id and F ab = f ab | Im π a Im π b when b ≤ a , F ( π ) is a presheaf from A to sub ( V ) that we shall call the canonical presheafassociated to π .Proof. For any ( a, b ) ∈ N ( A ) , Im f ab (Im π a ) ⊆ Im π b therefore F ba is well defined.Furthermore for a ≥ b ≥ c and v ∈ V , F bc F ab π a ( v ) = f bc π b ( v ) = π c ( v ) = F ac π a ( v );as π a is surjective on its image, F bc F ab = F ac .Let ( f ba , ( a, b ) ∈ N ( A ) ) that satisfies Equation (1), for b ≤ a , f ab π a = π b = f ab π b therefore F ab = F ab which justifies why we can call F canonical. (cid:3) Proposition 2.3.
Let A be a poset, π = ( π a , a ∈ A ) a collection of projectorsof V . π is presheafable in sub ( V ) if an only if for any for any a, b ∈ N ( A ) , π a π b = π a .Proof. Let us assume that π is presheafable, and let ( f ba , ( a, b ) ∈ N ( A ) ) be suchthat Equation (1) holds for any ( a, b ) ∈ N ( A ) . Let ( a, b ) ∈ N ( A ) and v ∈ Im π b then, π a π b = f ba π b π b = f ba π b = π a ; let π be such that for ( a, b ) ∈ N ( A ) , π a π b = π a then by definition π is presheafable ( f ba = π a ). (cid:3) Remark . A presheafable collection of projectors over A , is not necessarilysuch that Im π ∈ hom ( A , Gr V ) and a collection of projectors such that Im π ∈ hom ( A , Gr V ) is not necessarily presheafable: let π , π two projectors, π π = π is equivalent to having Im π ⊆ Im π , as one can rewrite π π = π as ( id − π ) π =0; π π = π is equivalent to ker π ⊆ ker π as one can rewrite π π = π as π ( id − π ) = 0 which says that Im( id − π ) ⊆ ker π and Im( id − π ) = ker π . Proposition 2.4.
Let I be a finite set, E = Q i ∈ I E i , P a probability measure on E ; ( π a = E [ | F a ] , a ∈ P ( I )) is such that Im π ∈ hom ( A , Gr V ) and is presheafable.Proof. For any f ∈ M b ( E ) and a, b ∈ A such that b ≤ a , E [ E [ f | F a ] | F b ] = E [ f | F b ] therefore by Proposition 2.3 π is presheafable; for f ∈ F b , E [ f | F a )] = f . ∈ Supp P b ], E [ E [ f | F b ] | F a ] = E [ f | F b ]1[ . ∈ Supp P b ] = E [ f | F b ], therefore byProposition 2.1. (cid:3) Notation . Let A be a poset, let A + = A ⊕ A and theone element poset 1, i.e. any a ∈ A + is in A or is equal to 1 and for any a ∈ A , a ≤ A + Definition 2.4.
Let V be a vector space, A be a finite poset and ( π a , a ∈ A ) acollection of endomorphisms of V . Let U( π ) be the application from A + to Gr V such that for any a ∈ A , U( a ) = Im π ( a ), U(1) = V . ( π a , a ∈ A ) is decomposableif and only if there is a collection of vector subspaces of V , ( S a , a ∈ A + ) such that NTERACTION DECOMPOSITION FOR PRESHEAVES 5 (1) L a ∈ A + i S a V : L a ∈ A + S a → V is an isomorphism; let us recall that for any w =( w a , a ∈ A ) ∈ L a ∈ A + S a , L a ∈ A + i S a V ( w ) = P a ∈ A + w a .(2) for any a ∈ A , π a ( v ) = P b ≤ a w b .We shall note L a ∈ A + i S a V as φ ; we shall say that ( S a , a ∈ A ) is the decompositionof π .For I finite, ( E i , i ∈ I ) a collection of finite sets and P ∈ P ( E ), we shall say that( E [ . | F a ] , a ∈ P ( I )) admits an interaction decomposition if it is decomposable.Let us recall what a decomposable collection of vector spaces is (Definition 2.1[11]). Definition 2.5. ( S a , a ∈ A ) is a decomposition of U ∈ hom( A , Gr ( V )) if andonly if,(1) for all a ∈ A , S a ∈ Gr (U( a )).(2) for all a ∈ A , L b ≤ a i S b U ( a ) : L b ∈ ˆ a S b → U( a ) are isomorphisms and L a ∈ A i S a V is anisomorphism on its image.We shall note i S b U ( a ) as φ a ; we shall say that U is decomposable. Proposition 2.5.
Let π = ( π a , a ∈ A ) be decomposable and ( S a , a ∈ A + ) thedecomposition of π , let for any a ∈ A + , φ a : L b ≤ a S a → U( π )( a ) be such that φ a ( L b ≤ a w b ) = P b ≤ a w b . Then ( φ a , a ∈ A ) is a natural transformation from L S a to G ( π ) and a natural transformation from L S a to F ( π ) , it is also an isomorphism, U( π ) is decomposable and ( π a , a ∈ A ) is presheafable.Proof. For any v ∈ L b ∈ A + S b , and a ∈ A , φ − π a ( φ ( v )) = i L b ≤ a S b L b ∈ A + S b pr L b ∈ A + S b L b ≤ a S b ( v ) therefore φ ( i L b ≤ a S b L b ∈ A + S b L b ≤ a S b ) = U( a ) and φ a = φi L b ≤ a S b L b ∈ A + S b | U( a ) is well defined and is an isomor-phism; from this remark one can conclude that ( φ a , a ∈ A ) is a natural transfor-mation from L S a to G ( π ) and a natural transformation from L S a to F ( π ), it isalso an isomorphism, U( π ) is decomposable. ( π a , a ∈ A ) is presheafable becausepr L b ∈ A + S b L b ≤ a S b is presheafable. (cid:3) Definition 2.6.
Let A be a poset, a, b ∈ A . We shall say that A possesses aninteresection for ( a, b ) when there is d such that, GRÉGOIRE SERGEANT-PERTHUIS ∀ c ∈ A , c ≤ a & c ≤ b = ⇒ c ≤ dd is unique and we shall note it a ∩ b . We shall say that A possesses all itsintersections when it possesses intersections for any couple. Proof.
Let d and d be two intersections for a, b , then d ≤ a , d ≤ b and d ≤ d andby exchanging d and d one gets d ≤ d and therefore d = d . (cid:3) Example . Let I be any set then P ( I ) possesses all its intersections. Proposition 2.6.
Let ( π b , b ∈ A ) be decomposable and ( S b , b ∈ A ) its decomposi-tion; let v ∈ V , w ∈ L b ∈ A + S b such that v = P b ∈ A + w b ; for any a ∈ A + pose s a : V → V as s a ( v ) = w a . For any a, b ∈ A , s a s b = δ a ( b ) s a and, π a π b = X c ≤ ac ≤ b s c If A possesses all its intersections then, (I) ∀ a, b ∈ A π a π b = π a ∩ b Proof.
By definition s a s b = δ a ( b ) s b , π a π b = X d ≤ a s d ( X c ≤ b s c ) = X d ≤ a X c ≤ b δ d ( c ) s c = X c ≤ ac ≤ b s c When A possesses all its intersections π a ∩ b = P c ≤ a ∩ b s c = P c ≤ ac ≤ b s c . (cid:3) Question. (1) When ( I ) holds for a collection of endomorphism of V is thiscollection decomposable?(2) For what P does ( E P [ . | F a ] , a ∈ P ( I )) admit an iteraction decomposition,i.e is decomposable?3. Necessary and sufficient condition for the interactiondecomposition of projectors from a finite poset to
VectDefinition 3.1.
A poset A is locally finite if for any b ≤ a , [ b, a ] = { c ∈ A : b ≤ c ≤ a } is finite. Definition 3.2. (Zeta function) Let A be a locally finite poset, let ζ : L a ∈ A R → L a ∈ A R be such that for any λ ∈ L a ∈ A R and a ∈ A , NTERACTION DECOMPOSITION FOR PRESHEAVES 7 ζ ( λ )( a ) = X b ≤ a λ b ζ is the zeta function of A . Proposition 3.1. (Mobius inversion) The zeta function of a locally fintie poset A is invertible, we shall note µ its inverse and there is f : A × A → Z such that forany λ ∈ P a ∈ A R and a ∈ A , µ ( λ )( a ) = X b ≤ a f ( a, b ) λ b We shall note f as µ A .Proof. By applying Poposition 2 [10]. (cid:3)
Definition 3.3.
Let A be a locally finite poset and M a ( R -)module; let M A = L a ∈ A M . The zeta function of A with values in M , ζ A ( M ) : M A → M A , is suchthat for any m ∈ M A ,(2) ζ A ( M )( m )( a ) = X b ≤ a m a We shall note ζ A ( M ) as ζ A making the reference to M implicit. Proposition 3.2.
Let A be a locally finite poset, M a module, ζ A ( M ) is invert-ible, we shall call its inverse the Mobius function with values in M and note is as µ A ( M ) : M A → M A . Furthermore,for any m ∈ M A and a ∈ A , (3) µ A ( m )( a ) = X b ≤ a µ A ( a, b ) m a Proof.
For any m ∈ M and b ∈ A , ζ A µ A ( m )( b ) = P c ≤ b P a ≤ c µ A ( c, a ) m a and P a P c : a ≤ c ≤ b µ A ( c, a ) m a = P a δ b ( a ) m a = m b and similarly µ A ζ A = id . (cid:3) Definition 3.4.
Let A be a finite poset, let ( π a , a ∈ A ) be a collection of en-domorphism of V . Let Π(( π a , a ∈ A )) : V → V A be such that for any v ∈ V ,Π( v )( a ) = π a ( v ). For any a ∈ A and v ∈ V let s a ( v ) = µ A ◦ Π( v )( a ), as ζ A µ A Π = Π,(4) π a ( v ) = X b ≤ a s b ( v ) GRÉGOIRE SERGEANT-PERTHUIS
Remark . For A locally finite, ( π a ( v ) , a ∈ A ) is in general not in V A , thereforewe decide to restrict our attention to A finite for the moment. Notation . For any poset, let U ( A ) denote the set of lower-sets of A , i.e subsetsof A such that for any a ∈ A , b ∈ B such that a ≤ b one has that a ∈ B . We shallalso call U ( A ) the poset topology of A . For a ∈ A , let ˆ a = { b ∈ A : b ≤ a } ,ˆ a ∈ U ( A ). Proposition 3.3.
Let A a locally finite poset, if B ∈ U ( A ) , then one has thefollowing commutation relations, (5) ζ B pr AB = pr AB ζ A (6) µ B pr AB = pr AB µ A Proof.
Let B ∈ U ( A ), b ∈ B and v ∈ V A , ζ B ( pr AB ( v ))( b ) = X c ∈ B c ≤ b v c = X c ∈ A c ≤ b v c = ζ A ( v )( b )Therefore ζ B pr AB = pr AB ζ A , therefore pr AB = µ B pr AB ζ A and pr AB µ A = µ B pr AB . (cid:3) Definition 3.5.
Let A be a any poset and B ∈ U ( A ),(7) W A ( B ) = { v ∈ V A : ∀ a, c ∈ A , ˆ a ∩ B = ˆ c ∩ B = ⇒ v a = v c } Example . Let A = { , , ′ } where 0 ≤ ≤ ′ ; W ˆ0 = { ( v, v, v ) : v ∈ V } . Proposition 3.4.
Let A be a locally finite poset, for any B ∈ U ( A ) , ζ A ( V B ) ⊆ W A ( B ) . If there is b ∈ A such that B = ˆ b and A has all its intersections ζ A ( V B ) = W B Proof.
Let v ∈ V B , for any a ∈ A , ζ A ( v )( a ) = P c ∈ ˆ a ∩ B v a . Therefore if ˆ a ∩ B = ˆ b ∩ B , ζ A ( v )( a ) = ζ A ( v )( b ).Let u ∈ W A (ˆ b ) and v ∈ V A such that u = ζ A ( v ), for any a ∈ A , ζ A ( v . ∈ ˆ b ])( a ) = P c ∈ ˆ a ∩ ˆ b v c = u a ∩ b = u a = ζ A ( v )( a ); therefore v = v . ∈ ˆ b ]. (cid:3) Remark . Let us remark that ζ A ( V B ) is not equal to W A ( B ) in general; let usconsider A = { a, b, c, d } , with, a ≤ c , a ≤ d , b ≤ c , b ≤ d . Let B = { a, b } , askingfor v ∈ W A ( B ) is the same than asking for v c = v d , which can always be fullfilled:let u = ( s a , s b , s c , s d ) elements in V such that s c = s d = 0, then ζ ( u ) ∈ W A ( B )but u / ∈ V B . NTERACTION DECOMPOSITION FOR PRESHEAVES 9
Definition 3.6.
Let A be any poset that possesses all its intersections, let V bea vector spaces, a collection ( π a ∈ L ( V ) , a ∈ A ) is said to verify the intersectionproperty if,(I) ∀ a, b ∈ A , π a π b = π a ∩ b Example . Let A = { , , ′ } , with 0 ≤
1, 0 ≤ ′ ; let ( V, <, > ) be a Hilbert space,such that there are three closed subspaces S , S , S ′ such that V = S ⊕ ⊥ S ⊕ ⊥ S ′ ;let V = S , V = S ⊕ S , V ′ = S ⊕ S ′ . Let π , π , π ′ be the orthogonal projectionon respectively V , V , V ′ , then π π ′ = π = π ∩ ′ . Proposition 3.5.
Let A be a finite poset that possesses all its intersections, andlet ( π a ∈ L ( V ) , a ∈ A ) that satisfies the intersection property, for any a, b ∈ A , s b π a = 1[ b ≤ a ] s b Proof.
Let us remark that for any b, c ∈ A such that c ≤ b , Π A ◦ π b ( c ) = Π A ( c );therefore for any b ≤ a , s b π a = pr ˆ ab pr A ˆ a µ A Π A π a = pr ˆ ab µ ˆ a pr A ˆ a Π A π a and as noted just before pr A ˆ a Π A π a = pr A ˆ a Π A so s b π a = s b . Furthermore for any b ∈ A , π b π a = π a ∩ b = π a ∩ b π a , therefore ImΠ A π a ⊆ W ˆ a and so s b π a = 1[ b ≤ a ] s b π a and therefore s a π b = 1[ a ≤ b ] s a . (cid:3) Theorem 3.1.
Let A be a finite poset that possesses all its intersections, V avector space and ( π a ∈ L ( V ) , a ∈ A ) a collection that verifies the intersectionproperty (I). For any a, b ∈ A , (8) s a s b = δ a ( b ) s a Proof.
For any a, b ∈ A , s a s b = s a X c ≤ b µ ( b, c ) π c ( v ) = X c ≤ b µ ( b, c ) s a π c ( v ) s a s b = X c ≤ b µ ( b, c )1[ a ≤ c ] s a = s a X a ≤ c ≤ b µ ( b, c )and P a ≤ c ≤ b µ ( b, c ) = δ a ( b ), by definition of µ . (cid:3) Proposition 3.5 simply relies on two remarks: firstly that for any A finite and( π a ∈ L ( V ) , a ∈ A ), if for any b ∈ A , µ A (ImΠ ◦ π b ) ⊆ V ˆ b then for any a, b ∈ A such that b ≤ a , s b π a = 1[ b ≤ a ] s b π a ; secondly that if ( π a , a ∈ A ) is a collection ofprojectors, for any b ≤ a , s b π a = s b , as shown in the proof of 3.5. For this reasonit seems natural to want to redefine the intersection property for a collection ofprojectors of V over any finite poset to be that µ A (ImΠ ◦ π b ) ⊆ V ˆ b ; this would still allow us to get Theorem 3.1. However showing (I) is in pratice much easier thanshowing the following (I’). Definition 3.7. (Intersection property) Let A be a finite poset, we shall say thata collection of projectors of V , ( π a , a ∈ A ), satisfies the intersection property if,(I’) ∀ b ∈ A , µ A (ImΠ ◦ π b ) ⊆ V ˆ b Remark . Condition (I’) is equivalent to asking that for any v ∈ V and any a, b ∈ A , π a π b = P c ∈ ˆ a ∩ ˆ b s c , and for A finite that possesses all its intersections and acollection of projectors, (I) is equivalent to (I’). Theorem 3.2.
Let A be a finite poset, ( π a , a ∈ A ) a collection of projectors of V that verifies the intersection property; for any a, b ∈ A , (9) s a s b = δ a ( b ) s a Proof. As µ A (ImΠ ◦ π b ) ⊆ V ˆ b , for any a ∈ A and v ∈ V , ( s b ( π a ( v )) ∈ V B ,therefore for any b ∈ B , s b π a = 1[ b ≤ a ] s b π a . For any b, c ∈ A such that c ≤ b ,Π A ◦ π b ( c ) = Π A ( c ); therefore for any b ≤ a , s b π a = pr ˆ ab pr A ˆ a µ A Π A π a = pr ˆ ab µ ˆ a pr A ˆ a Π A π a = pr A ˆ a Π A Therefore s b π a = s b ; the end of the proof is exactly the same than the one ofTheorem 3.1. (cid:3) Corollary 3.1. (Intersection property and decomposability)Let A be a finite poset, V a vector space and ( π a , a ∈ A ) a collection of projectorsof V that verifies the intersection property (I’). Let for b ∈ A , S b = Im s b , and let a ∈ A , one has that ζ A | L b ∈ A S b Im π and ζ ˆ a | L b ≤ a S b Im π a are isomorphisms.Proof. For any collection ( π a ∈ L ( V ) , a ∈ A ) one has that Im ζ A ⊆ L b ∈ A S b andfor a ∈ A , Im ζ ˆ a ⊆ L b ≤ a S b ; when ( π a , a ∈ A ) is a collection of projectors thatverifies (I’), from Theorem 3.2, for any b ≤ a , π a s b = P c ≤ a s c s b = s b and S b ⊆ Im π a ,so Im ζ A = L b ∈ A S b and Im ζ ˆ a = L b ≤ a S b . As ζ A , ζ ˆ a are injective (Proposition 3.1), ζ A | L b ∈ A S b Im π and ζ ˆ a | L b ≤ a S b Im π a are isomorphisms. (cid:3) Corollary 3.2.
Let A be a finite poset, V a vector space; a collection of projectionsof V , ( π a , a ∈ A ) , satisfies the intersection property if and only if it is decomposable. NTERACTION DECOMPOSITION FOR PRESHEAVES 11
Proof.
If ( π a , a ∈ A ) is decomposable Proposition 2.6 and Remark 3.3 imply thatit satisfies (I’).For any v ∈ V , let s ( v ) = v − P a ∈ A s a ( v ), for any a ∈ A , s a s ( v ) = s a ( v ) − s a ( v ) =0 = s s a ( v ). Let S = Im s then, ( S b , b ∈ A + ) is a decomposition of U and for any v ∈ V , and a ∈ A + , π a ( v ) = P b ≤ a s a ( v ), where π = id . (cid:3) Corollary 3.3. (Interaction Decomposition for factor spaces)Let I be a finite set, ( E i , i ∈ I ) a collection of finite sets, and P a probability measureon E , ( E a [ . | F a ] , a ∈ P ( I )) is decomposable if and only if P is a product measure,i.e there is ( p i ∈ P ( E i ) , i ∈ I ) such that P = ⊗ i ∈ I p i .Proof. P ( I ) possesses all its intersections; if ( π a = E a [ . | F a ] , a ∈ P ( I )) is decom-posable, then by Corollary 3.2, for any a, b ∈ A , π a π b = π a ∩ b . Therefore for any i ∈ I and f ∈ M b ( E, F ), π { i } π { i } c ( f ) = π ∅ ( f ) = E [ f ], and so P is a productmeasure.Let P be a product measure, then for any a, b ∈ A , π a π b = π a ∩ b and by Theorem3.1 ( E a [ . | F a ] , a ∈ P ( I )) is decomposable. (cid:3) Interaction Decomposition for presheaves in
ModDefinition 4.1.
Let
Split be the subcategory of ( R − ) Mod × ( R − ) Mod op thathas as objects ( M, M ) with M a ( R − )module and for any M, M two modules, Split (( M, M ) , ( M , M )) = { ( s, r ) ∈ Mod ( V, V ) × Mod ( V , V ) : rs = id } . Proof.
Let ( s, r ) ∈ Split (( M, M ) , ( M , M )), ( s , r ) ∈ Split (( M , M ) , ( M , M )), rr s s = id . (cid:3) Remark . Let π , π be the two projections from respectively Split → Mod , Split → Mod op defined as π ( V, V ) = π ( V, V ) = V for V and object of Split and for a morphism of
Split , π ( s, r ) = s , π ( s, r ) = r . Any functor H from aposet A to Mod × Mod op defines a couple of functor/presheaf ( π H, π H ) andfor any couple of functor/presheaf ( G, F ) there is a unique a functor from A to Mod × Mod op , H , such that π H = G , π H = F ; similarly any functor H froma poset A to Split defines a couple of functor/presheaf ( π H, π H ), any couple( G, F ) of functor/presheaf from A to Mod defines a functor from A to Split ifand only if for any a, b ∈ A such that b ≤ a , F ab G ba = id . From now on when werefer to a functor from A → Split we shall refer to its couple ( π H, π H ). Remark . Let (
G, F ), ( G , F ) be two functors from A to Split and φ ∈ Split A (( G, F ) , ( G , F )); for any a, b ∈ A such that b ≤ a , φ a G ba = G ba φ b and F ab φ a = φ b F ab , in other words π ⋆ φ is a morphism from G to G and π ⋆ φ from F to F . Proposition 4.1.
Let ( G, F ) , ( G , F ) be two functors from A to Split and φ :( G, F ) → ( G , F ) be a natural transformation. Let Im φ = ( Im π ⋆ φ, Im π ⋆ φ ) ,Im φ is a functor from A to Split .Proof.
Let g = Im π ⋆ φ , f = Im π ⋆ φ , for any a, b ∈ A such that a ≥ b , and v ∈ g ( a ), f ab g ba ( v ) = F ab G ba ( v ) = v . (cid:3) Proposition 4.2.
Let I be any set and (( G i , F i ) , i ∈ I ) a collection of functorsfrom A to Split ; ( L i ∈ I G i , L i ∈ I F i ) and ( Q i ∈ I G i , Q i ∈ I F i ) are functors from A to Split .Proof.
Let g = Q i ∈ I G i , f = Q i ∈ I F i , let a, b ∈ A such that b ≤ a , let v ∈ Q i ∈ I G i ( a ), forany j ∈ I , f ab g ba ( v )( j ) = F jab G jba ( v j ) = v j . (cid:3) Proposition 4.3.
Let ( G, F ) be a functor from a poset A to Split , for any a ≥ b ≥ c , F ab G ca = G cb Proof. F ab G ca = F ab G ba G cb = G cb . (cid:3) Proposition 4.4.
Let ( G, F ) be a functor from A to Split and ( G , F ) a functorfrom A to Mod × Mod op ; if there is an epimorphism φ from ( G, F ) to ( G , F ) then ( G , F ) is a functor from A to Split .Proof.
Let a, b ∈ A such that b ≤ a , one has that F ab G ba φ b = φ b F ab G ba = φ b andas φ b is an epimorphism, F ab G ba = id . (cid:3) Definition 4.2.
Let A be any poset and G : A → Mod be a functor. For any a, b, c ∈ A , let G a ≤ . ]( b ) = G ( b ) if b ≥ a and otherwise G a ≤ . ]( b ) = 0; for a ≤ c ≤ b , G a ≤ . ] cb = G cb if a c and c ≤ b , G a ≤ . ] cb = 0 (as 0 is initial in Mod ). G a ≤ . ] is a functor from A to Mod .Similarly let F be a presheaf from A to Mod , let for any a, b ∈ A and a ≤ b , F a ≤ . ]( b ) = F ( b ) and otherwise F a ≤ . ]( b ) = 0, for a ≤ c ≤ b , F a ≤ . ] bc = F bc . F a ≤ . ] is a presheaf. Proof.
Let a, b, c, d ∈ A such that d ≤ c ≤ b , if a ≤ d then G a ≤ . ] cb G a ≤ . ] dc = G cb G dc = G db = G a ≤ . ] db and F a ≤ . ] cd F a ≤ . ] bc = F cd F bc = F a ≤ . ] bd ; if a d then G a ≤ . ] cb G a ≤ . ] dc = G a ≤ . ] cb G a ≤ . ] db , F a ≤ . ] cd F a ≤ . ] bc = NTERACTION DECOMPOSITION FOR PRESHEAVES 13 F a ≤ . ] bc = F a ≤ . ] bd . (cid:3) Notation . Let M be an object of Mod and let Gp ( M ) be the groupoid thathas as object M and as morphisms the isomorphisms of Mod ( M, M ). Definition 4.3. (Decomposable) Let H be a functor from A to Split and let(( G a , F a ) , a ∈ A ) be a collection of functors from A to Mod × Mod op such thatfor any a ∈ A there is an module M a such that ( G a , F a ) is a functor from A to Gp ( M ) × Gp ( M ). We shall say that H is decomposable if there is such a collection(( G a , F a ) , a ∈ A ) for which H is isomorphic to ( Q a ∈ A G a a ≤ . ] , Q a ∈ A F a a ≤ . ]). When H is decomposable we shall call ( Q a ∈ A G a a ≤ . ] , Q a ∈ A F a a ≤ . ]) itsdecomposition and note it as ( Q a ∈ A S a , Q a ∈ A S a ). Corollary 4.1.
Let H be a decomposable functor from A to Split and ( Q a ∈ A S a , Q a ∈ A S a ) its decomposition, ( Q a ∈ A S a , Q a ∈ A S a ) is a functor from A to Split and for any a ∈ A , ( S a , S a ) is too.Proof. By Proposition 4.4, ( Q a ∈ A S a , Q a ∈ A S a ) is a functor from A to Split . Let a, b, c ∈ A such that c ≤ b and v ∈ Q d ∈ A S d ( c ), Q d ∈ A S dbc Q d ∈ A S dcb ( v )( a ) = v a = S abc S acb ( v a ). (cid:3) Remark . Let H be a decomposable functor from A to Split and ( Q a ∈ A S a , Q a ∈ A S a )its decomposition, for any a, b, c ∈ A such that c ≤ b , S acb − = S abc . Proposition 4.5.
Let A be a poset and ( G, F ) a functor from A to Split , for any a, b, c ∈ A such that c ≤ b ≤ a , (10) F bc ( kerG ba ) ⊆ ker G ca F ab ( Im G ca ) = Im G cb Proof.
For any a, b, c ∈ A such that c ≤ b ≤ a , F ab G ba = id , therefore ker G ba = 0; F ab G ca = G cb (Proposition 4.3) therefore F ab (Im G ca ) = Im G cb . (cid:3) Definition 4.4.
Let A be a poset and ( G, F ) a functor from A to Split , let R ( α, a ) = Im G aα , let α ≥ b ≥ c , let us call R αbαc : Im G bα → Im G cα the uniquemorphism that satisfies R αbαc G bα | R ( α,b ) = G cα | R ( α,c ) F bc and for α ≥ β ≥ a , let R αaβa : R ( α, a ) → R ( β, a ) be such that R αaβa = F αβ | R ( β,a ) R ( α,a ) .For α ≥ β ≥ a ≥ b , R ( α, b ) ⊆ R ( α, a ), we shall note the inclusion as L αbαa ; let L βaαa = G βα | R ( α,a ) R ( β,a ) .We shall call L ( G, F ) the left coupling of (
G, F ) and R ( G, F ) its right coupling.
Proof.
By Proposition 4.5 R is well defined. For any a ≥ b ≥ c , G ba G cb = G ca ,therefore R ( a, c ) = Im G ca ⊆ Im G ba = R ( a, b ), and G ba (Im G cb ) = Im G ca and L is welldefined. (cid:3) Remark . Let (
G, F ) be a functor from A to Split , for any α ∈ A , G | ˆ α in-duces a functor monomorphism G ˆ αα : G | ˆ α → G ( a ) and a presheaf monomorphism G ˆ ααF : F ˆ α → F ( α ); L ( α, . ) = Im G ˆ αα , R ( α, . ) = Im G ˆ ααF . Furthermore, G ˆ αα | L ( α,. ) and G ˆ ααF | R ( α,. ) are isomorphisms. Proposition 4.6.
Let ( G, F ) be a functor from A to Split , L its left coupling, R its right coupling; for any α, β, a, b ∈ A such that α ≥ β ≥ a ≥ b , (11) L βaαa L βbβa = L αbαa L βbαb (12) R βaβb R αaβa = R αbβb R αaαb Proof.
Let α ≥ β ≥ a ≥ b , for any v ∈ R ( β, b ), L βaαa L βbβa ( v ) = G βα ( v ) = L αbαa L βbαb ( v ).For v ∈ F ( a ), v ∈ F ( b ), R αaβa G aα ( v ) = F αβ G aα ( v ) = G aβ ( v ) and R αbβb G bα ( v ) = G bβ ( v ); R βaβb R αaβa G aα ( v ) = R βaβb G aβ ( v ), R αbβb R αaαb G aα ( v ) = R αbβb G bα F ab ( v ) = G bβ F ab ( v ), byconstruction G bβ F ab ( v ) = R βaβb G aβ .(13) F ( a ) R ( α, a ) R ( β, a ) F ( b ) R ( α, b ) R ( β, b ) G aα | R ( α,a ) R αaβa G bα | R ( α,b ) R αbβb F ab R αaαb R βaβb (cid:3) Definition 4.5.
Let A be the subposet of A × A constituted of couples ( α, a )such that a ≤ α . NTERACTION DECOMPOSITION FOR PRESHEAVES 15
Proposition 4.7.
Let A be a poset, C be any category; let M = { (( α, a ) , ( α, b )) :( α, a ) , ( α, b ) ∈ A and a ≥ b } , M = { (( α, a ) , ( β, a )) : ( α, a ) , ( β, a ) ∈ A and α ≥ β } , and let ( G ij ; i, j ∈ M ∪ M : i ≤ j ) be such that for any ( α, a ) , ( α, b ) , ( α, c ) ∈ A such that ( α, a ) ≥ ( α, b ) ≥ ( α, c ) , G αbαa G αcαb = G αcαa for any ( α, a ) , ( β, a ) , ( γ, a ) such that ( α, a ) ≥ ( β, a ) ≥ ( γ, a ) , G βaαa G γaβa = G γaαa and for any ( α, a ) , ( α, b ) , ( β, a ) , ( β, b ) such that ( α, a ) ≥ ( α, b ) ≥ ( β, b ) and ( α, a ) ≥ ( β, a ) ≥ ( β, b ) , i.e α ≥ β ≥ a ≥ b , G βaαa G βbβa = G αbαa G βbαb Then G extends into a unique functor G : A → C , we shall also denote thisextension as G .Proof. Let us remark that for any ( α, a ) , ( β, b ) ∈ A such that ( α, a ) ≥ ( β, b ),then ( α, a ) ≥ ( α, b ) ≥ ( β, b ). Let G : A → C be a functor such that for any( α, a ) ≥ ( α, b ), G αaαb = G αaαb and for any ( α, a ) ≥ ( β, a ), G βaαa = G βaαa , then forany ( α, a ) ≥ ( β, b ), G βbαa = G αbαa G βbαb . Therefore there can be only one functor thatextends G to A .Let for any ( α, a ) ≥ ( β, b ), G βbαa = G αbαa G βbαb .(14) G ( β, b ) G ( α, b ) G ( α, a ) G βbαb G αbαa G βbαa For any ( α, a ) ≥ ( β, b ) ≥ ( γ, c ), G βbαa G γcβb = G αbαa G βbαb G βcβb G γcβc = G αbαa G αcαb G βcαc G γcβc = G αcαa G γcαc (15) G ( γ, c ) G ( β, c ) G ( α, c ) G ( β, b ) G ( α, b ) G ( α, a ) G γcβc G βcαc G βcβb G βbαb G αcαb G αbαa G γcβb G βbαa (cid:3) Remark . Let A be a poset, if one applies Proposition 4 . A op then oneextends G to a presheaf. Corollary 4.2.
Let ( G, F ) be a functor from A to Split , let ( L, R ) be its left andright coupling; ( L, R ) has a unique extension into a functor from A to Split .Proof.
By construction for any ( α, a ) , ( β, a ) , ( γ, a ) ∈ A such that ( α, a ) ≥ ( β, a ) ≥ ( γ, a ), L βaαa L γaβa = L γaαa , and for any ( α, a ) , ( α, b ) , ( α, c ) ∈ A such that ( α, a ) ≥ ( α, b ) ≥ ( α, c ), L αbαa L αcαb = L αcαa . Therefore by Proposition 4.6 and Proposition 4.7, L extends into a unique functor from A to Mod .By construction for any ( α, a ) , ( β, a ) , ( γ, a ) ∈ A such that ( α, a ) ≥ ( β, a ) ≥ ( γ, a ), R βaγa R αaβa = R αaγa . Let ( α, a ) , ( α, b ) , ( α, c ) ∈ A such that ( α, a ) ≥ ( αb ) ≥ ( αc )and v ∈ G ( a ), R αbαc R αaαb G aα ( v ) = R αbαc G bα F ab ( v ) = G cα F ac ( v ) = R αaαc G aα ( v ); as G aα issurjective, R αbαc R αaαb = R αaαc . Therefore by Proposition 4.6 and Proposition 4.7, R extends into a unique presheaf from A to Mod .For α ≥ a ≥ b and v ∈ G ( b ), R αaαb L αbαa G bα ( v ) = R αaαb G aα G ba ( v ) = G bα F ab G ba ( v ) = G bα ( v ), as G bα is surjective, R αaαb L αbαa = id ; for α ≥ β ≥ a and v ∈ R ( β, a ), R αaβa L βaαa ( v ) = F αβ G βα ( v ) = v ; therefore for any ( α, a ) ≥ ( β, b ), R αaβb L βbαa = R αbβb R αaαb L αbαa L βbαb = R αbβb L βbαb = id . (cid:3) Remark . For (
G, F ) a functor from A to Split , and ( α, a ) ∈ A , L aaαα = G aα , R ααaa = F αa . Indeed, let us recall that R ( a, a ) = F ( a ), R αααa = L aaαa F αa and R ααaa = R αaaa R αααa = R αaaa L aaαa F αa ; R αaaa = F αa | R ( α,a ) = F αa L αaαα ; so R ααaa = F αa L αaαα L aaαa F αa = F αa L aaαα F αa = F αa G aα F αa = F αa . Corollary 4.3.
Let ( G, F ) be a functor from A to Split , for any α, β, a ∈ A suchthat α ≥ β ≥ a , L βaαa is an isomorphism, its inverse is R αaβa .Proof. For any α ≥ β ≥ a , L βaαa = G βα | Im G aα Im G aβ and as G βα is injective and G βα G aβ = G aα ,Im L βaαa = Im G aα = L ( α, a ); therefore L βaαa is an isomorphism. Furthermore byCorollary 4.2 R αaαb is the inverse of L αbαa . (cid:3) Definition 4.6.
Let A be any poset, let ( G, F ) be a functor from A to Split . Forany ( α, a ) ∈ A , let V ( α, a ) = Q b ≤ a G ( α ) (which in the previous section we wouldnote as G ( α ) ˆ a ). For any α, β, a, b such that α ≥ β ≥ a ≥ b let V rαaαb = pr Q c ∈ ˆ a G ( α ) Q c ∈ ˆ b G ( α ) , V lαbαa = i Q c ∈ ˆ b G ( α ) Q c ∈ ˆ a G ( α ) , V rαaβa : V ( α, a ) → V ( β, a ) be such that for any v ∈ V ( α, a ) and c ≤ NTERACTION DECOMPOSITION FOR PRESHEAVES 17 a , V rαaβa ( v )( c ) = F αβ ( v c ), V lβaαa : V ( β, a ) → V ( α, a ) be such that for any v ∈ V ( β, a )and c ≤ a , V lαaβa ( v )( c ) = G βα ( v c ). Proposition 4.8.
Let A , ( G, F ) be a functor from A to Split , ( Q a ∈ A G a ≤ . ] , Q a ∈ A F a ≤ . ]) is a functor from A to Split . ( V l , V r ) extends into a uniquefunctor form A to Split .Proof.
Let b, c ∈ A such that c ≤ b , let us note Q a ∈ A F a ≤ . ] bc as F and Q a ∈ A G a ≤ . ] cb as G ; for any v ∈ Q a ∈ A G a ≤ . ]( c ), and a ∈ A , F G ( v )( a ) = F bc G cb ( v a )1[ a ≤ b ]1[ a ≤ c ] = v a a ≤ c ] = id Q a ∈ A G a ≤ . ]( c ) ( v )( a )( ≃ id Q a ∈ A a ≤ c G ( c ) ( v )( a )) (this also showsthat ( G a ≤ . ] , F a ≤ ]) is a functor from A to Split ).For α ≥ a ≥ b ≥ c , V lαbαa V lαcαb = V lαcαa , V rαbαc V rαaαb = V rαaαc . For any α ≥ β ≥ γ ≥ a ,any v ∈ V ( α, c ), any c ≤ a , V lβaαa V lγaβa ( v )( c ) = G βα G γα ( v c ) = G γα ( v c ) = V lγaαa ( v )( c )and for any v ∈ V ( α, a ), and c ≤ a , V rβaγa V rαaβa ( v )( c ) = F βγ F αβ ( v c ) = F αγ ( v c ) = V rαaγa ( v )( c ).Let α ≥ β ≥ a ≥ b , v ∈ V ( β, b ); for any c ≤ a , V lβaαa V lβbβa ( v )( c ) = G βα ( V lβbβa ( v )( c )) = G βα ( v c c ≤ b ]) = G βα ( v c )1[ c ≤ b ] and V lαbαa V lβbαb ( v )( c ) = V lβbαb ( v )( c )1[ c ≤ b ] = G βα ( v c )1[ c ≤ b ]; similarly one has that V rαbβb V rαaαb = V rβaβb V rαaβa . Therefore by Propo-sition 4.7 V l extends to a functor from A to Mod and V r to a presheaf from A to Mod .For any α ≥ β ≥ a , v ∈ V ( β, a ), c ≤ b , V rαaβa V lβaαa ( v )( c ) = F αβ G βα ( v c ) = v c ;for any α ≥ a ≥ b , V rαaαb V lαbαa = id , therefore for any ( α, a ) ≥ ( β, b ), V rαaβb V lβbαa = V rαbβb V rαaαb V lαbαa V lβbαb = id . (cid:3) Until now in this subsection there was no constraint on A , in order to be ableto define ζ ˆ α ( G ( α )) on V ( α, α ) we will have to assume that ˆ α is finite for any α ∈ A . Notation . The class of posets that are such that ˆ a is finite for any a ∈ A willbe denoted as ˆ P f . Remark . Let A ∈ ˆ P f and ( G a , a ∈ A ) a collection of functors from A to Mod , Q a ∈ A G a a ≤ . ] = L a ∈ A G a a ≤ . ], indeed for any b ∈ A , Q a : a ≤ b G a ( b ) = L a : a ≤ b G a ( b ). Proposition 4.9.
Let ( G, F ) be a functor from A ∈ ˆ P f to Split , for any ( α, a ) ∈ A , let ζ ( α, a ) = ζ ˆ a ( G ( α )) : V ( α, a ) → V ( α, a ) and µ ( α, a ) = µ ˆ a ( G ( α )) : V ( α, a ) → V ( α, a ) ; ζ , µ ∈ Mod A op ( V r , V r ) . Lemma 4.1.
Let V , V be two modules, A a finite poset, and l : V → V a linearapplication; let L : V A → V A be such that L ( v )( a ) = l ( v a ) . ζ A ( V ) L = Lζ A ( V ) and µ A ( V ) L = Lµ A ( V ) .Proof. For any v ∈ V A , a ∈ A , L ( ζ A ( V )( v ))( a ) = l ( P b ≤ a v b ) = P b ≤ a l ( v b ) = ζ A ( V )( L ( v ))( a ).Futhermore µ A ( V ) ζ A ( V ) Lµ A ( V ) = µ A ( V ) Lζ A ( V ) µ A ( V ) so µ A ( V ) L = Lµ A ( V ). (cid:3) Proof.
Proof of Proposition 4.9.For any α ≥ a ≥ b , by Proposition 3.3 ζ ( α, b ) V rαaαb = V rαaαb ζ ( α, a ). By Lemma3.3, for any α ≥ β ≥ a , ζ ( β, a ) V rαaβa = V rαaβa ζ ( α, a ). Therefore for any ( α, a ) , ( β, b ) ∈ A such that ( α, a ) ≥ ( β, b ), V rαaβb ζ ( α, a ) = V rαbβb V αaαb ζ ( α, a ) = V rαbβb ζ ( α, b ) V rαaαb = ζ ( β, b ) V rαbβb V rαaαb . Therefore V rαaβb µ ( α, a ) = µ ( β, b ) V rαaβb . (cid:3) Remark . ζ , µ are in general not natural transformations from V l to V l becausefor α ≥ a ≥ b , V lαbαa ζ ( α, b ) = ζ ( α, a ) V lαbαa . Notation . For any α ∈ A , we shall note ζ ( α, α ) as ζ α . Definition 4.7. (Intersection property) Let (
G, F ) be a functor from A to Split .For any ( α, a ) ∈ A , let π αααa = L αaαα R αααa , π αααa is a projector. For a given α , we shalldenote this collection as π α .( G, F ) is said to satisfy the intersection property for α ∈ A if π α satisfies theintersection property (I’) and is said to satisfy the intersection property if is satisfiesit for any α ∈ A . Proof.
As (
L, R ) is a functor from A to Split (Corollary 4.2) π αααa = L αaαα R αααa L αaαα R αααa = L αaαα R αααa . (cid:3) Remark . Let (
G, F ) be a functor from A to Split ; let ( α, a ) ∈ A , let us remarkthat π αααa = L αaαα L aaαa F αa = L aaαα F αa = G aα F αa . Therefore the intersection propertyis equivalent to for any ( α, a ) , ∈ A , for any v ∈ G ( α ), Im( G bα F αb G aα F αa , b ∈ ˆ α ) ⊆ Im ζ α V lαaαα . If A has all its intersections, the intersection property is equivalent tofor any ( α, a ) , ( α, b ) ∈ A , G bα F αb G aα F αa = G b ∩ aα F αb ∩ a . Remark . Let (
G, F ) be a functor from A to Split , for any α ∈ A , ( π α,αα,a , a ∈ ˆ α )is presheafable as for any v ∈ R ( α, α ) and ( α, a ) , ( α, b ) ∈ A , such that b ≤ a , π αααb ( v ) = π αaαb π αααa ( v ). Remark . Let (
G, F ) be a functor from A to Split , for any ( α, a ) ∈ A ,Im π αααa = R ( α, a ). Proposition 4.10.
Let ( G, F ) be a functor from A to Split , if ( G, F ) verifies theintersection property for α then it verifies it for any β ≤ α . NTERACTION DECOMPOSITION FOR PRESHEAVES 19
Proposition 4.11.
Let ( G, F ) be a functor from A to Split , for any ( α, a ) ∈ A let us denote R | ( α, ˆ a ) as R α, ˆ a ; ( R αaαb , b ≤ a ) is a natural transformation from R ( α, a ) to R α, ˆ a , let us denote φ ( α, a ) : R ( α, a ) → lim b R α, ˆ a ( b ) its limit. One hasthat (lim b R α, ˆ a ( b ) , ( α, a ) ∈ A ) is a subobject of V r that we shall note as M and φ : R → M is a natural transformation, furthermore it is an isomorphism.Proof. By definition lim b R α, ˆ a ( b ) ⊆ V ( α, a ); let α, a, b, c, c ∈ A such that α ≥ a ≥ b ≥ c ≥ c and v ∈ lim c R α, ˆ a ( c ), one has that R αcαc ( V rαaαb ( v )( c )) = R αcαc ( v c ) = v c .Let α ≥ β ≥ a ≥ b ≥ b and v ∈ lim c R α, ˆ a ( c ), one has that V rαaβa ( v )( b ) ∈ R ( β, b ) and R βbβb ( V rαaβa ( v )( b )) = R βbβb F αβ ( v b ) = R βbβb R αbβb ( v b ) = R αb βb R αbαb ( v b ) = R αb βb ( v b ) = F αβ ( v b ) and therefore R βbβb ( V rαaβa ( v )( b )) = V rαaβa ( v )( b ). For any ( α, a ) , ( β, b ) ∈ A such that ( α, a ) ≥ ( β, b ), V rαaβb (lim c R α, ˆ a ( c )) = V rαbβb V rαaαb (lim c R α, ˆ a ( c )) ⊆ V rαbβb (lim c R α, ˆ b ( c )) ⊆ lim c R β, ˆ b ( c )).Let α ≥ a ≥ b ≥ c and v ∈ R ( α, a ), M αaαb φ ( α, a )( v )( c ) = R αaαc ( v ) = R αbαc R αaαb ( v ) = φ ( α, b ) R αaαb ( v )( c ). For α ≥ β ≥ a ≥ b , M αaβa φ ( α, a )( v )( b ) = F αβ ( R αaαb ( v )) = R βaβb R αaβa ( v ) = φ ( α, a ) R αaβa ( v )( b ). Therefore for any ( α, a ) , ( β, b ) ∈ A such that ( α, a ) ≥ ( β, b ), M αaβb φ ( α, a ) = φ ( β, b ) R αaβb .For any ( α, a ) ∈ A , ( α, a ) is initial in ( α, ˆ a ), therefore φ ( α, a ) is an isomorphism.therefore so is φ . (cid:3) Proposition 4.12.
Let ( G, F ) be a functor from A ∈ ˆ P f to Split , if ( G, F ) satis-fies the intersection property, j = µi MV r φ ∈ Split A (( L, R ) , ( V l , V r )) is a monomor-phism.Proof. Let ( α, a ) ∈ A and v ∈ R ( α, a ), for any b ≤ a , j ( α, α ) L αaαα ( v )( b ) = µ ( α, α )( R αααc ( v ) , c ≤ α )( b ) = µ ( α, a )( R αaαc ( v ) , c ≤ a )( b ) as µ is a endomorphismof V r . Furthermore as R ( α, a ) = Im π αααa , φ ( α, α ) L αaαα ( v ) = φ ( α, α )( v ) ∈ Im ζ α V lαaαα ,therefore for any b a , µ ( α, α )( φ ( α, α )( v ))( b ) = 0 and so j ( α, α ) L αaαα = V lαaαα j ( α, a ).Therefore j ( α, α ) L αaαα = V lαaαα j ( αa ).Let b ≤ a , V lαaαα j ( α, a ) L αbαa = j ( α, α ) L αbαα = V lαbαα j ( α, b ) = V lαaαα V lαbαa j ( α, b ). V lαaαα is a monomorphism therefore, j ( α, a ) L αbαa = V lαbαa j ( αb ).For α ≥ β ≥ a , j ( α, a ) L βaαa = V lβaαa j ( β, a ) holds even if ( G, F ) does not satisfythe intersection property. φ is a isomorphism from R to M , i MV r is a monomorphism and µ is an isomorphismfrom V r to V r therefore j is a monomorphism. (cid:3) Proposition 4.13.
Let ( G, F ) be a functor from A to Split , let b ∈ A . For any ( α, a ) ∈ A if a ≥ b let G b ( α, a ) = G ( α ) and if not G b ( α, a ) = 0 ; let α ≥ a ≥ a , if a ≥ b , let G bαaαa = G bαa αa = id , if not G bαaαa = G bαa αa = 0 , let α ≥ β ≥ a ,if a ≥ b let G bαaβa = F αβ and G bβaαa = G βα , if not, G bαaβa = 0 = G bβaαa . ( G b , G b ) extend into a unique functor from A to Split . For any ( α, a ) ∈ A , if a ≥ b let pr b ( α, a ) : V ( α, a ) → G ( α ) be such that for any v ∈ V ( α, a ) , pr b ( v ) = v b , if a b , pr b ( v ) = 0 . pr b ∈ Split A (( V l , V r ) , ( G b , G b )) .Proof. Let b ∈ A , α ∈ A , G b | ( α, ˆ α ) = G ( α )1[( α, b ) ≤ . ] (if b α ), 1[( α, b ) ≤ . ] = 0as the relation is taken in A ), and G b | ( α, ˆ α ) op = G ( α )1[ a ≤ . ], therefore for α ≥ a ≥ a ≥ a , G bαa αa G bαa αa = G bαa αa , G bαa αa G bαaαa = G bαaαa .Let α ≥ β ≥ γ ≥ a , if a ≥ b , G bβaαa G bγaβa = G βα G γβ = G bγaαa , G bβaγa G bαaβa = F βγ F αβ = G bαaγa , and if a b , G bβaαa G bγaβa = 0 = G bγaαa , G bβaγa G bαaβa = 0 = G bαaγa .Furthermore for α ≥ β ≥ a ≥ c , if c ≥ b , G bβaαa G bβcβa = G βα = G bαcαa G bβcαb , G bβaβc G bαaβa = F αβ = G bαcβc G bαaαc . Therefore by Proposition 4.7, G a , G a extend respec-tively to a functor and a presheaf from A to Mod .Let α ≥ a ≥ a , if b ≤ a , G bαaαa G bαa αa = id if b a , G bαaαa G bαa αa : 0 → G bαaαa G bαa αa = id ; let α ≥ β ≥ a , G bα,aβa, G bβ,aα,a = id ; therefore ( G b , G b ) is afunctor from A to Split .Let α ≥ a ≥ a , if b ≤ a , pr b ( α, a ) V lαa αa = pr b ( α, a ) = G bαa αa pr b ( α, a ), pr b ( α, a ) V rαaαa = pr b ( α, a ) = G bαaαa pr b ( α, a ), and if b a , for any v ∈ V l ( α, a ), G bαa αa pr b ( α, a )( v ) = 0 = v b b ≤ a ] = pr b ( α, a ) V lαa αa and pr b ( α, a ) V rαaαa =0 = G bαaαa pr b ( α, a ). For α ≥ β ≥ a , if b ≤ a , pr b ( α, a ) V lβaαa = G βα pr b ( β, a ) = G bβaαa pr b ( β, a ) and pr b ( β, a ) V rαaβa = F αβ pr b ( α, a ) = G bαaβa pr b ( α, a ); if b a , pr b ( β, a ) V rαaβa =0 = G bαaβa pr b ( α, a ) and pr b ( α, a ) V lβaαa = 0 = G bβaαa pr b ( β, a ). (cid:3) Definition 4.8.
Let (
G, F ) be a functor from A ∈ ˆ P f to Split . For any a ∈ A let ( S a , S a ) = Im pr a ◦ j : A → SplitTheorem 4.1.
Let ( G, F ) a functor from A ∈ ˆ P f to Split that satisfies theintersection property, j | Im j ( L, R ) → ( L a ∈ A S a , L a ∈ A S a ) is an isomorphism. We shallnote j | Im j as ψ .Proof. From Proposition 4.12 j is a monomorphism furthermore from Corollary 3.1for any ( α, a ) ∈ A , Im j ( α, a ) = L b ≤ a S b ( α, a ) = L b ∈ A S b ( α, a ). (cid:3) Corollary 4.4.
Let ( G, F ) be a functor from A ∈ ˆ P f to Split , ( G, F ) statisfiesthe intersection property if and only if ( G, F ) is decomposable.Proof. Let (
G, F ) be a functor from A ∈ ˆ P f to Split that satisfies the intersec-tion property; let us recall that (
L, R ) is a functor from A to Split (Corollary 4.2)and that for any α ≥ β ≥ a , L βaαa is an isomorphism. Let us recall that for any NTERACTION DECOMPOSITION FOR PRESHEAVES 21 α ≥ β ≥ a , L b ∈ A S bβaαa ψ ( β, a ) = ψ ( α, a ) L βaαa , where ψ ( α, a ), ψ ( β, a ), L αaαa are isomor-phisms; therefore L b ∈ A S bβaαa is an isomorphism and as ( L b ∈ A S b , L b ∈ A S b ) is a functorfrom A to Split its inverse is L b ∈ A S bαaβa and S aβaαa is an isomorphism (its inverse is S aαaβa ); for α ≥ a ≥ a ≥ b , S aαa αa is also an isomorphism.Let us remark that { ( a, a ) ∈ A : a ∈ A } is isomorphic to A ; let for any a ∈ A , C a = S a | A and C a = S a | A ; by definition ( C a , C a ) is a functor from A to Split ;for any a ≥ a ≥ a , C aa a = S aa a a ,a = S aa a a a S aa a a a = S aa a a a is an isomorphism.Furthermore ( G, F ) ∼ = ( L, R ) | A (Remark 4.6), therefore ( G, F ) is decomposableand its decomposition is ( L a ∈ A C a , L a ∈ A C a ).Let A ∈ ˆ P f , ( G, F ) = ( L a ∈ A C a , L a ∈ A C a ), let α ∈ A , for any a ≤ α and c ≤ α and v ∈ G ( α ), π αa ( v )( c ) = G aα F αa ( v )( c ) = v c c ≤ a ]. Let us denote π α simplyas π and µ ˆ α as µ ; for any v ∈ ImΠ( π ) and a ∈ ˆ α , µ ( v )( a ) = P b ≤ a µ ( a, b ) L c ≤ b v c = L c ≤ a P b : c ≤ b ≤ a µ ( a, b ) v c = v a × a .Furthermore for a ≤ α and b ≤ α and c ≤ α and v ∈ G ( α ), π a π b ( v )( c ) = v c c ≤ a & c ≤ b ], π a π b ( v ) = P c ≤ ac ≤ b v c × c = P c ≤ ac ≤ b µ ( v )( c ). Therefore ( G, F ) satisfies theintersection property. (cid:3) Conlusion
Several applications of Theorem 4.1 can be considered, for example one can givea process that extends the Gram-Schmidt process from N to any finite poset, andif one extends Theorem 4.1 to the case of separable Hilbert spaces then one getsthis process for separable Hilbert spaces and countable posets. Notations (1) For I any set we shall denote P ( I ) the set of subsets of I .(2) For A a poset, A + is the poset sum of A and the one element poset 1.(3) The class of posets that are such that ˆ a is finite for any a ∈ A will bedenoted as ˆ P f .(4) k − Vect is the category of k -vector spaces, R − Mod is the category of R modules; if k and R are implicit, we shall simply note them as Vect and
Mod .(5) Let B be a subset of A , we shall note its complementary B c .(6) Let C be a locally small category, for any two object, A, B , of C we shallnote hom C ( A, B ) as C ( A, B ).(7) A collection of element over a set I shall be noted as ( x i , i ∈ I ),( x i ; i ∈ I )or ( x i ∈ X i , i ∈ I ) if one wants to precise that for any i ∈ I , x i ∈ X i . (8) Let A, B , B, C be sets such that B ⊆ B , f : A → B , g : B → C be twoapplications, for any v ∈ A , we shall note gf ( v ) = g ( f ( v )).(9) Let I be any set, ( V i , i ∈ I ) a collection of modules over a fixed ring R and v ∈ Q i ∈ I V i such that for any i = j , v ( i ) = 0, we shall denote v as v j × j .(10) Let C be any category, A , A , B , B object of C with i : A → A a subojectof A and j : B → B a suboject of B . Let φ : A → B be morphism whenit exists we shall note φ | B A the unique morphisms such that φi = jφ | B A .(11) Let C , C , C be three categories, let F, G : C → C be two functor, H : C → C a functor and φ : F → G a natural transformation. We shallnote H ⋆ φ their whiskering (Appendix A Definition A.3.5 [5]).
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