# Partial Inner Product Spaces: Some Categorical Aspects

Authors: Jean-Pierre Antoine; Dominique Lambert; Camillo Trapani
Journal: Advances in Mathematical Physics
 (2011)

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## Abstract

We make explicit in terms of categories a number of statements from the theory of partial inner product spaces (PIP-spaces) and operators on them. In particular, we construct sheaves and cosheaves of operators on certain PIP-spaces of practical interest.


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## Body

## 1. Motivation

Partial inner product spaces (PIP-spaces) were introduced some time ago by Grossmann and one of us (JPA) as a structure unifying many constructions introduced in functional analysis, such as distributions or generalized functions, scales of Hilbert or Banach spaces, interpolation couples, and so forth [1–4]. Since these structures have regained a new interest in many aspects of mathematical physics and in modern signal processing, a comprehensive monograph was recently published by two of us [5], as well as a review paper [6].Roughly speaking, aPIP-space is a vector space equipped with a partial inner product, that is, an inner product which is not defined everywhere, but only for specific pairs of vectors. Given such an object, operators can be defined on it, which generalize the familiar notions of operators on a  Hilbert space, while admitting extremely singular ones.Now, in the previous work, many statements have a categorical “flavor”, but the corresponding technical language was not used; only some hints in that direction were given in [7]. Here we fill the gap and proceed systematically. We introduce the category PIP of (indexed) PIP-spaces, with homomorphisms as arrows (they are defined precisely to play that role), as well as several other categories of   PIP-spaces.In a second part, we consider a singlePIP-space  VI as a category by itself, called VI, with natural embeddings as arrows. For this category VI, we show, in Sections 4 and 5, respectively, that one can construct sheaves and cosheaves of operators. There are some restrictions on the PIP-space  VI, but the cases covered by our results are the most useful ones for applications. Then, in Section 6, we describe the cohomology of these (co)sheaves and prove that, in many cases, the sheaves of operators are acyclic; that is, all cohomology groups of higher order are trivial.Although sheaves are quite common in many areas of mathematics, the same cannot be said of cosheaves, the dual concept of sheaves, for which very few concrete examples are known. Actually, cosheaves were recently introduced in the context of nonclassical logic (see Section7) and seem to be related to certain aspects of quantum gravity. Hence the interest of having at one's disposal new, concrete examples of cosheaves, namely, cosheaves of operators on certain types of  PIP-spaces.


## 2. Preliminaries

### 2.1. Partial Inner Product Spaces

We begin by fixing the terminology and notations, following our monograph [5], to which we refer for a full information. For the convenience of the reader, we have collected in Appendix A the main features of partial inner product spaces and operators on them.Throughout the paper, we will consider an  indexedPIP-space   VI={Vr,r∈I}, corresponding to the linear compatibility #. The assaying subspaces are denoted by Vp,Vr,…,p,r∈I. The set I indexes a generating involutive sublattice of the complete lattice ℱ(V,#) of all assaying subspaces defined by #; that is,  (2.1)f#g⟺∃r∈Isuch thatf∈Vr,g∈Vr¯.
The lattice properties are the following: (i)
involution:Vr↔Vr¯:=(Vr)#,(ii)
infimum:Vp∧q:=Vp∧Vq=Vp∩Vq,(p,q,r∈I),(iii)
supremum:Vp∨q:=Vp∨Vq=(Vp+Vq)##.The smallest element ofℱ(V,#) is V#=⋂rVr, and the greatest element is V=⋃rVr, but often they do not belong to VI.Each assaying subspaceVr carries its Mackey topology τ(Vr,Vr¯), and V# is dense in every Vr, since the  indexed  PIP-space  VI is assumed to be nondegenerate. In the sequel, we consider projective and additive  indexed  PIP-spaces (see Appendix A) and, in particular, lattices of the Banach or Hilbert spaces (LBS/LHS).Given two  indexedPIP-spaces VI,YK, an operator A:VI→YK may be identified with the coherent collection of its representatives A≃{Aur}, where each Aur:Vr→Yu is a continuous operator from Vr into Yu. We will also need the set d(A)={r∈I:there isau∈Ksuch thatAurexists}. Every operator A has an adjoint A×, and a partial multiplication between operators is defined.A crucial role is played by homomorphisms, in particular, mono-, epi-, and isomorphisms. The set of all operators fromVI into YK is denoted by Op(VI,YK) and the set of all homomorphisms by Hom(VI,YK).For more details and references to related work, see AppendixA or our monograph [5].


### 2.2. Categories

According to the standard terminology [8], a (small) category C is a collection of objects X,Y,Z,… and arrows or morphisms α,β,…, where we note α∈hom(X,Y) or X→αY, satisfying the following axioms.   (i)
Identity: for any object X, there is a unique arrow 1X:X→X.(ii)
Composition: whenever X→αY→βZ, there is a unique arrow β∘α such that X→β∘αZ.(iii)
Associativity: whenever X→αY→βZ→γT, one has (γ∘β)∘α=γ∘(β∘α).(iv)
Unit law: whenever X→αY, one has α∘1X=α and 1Y∘α=α.In a category, an objectS is initial if, for each object X, there is exactly one arrow S→X. An object T is final or terminal if, for each object X, there is exactly one arrow X→T. Two terminal objects are necessarily isomorphic (isomorphisms in categories are defined exactly as for  indexed  PIP-spaces, see Appendix A).Given two categoriesC and D, a covariant functorF:C→D is a morphism between the two categories. To each object X of C, it associates an object F(X) of D, and, to each arrow α:X→Y in C, it associates an arrow F(α):F(X)→F(Y) of D in such a way that (2.2)F(1X)=1F(X),F(β∘α)=F(β)∘F(α),
whenever the arrow β∘α is defined in C.Given a categoryC, the opposite category Cop has the same objects as C and all arrows reversed: to each arrow α:X→Y, there is an arrow αop:Y→X, so that αop∘βop=(β∘α)op.Acontravariant functorF¯:C→D may be defined as a functor F:Cop→D, or directly on C, by writing F¯(α)=F(αop). Thus, we have (2.3)F¯(1X)=1F¯(X),F¯(β∘α)=F¯(α)∘F¯(β).
Some standard examples of categories are(i)
Set, the category of sets with functions as arrows.(ii)
Top, the category of topological spaces with continuous functions as arrows.(iii)
Grp, the category of groups with group homomorphisms as arrows.For more details, we refer to standard texts, such as Mac Lane [8].


## 2.1. Partial Inner Product Spaces

We begin by fixing the terminology and notations, following our monograph [5], to which we refer for a full information. For the convenience of the reader, we have collected in Appendix A the main features of partial inner product spaces and operators on them.Throughout the paper, we will consider an  indexedPIP-space   VI={Vr,r∈I}, corresponding to the linear compatibility #. The assaying subspaces are denoted by Vp,Vr,…,p,r∈I. The set I indexes a generating involutive sublattice of the complete lattice ℱ(V,#) of all assaying subspaces defined by #; that is,  (2.1)f#g⟺∃r∈Isuch thatf∈Vr,g∈Vr¯.
The lattice properties are the following: (i)
involution:Vr↔Vr¯:=(Vr)#,(ii)
infimum:Vp∧q:=Vp∧Vq=Vp∩Vq,(p,q,r∈I),(iii)
supremum:Vp∨q:=Vp∨Vq=(Vp+Vq)##.The smallest element ofℱ(V,#) is V#=⋂rVr, and the greatest element is V=⋃rVr, but often they do not belong to VI.Each assaying subspaceVr carries its Mackey topology τ(Vr,Vr¯), and V# is dense in every Vr, since the  indexed  PIP-space  VI is assumed to be nondegenerate. In the sequel, we consider projective and additive  indexed  PIP-spaces (see Appendix A) and, in particular, lattices of the Banach or Hilbert spaces (LBS/LHS).Given two  indexedPIP-spaces VI,YK, an operator A:VI→YK may be identified with the coherent collection of its representatives A≃{Aur}, where each Aur:Vr→Yu is a continuous operator from Vr into Yu. We will also need the set d(A)={r∈I:there isau∈Ksuch thatAurexists}. Every operator A has an adjoint A×, and a partial multiplication between operators is defined.A crucial role is played by homomorphisms, in particular, mono-, epi-, and isomorphisms. The set of all operators fromVI into YK is denoted by Op(VI,YK) and the set of all homomorphisms by Hom(VI,YK).For more details and references to related work, see AppendixA or our monograph [5].


## 2.2. Categories

According to the standard terminology [8], a (small) category C is a collection of objects X,Y,Z,… and arrows or morphisms α,β,…, where we note α∈hom(X,Y) or X→αY, satisfying the following axioms.   (i)
Identity: for any object X, there is a unique arrow 1X:X→X.(ii)
Composition: whenever X→αY→βZ, there is a unique arrow β∘α such that X→β∘αZ.(iii)
Associativity: whenever X→αY→βZ→γT, one has (γ∘β)∘α=γ∘(β∘α).(iv)
Unit law: whenever X→αY, one has α∘1X=α and 1Y∘α=α.In a category, an objectS is initial if, for each object X, there is exactly one arrow S→X. An object T is final or terminal if, for each object X, there is exactly one arrow X→T. Two terminal objects are necessarily isomorphic (isomorphisms in categories are defined exactly as for  indexed  PIP-spaces, see Appendix A).Given two categoriesC and D, a covariant functorF:C→D is a morphism between the two categories. To each object X of C, it associates an object F(X) of D, and, to each arrow α:X→Y in C, it associates an arrow F(α):F(X)→F(Y) of D in such a way that (2.2)F(1X)=1F(X),F(β∘α)=F(β)∘F(α),
whenever the arrow β∘α is defined in C.Given a categoryC, the opposite category Cop has the same objects as C and all arrows reversed: to each arrow α:X→Y, there is an arrow αop:Y→X, so that αop∘βop=(β∘α)op.Acontravariant functorF¯:C→D may be defined as a functor F:Cop→D, or directly on C, by writing F¯(α)=F(αop). Thus, we have (2.3)F¯(1X)=1F¯(X),F¯(β∘α)=F¯(α)∘F¯(β).
Some standard examples of categories are(i)
Set, the category of sets with functions as arrows.(ii)
Top, the category of topological spaces with continuous functions as arrows.(iii)
Grp, the category of groups with group homomorphisms as arrows.For more details, we refer to standard texts, such as Mac Lane [8].


## 3. Categories ofPIP-Spaces

### 3.1. A SinglePIP-Space  as Category

We begin by a trivial example.A single-indexedPIP-space  VI={Vr,r∈I} may be considered as a category VI, where   (i)
the objects are the assaying subspaces{Vr,r∈I,V#,V};(ii)
the arrows are the natural embeddings{Ers:Vr→Vs,r⩽s}, that is, the representatives of the identity operator on VI.The axioms of categories are readily checked as follows.(i)
For everyVr, there exists an identity, Err:Vr→Vr, the identity map.(ii)
For everyVr,Vs with r⩽s, one has Ess∘Esr=Esr and Esr∘Err=Esr.(iii)
For everyVr,Vs,Vt with r⩽s⩽t, one has Ets∘Esr=Etr.(iv)
For everyVr,Vs,Vt,Vu with r⩽s⩽t⩽u, one has (Eut∘Ets)∘Esr=Eut∘(Ets∘Esr).In the categoryVI, we have: (i)
V∞¯:=V#=⋂r∈I is an initial object: for every Vr∈VI, there is a unique arrow Er∞¯:V#→Vr.(ii)
V∞:=V=∑r∈I is a terminal object: for every Vr∈VI, there is a unique arrow E∞r:Vr→V.(iii)
The compatibility#:Vr↦(Vr)#=Vr¯ defines a contravariant functor VI→VI.Although this category seems rather trivial, it will allow us to define sheaves and cosheaves of operators, a highly nontrivial (and desirable) result.


### 3.2. A Category Generated by a Single Operator

In the  indexedPIP-space  VI={Vr,r∈I}, take a single totally regular operator, that is, an operator A that leaves every Vr invariant. Hence so does each power An,n∈ℕ. Then this operator induces a category A(VI), as follows.   (i)
The objects are the assaying subspaces{Vr,r∈I}.(ii)
The arrows are the operatorsApqn:Vq→Vp,q⩽p,n∈ℕ.The axioms of categories are readily checked as follows.(i)
For everyVr, there exists an identity, Arr:Vr→Vr, since A is totally regular.(ii)
For everyVr,Vs with r⩽s, one has Ass∘Asrn=Asrn+1 and Asrn∘Arr=Asrn+1,foralln∈ℕ.(iii)
For everyVr,Vs,Vt with r⩽s⩽t, one has Atsm∘Asrn=Atrm+n,forallm,n∈ℕ.(iv)
For everyVr,Vs,Vt,Vu with r⩽s⩽t⩽u, one has (Autm∘Atsn)∘Asrk=Autm∘(Atsn∘Asrk),forallm,n,k∈ℕ.As forVI, the space V∞¯:=V# is an initial object in A(VI) and V∞:=V is a terminal object.The adjunctionA↦A# defines a contravariant functor from A(VI) into A(VI)#, where the latter is the category induced by A#. The proof is immediate.


### 3.3. The Category PIP of  IndexedPIP-Spaces

The collection of all  indexedPIP-spaces constitutes a category, which we callPIP, where   (i)
objects are  indexedPIP-spaces {VI};(ii)
arrows are homomorphismsA:VI→YK, where an operator A∈Op(VI,YK) is called a homomorphism if
(a)
for everyr∈I there exists u∈K such that both Aur and Au¯r¯ exist;(b)
for everyu∈K there exists r∈I such that both Aur and Au¯r¯ exist.For making the notation less cumbersome (and more automatic), we will, henceforth, denote byAKI an element A∈Hom(VI,YK). Then the axioms of a category are obviously satisfied as follows.   (i)
For everyVI, there exists an identity, 1I∈Hom(VI,VI), the identity operator on VI.(ii)
For everyVI,YK, one has 1K∘AKI=AKI and AKI∘1I=AKI.(iii)
For everyVI,YK,WL, one has BLK∘AKI=CLI∈Hom(VI,WL).(iv)
For everyVI,YK,WL,ZM, one has (CML∘BLK)∘AKI=CML∘(BLK∘AKI)∈Hom(VI,ZM).The categoryPIP has no initial object and no terminal object; hence, it is not a topos.One can define in the same way smaller categoriesLBS andLHS, whose objects are, respectively, lattices of the Banach spaces (LBS) and lattices of the Hilbert spaces (LHS), the arrows being still the corresponding homomorphisms.


#### 3.3.1. Subobjects

We recall that a homomorphismMKI∈Hom(VI,YK) is a monomorphism if MKLALI=MKLBLI implies ALI=BLI, for any pair ALI,BLI and any  indexed  PIP-space  WL (a typical example is given in Appendix A). Two monomorphisms MLI,NLK with the same codomain WL are equivalent if there exists an isomorphism UKI such that NLKUKI=MLI. Then a subobject of VI is an equivalence class of monomorphisms into VI. A   PIP-subspace  W of an  indexed  PIP-space  V is defined as an orthocomplemented subspace of V, and this holds if and only if W is the range of an orthogonal projection, WI=PVI. Now the embedding M:WI=PVI↦VI is a monomorphism; thus, orthocomplemented subspaces are subobjects of PIP.However, the converse is not true, at least for a general  indexedPIP-space. Take the case where V is a noncomplete prehilbert space (i.e., V=V#). Then every subspace is a subobject, but need not to be the range of a projection. To give a concrete example [5, Section 3.4.5], take V=𝒮(ℝ), the Schwartz space of test functions. Let W=𝒮+:={φ∈𝒮:φ(x)=0forx⩽0}. Then W⊥=𝒮-:={ψ∈𝒮:ψ(x)=0forx⩾0}; hence, W⊥⊥=W. However, W is not orthocomplemented, since every χ∈W+W⊥ satisfies χ(0)=0, so that W+W⊥≠𝒮. Yet W is the range of a monomorphism (the injection), hence, a subobject. However, this example addresses an  indexed  PIP-space  which is not a LBS/LHS.Take now a LBS/LHSVI={Vr,r∈I} and a vector subspace W. In order that W becomes a LBS/LHS WI in its own right, we must require that, for every r∈I, Wr=W∩Vr and Wr¯=W∩Vr¯ are a dual pair with respect to their respective Mackey topologies and that the intrinsic Mackey topology τ(Wr,Wr¯) coincides with the norm topology induced by Vr. In other words, W must be topologically regular, which is equivalent that it be orthocomplemented [5, Section 3.4.2]. Now the injection MI:WI→VI is clearly a monomorphism, and WI is a subobject of VI. Thus, we have shown that, in a LBS/LHS, the subobjects are precisely the orthocomplemented subspaces.Coming back to the previous example of a noncomplete prehilbert space, we see that an arbitrary subspaceW need not be orthocomplemented, because it may fail to be topologically regular. Indeed the intrinsic topology τ(W,W) does not coincide with the norm topology, unless W is orthocomplemented (see the discussion in [5, Section 3.4.5]). In the Schwartz example above, one has W⊥⊥=W, which means that W is τ(W,W) closed, hence, norm closed, but it is not orthocomplemented.Remark 3.1.
Homomorphisms are defined between arbitraryPIP-spaces. However, when it comes to indexedPIP-spaces, the discussion above shows that the notion of homomorphism is more natural between two  indexed  PIP-spaces of the same type, for instance, two LBSs or two LHSs. This is true, in particular, when trying to identify subobjects. This suggests to define categories LBS and LHS, either directly as above or as subcategories within PIP, and then define properly subobjects in that context.


#### 3.3.2. Superobjects

Dually, one may define superobjects in terms of epimorphisms. We recall that a homomorphismNKL∈Hom(VI,WK) is an epimorphism if AIKNKL=BIKNKL implies AIK=BIK, for any pair AIK,BIK and any  indexed  PIP-space  YL. Then a superobject is an equivalence class of epimorphisms, where again equivalence means modulo isomorphisms.Whereas monomorphisms are natural in the context of sheaves; epimorphisms are natural in the dual structure, that is, cosheaves.


## 3.1. A SinglePIP-Space  as Category

We begin by a trivial example.A single-indexedPIP-space  VI={Vr,r∈I} may be considered as a category VI, where   (i)
the objects are the assaying subspaces{Vr,r∈I,V#,V};(ii)
the arrows are the natural embeddings{Ers:Vr→Vs,r⩽s}, that is, the representatives of the identity operator on VI.The axioms of categories are readily checked as follows.(i)
For everyVr, there exists an identity, Err:Vr→Vr, the identity map.(ii)
For everyVr,Vs with r⩽s, one has Ess∘Esr=Esr and Esr∘Err=Esr.(iii)
For everyVr,Vs,Vt with r⩽s⩽t, one has Ets∘Esr=Etr.(iv)
For everyVr,Vs,Vt,Vu with r⩽s⩽t⩽u, one has (Eut∘Ets)∘Esr=Eut∘(Ets∘Esr).In the categoryVI, we have: (i)
V∞¯:=V#=⋂r∈I is an initial object: for every Vr∈VI, there is a unique arrow Er∞¯:V#→Vr.(ii)
V∞:=V=∑r∈I is a terminal object: for every Vr∈VI, there is a unique arrow E∞r:Vr→V.(iii)
The compatibility#:Vr↦(Vr)#=Vr¯ defines a contravariant functor VI→VI.Although this category seems rather trivial, it will allow us to define sheaves and cosheaves of operators, a highly nontrivial (and desirable) result.


## 3.2. A Category Generated by a Single Operator

In the  indexedPIP-space  VI={Vr,r∈I}, take a single totally regular operator, that is, an operator A that leaves every Vr invariant. Hence so does each power An,n∈ℕ. Then this operator induces a category A(VI), as follows.   (i)
The objects are the assaying subspaces{Vr,r∈I}.(ii)
The arrows are the operatorsApqn:Vq→Vp,q⩽p,n∈ℕ.The axioms of categories are readily checked as follows.(i)
For everyVr, there exists an identity, Arr:Vr→Vr, since A is totally regular.(ii)
For everyVr,Vs with r⩽s, one has Ass∘Asrn=Asrn+1 and Asrn∘Arr=Asrn+1,foralln∈ℕ.(iii)
For everyVr,Vs,Vt with r⩽s⩽t, one has Atsm∘Asrn=Atrm+n,forallm,n∈ℕ.(iv)
For everyVr,Vs,Vt,Vu with r⩽s⩽t⩽u, one has (Autm∘Atsn)∘Asrk=Autm∘(Atsn∘Asrk),forallm,n,k∈ℕ.As forVI, the space V∞¯:=V# is an initial object in A(VI) and V∞:=V is a terminal object.The adjunctionA↦A# defines a contravariant functor from A(VI) into A(VI)#, where the latter is the category induced by A#. The proof is immediate.


## 3.3. The Category PIP of  IndexedPIP-Spaces

The collection of all  indexedPIP-spaces constitutes a category, which we callPIP, where   (i)
objects are  indexedPIP-spaces {VI};(ii)
arrows are homomorphismsA:VI→YK, where an operator A∈Op(VI,YK) is called a homomorphism if
(a)
for everyr∈I there exists u∈K such that both Aur and Au¯r¯ exist;(b)
for everyu∈K there exists r∈I such that both Aur and Au¯r¯ exist.For making the notation less cumbersome (and more automatic), we will, henceforth, denote byAKI an element A∈Hom(VI,YK). Then the axioms of a category are obviously satisfied as follows.   (i)
For everyVI, there exists an identity, 1I∈Hom(VI,VI), the identity operator on VI.(ii)
For everyVI,YK, one has 1K∘AKI=AKI and AKI∘1I=AKI.(iii)
For everyVI,YK,WL, one has BLK∘AKI=CLI∈Hom(VI,WL).(iv)
For everyVI,YK,WL,ZM, one has (CML∘BLK)∘AKI=CML∘(BLK∘AKI)∈Hom(VI,ZM).The categoryPIP has no initial object and no terminal object; hence, it is not a topos.One can define in the same way smaller categoriesLBS andLHS, whose objects are, respectively, lattices of the Banach spaces (LBS) and lattices of the Hilbert spaces (LHS), the arrows being still the corresponding homomorphisms.


### 3.3.1. Subobjects

We recall that a homomorphismMKI∈Hom(VI,YK) is a monomorphism if MKLALI=MKLBLI implies ALI=BLI, for any pair ALI,BLI and any  indexed  PIP-space  WL (a typical example is given in Appendix A). Two monomorphisms MLI,NLK with the same codomain WL are equivalent if there exists an isomorphism UKI such that NLKUKI=MLI. Then a subobject of VI is an equivalence class of monomorphisms into VI. A   PIP-subspace  W of an  indexed  PIP-space  V is defined as an orthocomplemented subspace of V, and this holds if and only if W is the range of an orthogonal projection, WI=PVI. Now the embedding M:WI=PVI↦VI is a monomorphism; thus, orthocomplemented subspaces are subobjects of PIP.However, the converse is not true, at least for a general  indexedPIP-space. Take the case where V is a noncomplete prehilbert space (i.e., V=V#). Then every subspace is a subobject, but need not to be the range of a projection. To give a concrete example [5, Section 3.4.5], take V=𝒮(ℝ), the Schwartz space of test functions. Let W=𝒮+:={φ∈𝒮:φ(x)=0forx⩽0}. Then W⊥=𝒮-:={ψ∈𝒮:ψ(x)=0forx⩾0}; hence, W⊥⊥=W. However, W is not orthocomplemented, since every χ∈W+W⊥ satisfies χ(0)=0, so that W+W⊥≠𝒮. Yet W is the range of a monomorphism (the injection), hence, a subobject. However, this example addresses an  indexed  PIP-space  which is not a LBS/LHS.Take now a LBS/LHSVI={Vr,r∈I} and a vector subspace W. In order that W becomes a LBS/LHS WI in its own right, we must require that, for every r∈I, Wr=W∩Vr and Wr¯=W∩Vr¯ are a dual pair with respect to their respective Mackey topologies and that the intrinsic Mackey topology τ(Wr,Wr¯) coincides with the norm topology induced by Vr. In other words, W must be topologically regular, which is equivalent that it be orthocomplemented [5, Section 3.4.2]. Now the injection MI:WI→VI is clearly a monomorphism, and WI is a subobject of VI. Thus, we have shown that, in a LBS/LHS, the subobjects are precisely the orthocomplemented subspaces.Coming back to the previous example of a noncomplete prehilbert space, we see that an arbitrary subspaceW need not be orthocomplemented, because it may fail to be topologically regular. Indeed the intrinsic topology τ(W,W) does not coincide with the norm topology, unless W is orthocomplemented (see the discussion in [5, Section 3.4.5]). In the Schwartz example above, one has W⊥⊥=W, which means that W is τ(W,W) closed, hence, norm closed, but it is not orthocomplemented.Remark 3.1.
Homomorphisms are defined between arbitraryPIP-spaces. However, when it comes to indexedPIP-spaces, the discussion above shows that the notion of homomorphism is more natural between two  indexed  PIP-spaces of the same type, for instance, two LBSs or two LHSs. This is true, in particular, when trying to identify subobjects. This suggests to define categories LBS and LHS, either directly as above or as subcategories within PIP, and then define properly subobjects in that context.


### 3.3.2. Superobjects

Dually, one may define superobjects in terms of epimorphisms. We recall that a homomorphismNKL∈Hom(VI,WK) is an epimorphism if AIKNKL=BIKNKL implies AIK=BIK, for any pair AIK,BIK and any  indexed  PIP-space  YL. Then a superobject is an equivalence class of epimorphisms, where again equivalence means modulo isomorphisms.Whereas monomorphisms are natural in the context of sheaves; epimorphisms are natural in the dual structure, that is, cosheaves.


## 3.3.1. Subobjects

We recall that a homomorphismMKI∈Hom(VI,YK) is a monomorphism if MKLALI=MKLBLI implies ALI=BLI, for any pair ALI,BLI and any  indexed  PIP-space  WL (a typical example is given in Appendix A). Two monomorphisms MLI,NLK with the same codomain WL are equivalent if there exists an isomorphism UKI such that NLKUKI=MLI. Then a subobject of VI is an equivalence class of monomorphisms into VI. A   PIP-subspace  W of an  indexed  PIP-space  V is defined as an orthocomplemented subspace of V, and this holds if and only if W is the range of an orthogonal projection, WI=PVI. Now the embedding M:WI=PVI↦VI is a monomorphism; thus, orthocomplemented subspaces are subobjects of PIP.However, the converse is not true, at least for a general  indexedPIP-space. Take the case where V is a noncomplete prehilbert space (i.e., V=V#). Then every subspace is a subobject, but need not to be the range of a projection. To give a concrete example [5, Section 3.4.5], take V=𝒮(ℝ), the Schwartz space of test functions. Let W=𝒮+:={φ∈𝒮:φ(x)=0forx⩽0}. Then W⊥=𝒮-:={ψ∈𝒮:ψ(x)=0forx⩾0}; hence, W⊥⊥=W. However, W is not orthocomplemented, since every χ∈W+W⊥ satisfies χ(0)=0, so that W+W⊥≠𝒮. Yet W is the range of a monomorphism (the injection), hence, a subobject. However, this example addresses an  indexed  PIP-space  which is not a LBS/LHS.Take now a LBS/LHSVI={Vr,r∈I} and a vector subspace W. In order that W becomes a LBS/LHS WI in its own right, we must require that, for every r∈I, Wr=W∩Vr and Wr¯=W∩Vr¯ are a dual pair with respect to their respective Mackey topologies and that the intrinsic Mackey topology τ(Wr,Wr¯) coincides with the norm topology induced by Vr. In other words, W must be topologically regular, which is equivalent that it be orthocomplemented [5, Section 3.4.2]. Now the injection MI:WI→VI is clearly a monomorphism, and WI is a subobject of VI. Thus, we have shown that, in a LBS/LHS, the subobjects are precisely the orthocomplemented subspaces.Coming back to the previous example of a noncomplete prehilbert space, we see that an arbitrary subspaceW need not be orthocomplemented, because it may fail to be topologically regular. Indeed the intrinsic topology τ(W,W) does not coincide with the norm topology, unless W is orthocomplemented (see the discussion in [5, Section 3.4.5]). In the Schwartz example above, one has W⊥⊥=W, which means that W is τ(W,W) closed, hence, norm closed, but it is not orthocomplemented.Remark 3.1.
Homomorphisms are defined between arbitraryPIP-spaces. However, when it comes to indexedPIP-spaces, the discussion above shows that the notion of homomorphism is more natural between two  indexed  PIP-spaces of the same type, for instance, two LBSs or two LHSs. This is true, in particular, when trying to identify subobjects. This suggests to define categories LBS and LHS, either directly as above or as subcategories within PIP, and then define properly subobjects in that context.


## 3.3.2. Superobjects

Dually, one may define superobjects in terms of epimorphisms. We recall that a homomorphismNKL∈Hom(VI,WK) is an epimorphism if AIKNKL=BIKNKL implies AIK=BIK, for any pair AIK,BIK and any  indexed  PIP-space  YL. Then a superobject is an equivalence class of epimorphisms, where again equivalence means modulo isomorphisms.Whereas monomorphisms are natural in the context of sheaves; epimorphisms are natural in the dual structure, that is, cosheaves.


## 4. Sheaves of Operators onPIP-Spaces

### 4.1. Presheaves and Sheaves

LetX be a topological space, and let C be a (concrete) category. Usually C is the category of sets, the category of groups, the category of abelian groups, or the category of commutative rings. In the standard fashion [8, 9], we proceed in two steps.Definition 4.1.
ApresheafF on X with values in C is a map F defined on the family of open subsets of X such that  (PS1)
for each open setU of X, there is an object F(U) in C;(PS2)
for each inclusion of open setsT⊆U, there is given a restriction morphismρVT:F(U)→F(T) in the category C, such that ρUU is the identity for every open set U and ρST∘ρTU=ρSU whenever S⊆T⊆U.Definition 4.2.
LetF be a presheaf on X, and let U be an open set of X. Every element s∈F(U) is called a section of F over U. A section over X is called a global section.Example 4.3.
LetX be a topological space; let C be the category of vector spaces. Let F associate to each open set U the vector space of continuous functions on U with values in ℂ. If T⊆U, ρTU associates to each continuous function on U its restriction to T. This is a presheaf. Any continuous function on U is a section of F on U.Definition 4.4.
LetF be a presheaf on the topological space X. One says that F is a sheaf if, for every open set U⊂X and for every open covering {Ui}i∈I of U, the following conditions are fulfilled:  (S1)
givens,s′∈F(U) such that ρUiU(s)=ρUiU(s′), for every i∈I, then s=s′ (local identity);(S2)
givensi∈F(Ui) such that ρUi∩UjUi(si)=ρUi∩UjUj(sj), for every i,j∈I, then there exists a section s∈F(U) such that ρUiU(s)=si, for every i∈I (gluing).The sections whose existence is guaranteed by axiom (S2) is called the gluing, concatenation, or collation of the sections si. By axiom (S1), it is unique. The sheaf F may be seen as a contravariant functor from the category of open sets of X into C:=({F(Ui)},{ρUjUi}).


### 4.2. A Sheaf of Operators on an  IndexedPIP-Space

LetVI={Vr,r∈I} be an  indexed  PIP-space, and let VI be the corresponding category defined in Section 3.1. If we put on I the discrete topology, then I defines an open covering of V. Each Vr carries its Mackey’s topology τ(Vr,Vr¯).We define a sheaf onVI by the contravariant functor F:VI→Set given by (4.1)F:Vr⟼Opr:={A↾Vr:A∈Op(VI),r∈d(A)}.
This means that an element of Opr is a representative Awr from Vr into some Vw. In the sequel, we will use the notation A*r whenever the dependence on the first index may be neglected without creating ambiguities. By analogy with functions, the elements of Opr may be called germs of operators. Opr is the restriction to Vr of the set Lr of left multipliers [5, Section 6.2.3]: (4.2)Lr:={A∈Op(VI):∃wsuch thatAwrexists}={A∈Op(VI):r∈d(A)}.Then we have:(i)
WhenVq⊆Vp, define ρqp:Opp→Opq by ρqp(A*p):=A*q, for A*p∈Opp. Clearly, ρpp=idOpp and ρrq∘ρqp=ρrp if Vr⊂Vq⊂Vp. Hence, F is a presheaf.(ii)
(S1) is clearly satisfied. As for (S2), if A*r∈Opr and A*s′∈Ops are such that ρr∧sr(A*r)=ρr∧ss(A*s′); that is, A*r∧s=A*r∧s′, then these two operators are the (r∧s,*) representative of a unique operator A∈Op(VI). It remains to prove that A extends to Vr∨s and that its representative A*r∨s extends both A*r and A*s′.Proposition 4.5.
Let the  indexedPIP-space   VI be additive; that is,  Vr∨s=Vr+Vs,forallr,s∈I. Then the map  F given in (4.1) is a sheaf of operators on  VI.Proof.
By linearity,A*r and A*s can be extended to an operator A(r∨s) on Vr+Vs as follows:
(4.3)A(r∨s)(fr+fs):=A*rfr+A*sfs.
This operator is well defined. Let indeed f=fr+fs=fr′+fs′ be two decompositions of f∈Vr+Vs, so that fr-fr′=fs′-fs∈Vr∩Vs. Then A(r∨s)f=A*rfr+A*sfs=A*rfr′+A*sfs′. Hence, A*r(fr-fr′)=A*s(fs′-fs). Taking the restriction to Vr∩Vs=Vr∧s, this relation becomes A*r∧s(f-f′)r∧s=A*r∧s(f′-f)r∧s=0, so that the vector A(r∨s)f is uniquely defined.
Next, by additivity,Vr+Vs, with its inductive topology, coincides with Vr∨s, and, thus, A*r∨s is the (r∨s,*)-representative of the operator A∈Op(VI). Therefore, F is a sheaf.We recall that the most interesting classes of  indexedPIP-spaces are additive, namely, the projective ones and, in particular, LBSs and LHSs. Thus the proposition just proven has a widely applicable range.


## 4.1. Presheaves and Sheaves

LetX be a topological space, and let C be a (concrete) category. Usually C is the category of sets, the category of groups, the category of abelian groups, or the category of commutative rings. In the standard fashion [8, 9], we proceed in two steps.Definition 4.1.
ApresheafF on X with values in C is a map F defined on the family of open subsets of X such that  (PS1)
for each open setU of X, there is an object F(U) in C;(PS2)
for each inclusion of open setsT⊆U, there is given a restriction morphismρVT:F(U)→F(T) in the category C, such that ρUU is the identity for every open set U and ρST∘ρTU=ρSU whenever S⊆T⊆U.Definition 4.2.
LetF be a presheaf on X, and let U be an open set of X. Every element s∈F(U) is called a section of F over U. A section over X is called a global section.Example 4.3.
LetX be a topological space; let C be the category of vector spaces. Let F associate to each open set U the vector space of continuous functions on U with values in ℂ. If T⊆U, ρTU associates to each continuous function on U its restriction to T. This is a presheaf. Any continuous function on U is a section of F on U.Definition 4.4.
LetF be a presheaf on the topological space X. One says that F is a sheaf if, for every open set U⊂X and for every open covering {Ui}i∈I of U, the following conditions are fulfilled:  (S1)
givens,s′∈F(U) such that ρUiU(s)=ρUiU(s′), for every i∈I, then s=s′ (local identity);(S2)
givensi∈F(Ui) such that ρUi∩UjUi(si)=ρUi∩UjUj(sj), for every i,j∈I, then there exists a section s∈F(U) such that ρUiU(s)=si, for every i∈I (gluing).The sections whose existence is guaranteed by axiom (S2) is called the gluing, concatenation, or collation of the sections si. By axiom (S1), it is unique. The sheaf F may be seen as a contravariant functor from the category of open sets of X into C:=({F(Ui)},{ρUjUi}).


## 4.2. A Sheaf of Operators on an  IndexedPIP-Space

LetVI={Vr,r∈I} be an  indexed  PIP-space, and let VI be the corresponding category defined in Section 3.1. If we put on I the discrete topology, then I defines an open covering of V. Each Vr carries its Mackey’s topology τ(Vr,Vr¯).We define a sheaf onVI by the contravariant functor F:VI→Set given by (4.1)F:Vr⟼Opr:={A↾Vr:A∈Op(VI),r∈d(A)}.
This means that an element of Opr is a representative Awr from Vr into some Vw. In the sequel, we will use the notation A*r whenever the dependence on the first index may be neglected without creating ambiguities. By analogy with functions, the elements of Opr may be called germs of operators. Opr is the restriction to Vr of the set Lr of left multipliers [5, Section 6.2.3]: (4.2)Lr:={A∈Op(VI):∃wsuch thatAwrexists}={A∈Op(VI):r∈d(A)}.Then we have:(i)
WhenVq⊆Vp, define ρqp:Opp→Opq by ρqp(A*p):=A*q, for A*p∈Opp. Clearly, ρpp=idOpp and ρrq∘ρqp=ρrp if Vr⊂Vq⊂Vp. Hence, F is a presheaf.(ii)
(S1) is clearly satisfied. As for (S2), if A*r∈Opr and A*s′∈Ops are such that ρr∧sr(A*r)=ρr∧ss(A*s′); that is, A*r∧s=A*r∧s′, then these two operators are the (r∧s,*) representative of a unique operator A∈Op(VI). It remains to prove that A extends to Vr∨s and that its representative A*r∨s extends both A*r and A*s′.Proposition 4.5.
Let the  indexedPIP-space   VI be additive; that is,  Vr∨s=Vr+Vs,forallr,s∈I. Then the map  F given in (4.1) is a sheaf of operators on  VI.Proof.
By linearity,A*r and A*s can be extended to an operator A(r∨s) on Vr+Vs as follows:
(4.3)A(r∨s)(fr+fs):=A*rfr+A*sfs.
This operator is well defined. Let indeed f=fr+fs=fr′+fs′ be two decompositions of f∈Vr+Vs, so that fr-fr′=fs′-fs∈Vr∩Vs. Then A(r∨s)f=A*rfr+A*sfs=A*rfr′+A*sfs′. Hence, A*r(fr-fr′)=A*s(fs′-fs). Taking the restriction to Vr∩Vs=Vr∧s, this relation becomes A*r∧s(f-f′)r∧s=A*r∧s(f′-f)r∧s=0, so that the vector A(r∨s)f is uniquely defined.
Next, by additivity,Vr+Vs, with its inductive topology, coincides with Vr∨s, and, thus, A*r∨s is the (r∨s,*)-representative of the operator A∈Op(VI). Therefore, F is a sheaf.We recall that the most interesting classes of  indexedPIP-spaces are additive, namely, the projective ones and, in particular, LBSs and LHSs. Thus the proposition just proven has a widely applicable range.


## 5. Cosheaves of Operators onPIP-Spaces

### 5.1. Pre-Cosheaves and Cosheaves

Pre-cosheaves and cosheaves are the dual notions of presheaves and sheaves, respectively. Let againX be a topological space, with closed sets Wi so that X=⋃i∈IWi, and let C be a (concrete) category.Definition 5.1.
Apre-cosheafG on X with values in C is a map G defined on the family of closed subsets of X such that  (PC1)
for each closed setW of X, there is an object G(W) in C;(PC2)
for each inclusion of closed setsZ⊇W, there is given a extension morphismδWZ:G(W)→G(Z) in the category C, such that δWW is the identity for every closed set W and δZT∘δWZ=δWT whenever T⊇Z⊇W.Definition 5.2.
LetG be a pre-cosheaf on the topological space X. We say that G is a cosheaf if, for every nonempty closed set W=⋂j∈JWj,J⊆I and for every family of (local) sections {tj∈G(Wj)}j∈J, the following conditions are fulfilled:  (CS1)
givent,t'∈G(W) such that δWWj(t)=δWWj(t′), for every j∈I such that W⊆Wj, then t=t′;(CS2)
ifδWjWj∪Wk(tj)=δWkWj∪Wk(tk), for every j,k∈J, then there exists a unique section t∈G(W) such that δWWj(s)=tj, for every j∈J.The cosheafG may be seen as a covariant functor from the category of closed sets of X into C:=({G(Wi)},{δWjWi}).Now, there are situations where extensions do not exist for all inclusionsZ⊇W but only for certain pairs. This will be the case for operators on a PIP-space, as will be seen below. Thus, we may generalize the notions of (pre-) cosheaf as follows (there could be several variants).Definition 5.3.
LetX be a topological space, let 𝒞l(X) be the family of closed subsets of X, and let ≼ a be coarsening of the set inclusion in 𝒞l(X); that is, W≼Z implies W⊆Z, but not necessarily the opposite. A partial  pre-cosheafG with values in C is a map G defined on 𝒞l(X), satisfying conditions (PC1) and  (pPC2)
ifZ≽W, there is given an extension morphismδWZ:G(W)→G(Z) in the category C, such that δWW is the identity for every closed set W and δZT∘δWZ=δWT whenever T≽Z≽W.We may use the term “partial,” since here not all pairsw⩽z admit extensions, but only certain pairs, namely, those that satisfy w≼z. This is analogous to the familiar situation of a compatibility.Definition 5.4.
LetG be a partial pre-cosheaf on the topological space X. We say that G is a partial cosheaf if, for every nonempty closed set W=⋂j∈JWj,J⊆I and for every family of sections {tj∈G(Wj)}j∈J, the following conditions are fulfilled:  (pCS1)
givent,t'∈G(W) such that δWWj(t) and δWWj(t′) exist and are equal, for every j∈J, then t=t′;(pCS2)
ifδWjWj∪Wk(tj) and δWkWj∪Wk(tk) exist and are equal, for every j,k∈J, then δWWj exists for every j∈J and there exists a unique section t∈G(W) such that δWWj(t)=tj, for every j∈J.


### 5.2. Cosheaves of Operators on  IndexedPIP-Spaces

Let againVI={Vr,r∈I} be an  indexed  PIP-space. If we put on I the discrete topology, then the assaying subspaces may be taken as closed sets in V. In order to build a cosheaf, we consider the same map as before in (4.1): (5.1)F:Vr⟼Opr:={Awr:Vr⟶Vw,for somew∈I,A∈Op(VI)}.
Then we immediately face a problem. Given Vq, there is no Vp, with Vq⊆Vp, such that δqp:Opp→Opq exists. One can always find an operator A such that q∈d(A) and p∉d(A); in other words, A*q exists, but it cannot be extended to Vp. This happens, for instance, each time d(A) has a maximal element (see Figure 3.1 in [5]). There are three ways out of this situation.


#### 5.2.1. General Operators: The Additive Case

We still consider the set of all operatorsOp(VI). Take two assaying subspaces Vr,Vs and two operators A(r)∈Opr,B(s)∈Ops, and assume that they have a common extension C(r∨s) to Opr∨s. This means that, for any suitable w, C(r∨s):Vr∨s→Vw is the (r∨s,w)-representative Cw,r∨s of a unique operator C∈Op(VI), and C=A=B. A fortiori, A, and B coincide on Vr∧s=Vr∩Vs; that is, condition (pCS2) is satisfied.Assume now thatVI is additive; that is, Vr∨s coincides with Vr+Vs, with its inductive topology. Then we can proceed as in the case of a sheaf, by defining the extension C of A and B by linearity. In other words, since C(r∨s)fr=A*rfr and C(r∨s)fs=B*sfs, we may write, for any f=fr+fs∈Vr+Vs, (5.2)C(r∨s)f=C*r∨sfr∨s=A*rfr+B*sfs.
Here too, this operator is well defined. Let indeed f=fr+fs=fr′+fs′ be two decompositions of f∈Vr+Vs, so that fr-fr′=fs′-fs∈Vr∩Vs. Then necessarily C(r∨s)f=A*rfr+B*sfs=A*rfr′+B*sfs′. Hence, A*r(fr-fr′)=B*s(fs′-fs). Taking the restriction to Vr∩Vs=Vr∧s, this relation becomes A*r∧s(f-f′)r∧s=B*r∧s(f′-f)r∧s, and this equals 0, since A=B on Vr∧s, by the condition (pcS2).The conclusion is that, for any pair of assaying subspacesVr,Vs, the extension maps δrr∨s and δsr∨s always exist, and the condition (pCS2) is satisfied for that pair. However, this is not necessarily true for any comparable pair, and this motivates the coarsening of the order given in Proposition 5.5 below. Condition (pPC2) is also satisfied, as one can see by taking supremums (= sums) of successive pairs within three assaying subspaces Vr,Vs,Vt (and using the associativity of ∨). Thus, me may state the following.Proposition 5.5.
LetVI={Vr,r∈I} be an additive  indexed  PIP-space. Then the map  F given in (4.1) is a partial cosheaf with respect to the partial order on  I:
(5.3)r≼s⟺∃w∈Isuch thats=r∨w.


#### 5.2.2. Universal Left Multipliers

We consider only the operatorsA which are everywhere defined; that is, d(A)=I. This is precisely the set of universal left multipliers: (5.4)LOp(VI)={C∈Op(VI):∀r,∃wsuch thatCwrexists}={C∈Op(VI):d(C)=I}=⋂r∈ILr,
where Lr:={C∈Op(VI):r∈d(C)}. Correspondingly, we consider the map  (5.5)G:Vr⟼LOpr:={A↾Vr:A∈LOp(VI)}.
Here again, elements of LOpr are representatives A*r. In that case, extensions always exist. When Vq⊆Vp, define δqp:LOpq→LOpp by δqp(A*q):=A*p for A*q∈LOpq. Clearly, δpp=idLOpp and δqp∘δrq=δrp if Vr⊂Vq⊂Vp. Hence, G is a pre-cosheaf. Moreover, if δqp∨q(A*q)=δpp∨q(A*p′); that is, A*,p∨q=A*,p∨q′, these two operators are the (p∨q,*) representative of a unique operator A∈LOp(VI). Then, of course, one has δp∧qq(A℘∧q)=A*q and δp∧qp(A*p∧q)=A*p; thus, G is a cosheaf. Therefore,Proposition 5.6.
LetVI={Vr,r∈I} be an arbitrary  indexed  PIP-space. Then the map given in (5.5) is a cosheaf on  VI with values in  LOp(VI), the set of universal left multipliers, with extensions given by  δqp(A*q)=A*p,q⩽p,A*q∈LOpq.


#### 5.2.3. General Operators: The Projective Case

We consider again all operators inOp(VI). The fact is that the set ⋃r∈IOpr is an initial set in Op(VI), like every  d(A); that is, it contains all predecessors of any of its elements, and this is natural for constructing sheaves into it. But it is not when considering cosheaves. By duality, one should rather take a final set, that is, containing all successors of any of its elements, just like any  i(A). Hence, we define the map: (5.6)Ĝ:Vr⟼Op̂r:={Ar*:A∈Op(VI),r∈i(A)}.
Now indeed ⋁r∈IOp̂r is a final set. Elements of Op̂r have been denoted as Ar*, but, for definiteness, we could also replace Ar* by Ar∞¯ where V∞¯:=V#.We claim that the mapĜ defines a cosheaf, with extensions δqp(Aq*):=Ap*, for Aq*∈Op̂q, which exist whenever q⩽p.As in the case of Section5.2.1, assume that Ar*∈Op̂r,Bs*∈Op̂s have a common extension belonging to Op̂r∨s; that is, Ar∨s*:=Er∨s,rAr*=Br∨s*:=Er∨s,sBs*. Call this operator Cr∨s*. Thus, for any suitable w, Cr∨s,w is the (w,r∨s) representative of a unique operator C∈Op(VI), which extends A and B. Thus C=A=B∈Op(VI). Assume now that VI is projective; that is, Vr∧s=Vr∩Vs,forallr,s∈I, with the projective topology. Then, Cr∧s*=Ar∧s*=Br∧s*; that is, condition (CS2) is satisfied with extensions δr∧sr(Cr∧s*)=Ar* and δr∧ss(Cr∧s*)=Bs*. The rest is obvious. As in the case of sheaves, extensions from Vr,Vs to Vr∨s exist by linearity, since VI is projective, hence, additive. Thus, we may state the following.Proposition 5.7.
Let the  indexedPIP-space   VI be projective that is,  Vr∧s=Vr∩Vs,forallr,s∈I, with the projective topology. Then the map  Ĝ given in (5.6) is a cosheaf of operators on  VI.Corollary 5.8.
LetVI={Vr,r∈I} be a projective  indexed  PIP-space. Then  VI generates in a natural fashion a sheaf and a cosheaf of operators. If  VI is additive, but not projective; it generates a sheaf and a partial cosheaf of operators.


## 5.1. Pre-Cosheaves and Cosheaves

Pre-cosheaves and cosheaves are the dual notions of presheaves and sheaves, respectively. Let againX be a topological space, with closed sets Wi so that X=⋃i∈IWi, and let C be a (concrete) category.Definition 5.1.
Apre-cosheafG on X with values in C is a map G defined on the family of closed subsets of X such that  (PC1)
for each closed setW of X, there is an object G(W) in C;(PC2)
for each inclusion of closed setsZ⊇W, there is given a extension morphismδWZ:G(W)→G(Z) in the category C, such that δWW is the identity for every closed set W and δZT∘δWZ=δWT whenever T⊇Z⊇W.Definition 5.2.
LetG be a pre-cosheaf on the topological space X. We say that G is a cosheaf if, for every nonempty closed set W=⋂j∈JWj,J⊆I and for every family of (local) sections {tj∈G(Wj)}j∈J, the following conditions are fulfilled:  (CS1)
givent,t'∈G(W) such that δWWj(t)=δWWj(t′), for every j∈I such that W⊆Wj, then t=t′;(CS2)
ifδWjWj∪Wk(tj)=δWkWj∪Wk(tk), for every j,k∈J, then there exists a unique section t∈G(W) such that δWWj(s)=tj, for every j∈J.The cosheafG may be seen as a covariant functor from the category of closed sets of X into C:=({G(Wi)},{δWjWi}).Now, there are situations where extensions do not exist for all inclusionsZ⊇W but only for certain pairs. This will be the case for operators on a PIP-space, as will be seen below. Thus, we may generalize the notions of (pre-) cosheaf as follows (there could be several variants).Definition 5.3.
LetX be a topological space, let 𝒞l(X) be the family of closed subsets of X, and let ≼ a be coarsening of the set inclusion in 𝒞l(X); that is, W≼Z implies W⊆Z, but not necessarily the opposite. A partial  pre-cosheafG with values in C is a map G defined on 𝒞l(X), satisfying conditions (PC1) and  (pPC2)
ifZ≽W, there is given an extension morphismδWZ:G(W)→G(Z) in the category C, such that δWW is the identity for every closed set W and δZT∘δWZ=δWT whenever T≽Z≽W.We may use the term “partial,” since here not all pairsw⩽z admit extensions, but only certain pairs, namely, those that satisfy w≼z. This is analogous to the familiar situation of a compatibility.Definition 5.4.
LetG be a partial pre-cosheaf on the topological space X. We say that G is a partial cosheaf if, for every nonempty closed set W=⋂j∈JWj,J⊆I and for every family of sections {tj∈G(Wj)}j∈J, the following conditions are fulfilled:  (pCS1)
givent,t'∈G(W) such that δWWj(t) and δWWj(t′) exist and are equal, for every j∈J, then t=t′;(pCS2)
ifδWjWj∪Wk(tj) and δWkWj∪Wk(tk) exist and are equal, for every j,k∈J, then δWWj exists for every j∈J and there exists a unique section t∈G(W) such that δWWj(t)=tj, for every j∈J.


## 5.2. Cosheaves of Operators on  IndexedPIP-Spaces

Let againVI={Vr,r∈I} be an  indexed  PIP-space. If we put on I the discrete topology, then the assaying subspaces may be taken as closed sets in V. In order to build a cosheaf, we consider the same map as before in (4.1): (5.1)F:Vr⟼Opr:={Awr:Vr⟶Vw,for somew∈I,A∈Op(VI)}.
Then we immediately face a problem. Given Vq, there is no Vp, with Vq⊆Vp, such that δqp:Opp→Opq exists. One can always find an operator A such that q∈d(A) and p∉d(A); in other words, A*q exists, but it cannot be extended to Vp. This happens, for instance, each time d(A) has a maximal element (see Figure 3.1 in [5]). There are three ways out of this situation.


### 5.2.1. General Operators: The Additive Case

We still consider the set of all operatorsOp(VI). Take two assaying subspaces Vr,Vs and two operators A(r)∈Opr,B(s)∈Ops, and assume that they have a common extension C(r∨s) to Opr∨s. This means that, for any suitable w, C(r∨s):Vr∨s→Vw is the (r∨s,w)-representative Cw,r∨s of a unique operator C∈Op(VI), and C=A=B. A fortiori, A, and B coincide on Vr∧s=Vr∩Vs; that is, condition (pCS2) is satisfied.Assume now thatVI is additive; that is, Vr∨s coincides with Vr+Vs, with its inductive topology. Then we can proceed as in the case of a sheaf, by defining the extension C of A and B by linearity. In other words, since C(r∨s)fr=A*rfr and C(r∨s)fs=B*sfs, we may write, for any f=fr+fs∈Vr+Vs, (5.2)C(r∨s)f=C*r∨sfr∨s=A*rfr+B*sfs.
Here too, this operator is well defined. Let indeed f=fr+fs=fr′+fs′ be two decompositions of f∈Vr+Vs, so that fr-fr′=fs′-fs∈Vr∩Vs. Then necessarily C(r∨s)f=A*rfr+B*sfs=A*rfr′+B*sfs′. Hence, A*r(fr-fr′)=B*s(fs′-fs). Taking the restriction to Vr∩Vs=Vr∧s, this relation becomes A*r∧s(f-f′)r∧s=B*r∧s(f′-f)r∧s, and this equals 0, since A=B on Vr∧s, by the condition (pcS2).The conclusion is that, for any pair of assaying subspacesVr,Vs, the extension maps δrr∨s and δsr∨s always exist, and the condition (pCS2) is satisfied for that pair. However, this is not necessarily true for any comparable pair, and this motivates the coarsening of the order given in Proposition 5.5 below. Condition (pPC2) is also satisfied, as one can see by taking supremums (= sums) of successive pairs within three assaying subspaces Vr,Vs,Vt (and using the associativity of ∨). Thus, me may state the following.Proposition 5.5.
LetVI={Vr,r∈I} be an additive  indexed  PIP-space. Then the map  F given in (4.1) is a partial cosheaf with respect to the partial order on  I:
(5.3)r≼s⟺∃w∈Isuch thats=r∨w.


### 5.2.2. Universal Left Multipliers

We consider only the operatorsA which are everywhere defined; that is, d(A)=I. This is precisely the set of universal left multipliers: (5.4)LOp(VI)={C∈Op(VI):∀r,∃wsuch thatCwrexists}={C∈Op(VI):d(C)=I}=⋂r∈ILr,
where Lr:={C∈Op(VI):r∈d(C)}. Correspondingly, we consider the map  (5.5)G:Vr⟼LOpr:={A↾Vr:A∈LOp(VI)}.
Here again, elements of LOpr are representatives A*r. In that case, extensions always exist. When Vq⊆Vp, define δqp:LOpq→LOpp by δqp(A*q):=A*p for A*q∈LOpq. Clearly, δpp=idLOpp and δqp∘δrq=δrp if Vr⊂Vq⊂Vp. Hence, G is a pre-cosheaf. Moreover, if δqp∨q(A*q)=δpp∨q(A*p′); that is, A*,p∨q=A*,p∨q′, these two operators are the (p∨q,*) representative of a unique operator A∈LOp(VI). Then, of course, one has δp∧qq(A℘∧q)=A*q and δp∧qp(A*p∧q)=A*p; thus, G is a cosheaf. Therefore,Proposition 5.6.
LetVI={Vr,r∈I} be an arbitrary  indexed  PIP-space. Then the map given in (5.5) is a cosheaf on  VI with values in  LOp(VI), the set of universal left multipliers, with extensions given by  δqp(A*q)=A*p,q⩽p,A*q∈LOpq.


### 5.2.3. General Operators: The Projective Case

We consider again all operators inOp(VI). The fact is that the set ⋃r∈IOpr is an initial set in Op(VI), like every  d(A); that is, it contains all predecessors of any of its elements, and this is natural for constructing sheaves into it. But it is not when considering cosheaves. By duality, one should rather take a final set, that is, containing all successors of any of its elements, just like any  i(A). Hence, we define the map: (5.6)Ĝ:Vr⟼Op̂r:={Ar*:A∈Op(VI),r∈i(A)}.
Now indeed ⋁r∈IOp̂r is a final set. Elements of Op̂r have been denoted as Ar*, but, for definiteness, we could also replace Ar* by Ar∞¯ where V∞¯:=V#.We claim that the mapĜ defines a cosheaf, with extensions δqp(Aq*):=Ap*, for Aq*∈Op̂q, which exist whenever q⩽p.As in the case of Section5.2.1, assume that Ar*∈Op̂r,Bs*∈Op̂s have a common extension belonging to Op̂r∨s; that is, Ar∨s*:=Er∨s,rAr*=Br∨s*:=Er∨s,sBs*. Call this operator Cr∨s*. Thus, for any suitable w, Cr∨s,w is the (w,r∨s) representative of a unique operator C∈Op(VI), which extends A and B. Thus C=A=B∈Op(VI). Assume now that VI is projective; that is, Vr∧s=Vr∩Vs,forallr,s∈I, with the projective topology. Then, Cr∧s*=Ar∧s*=Br∧s*; that is, condition (CS2) is satisfied with extensions δr∧sr(Cr∧s*)=Ar* and δr∧ss(Cr∧s*)=Bs*. The rest is obvious. As in the case of sheaves, extensions from Vr,Vs to Vr∨s exist by linearity, since VI is projective, hence, additive. Thus, we may state the following.Proposition 5.7.
Let the  indexedPIP-space   VI be projective that is,  Vr∧s=Vr∩Vs,forallr,s∈I, with the projective topology. Then the map  Ĝ given in (5.6) is a cosheaf of operators on  VI.Corollary 5.8.
LetVI={Vr,r∈I} be a projective  indexed  PIP-space. Then  VI generates in a natural fashion a sheaf and a cosheaf of operators. If  VI is additive, but not projective; it generates a sheaf and a partial cosheaf of operators.


## 5.2.1. General Operators: The Additive Case

We still consider the set of all operatorsOp(VI). Take two assaying subspaces Vr,Vs and two operators A(r)∈Opr,B(s)∈Ops, and assume that they have a common extension C(r∨s) to Opr∨s. This means that, for any suitable w, C(r∨s):Vr∨s→Vw is the (r∨s,w)-representative Cw,r∨s of a unique operator C∈Op(VI), and C=A=B. A fortiori, A, and B coincide on Vr∧s=Vr∩Vs; that is, condition (pCS2) is satisfied.Assume now thatVI is additive; that is, Vr∨s coincides with Vr+Vs, with its inductive topology. Then we can proceed as in the case of a sheaf, by defining the extension C of A and B by linearity. In other words, since C(r∨s)fr=A*rfr and C(r∨s)fs=B*sfs, we may write, for any f=fr+fs∈Vr+Vs, (5.2)C(r∨s)f=C*r∨sfr∨s=A*rfr+B*sfs.
Here too, this operator is well defined. Let indeed f=fr+fs=fr′+fs′ be two decompositions of f∈Vr+Vs, so that fr-fr′=fs′-fs∈Vr∩Vs. Then necessarily C(r∨s)f=A*rfr+B*sfs=A*rfr′+B*sfs′. Hence, A*r(fr-fr′)=B*s(fs′-fs). Taking the restriction to Vr∩Vs=Vr∧s, this relation becomes A*r∧s(f-f′)r∧s=B*r∧s(f′-f)r∧s, and this equals 0, since A=B on Vr∧s, by the condition (pcS2).The conclusion is that, for any pair of assaying subspacesVr,Vs, the extension maps δrr∨s and δsr∨s always exist, and the condition (pCS2) is satisfied for that pair. However, this is not necessarily true for any comparable pair, and this motivates the coarsening of the order given in Proposition 5.5 below. Condition (pPC2) is also satisfied, as one can see by taking supremums (= sums) of successive pairs within three assaying subspaces Vr,Vs,Vt (and using the associativity of ∨). Thus, me may state the following.Proposition 5.5.
LetVI={Vr,r∈I} be an additive  indexed  PIP-space. Then the map  F given in (4.1) is a partial cosheaf with respect to the partial order on  I:
(5.3)r≼s⟺∃w∈Isuch thats=r∨w.


## 5.2.2. Universal Left Multipliers

We consider only the operatorsA which are everywhere defined; that is, d(A)=I. This is precisely the set of universal left multipliers: (5.4)LOp(VI)={C∈Op(VI):∀r,∃wsuch thatCwrexists}={C∈Op(VI):d(C)=I}=⋂r∈ILr,
where Lr:={C∈Op(VI):r∈d(C)}. Correspondingly, we consider the map  (5.5)G:Vr⟼LOpr:={A↾Vr:A∈LOp(VI)}.
Here again, elements of LOpr are representatives A*r. In that case, extensions always exist. When Vq⊆Vp, define δqp:LOpq→LOpp by δqp(A*q):=A*p for A*q∈LOpq. Clearly, δpp=idLOpp and δqp∘δrq=δrp if Vr⊂Vq⊂Vp. Hence, G is a pre-cosheaf. Moreover, if δqp∨q(A*q)=δpp∨q(A*p′); that is, A*,p∨q=A*,p∨q′, these two operators are the (p∨q,*) representative of a unique operator A∈LOp(VI). Then, of course, one has δp∧qq(A℘∧q)=A*q and δp∧qp(A*p∧q)=A*p; thus, G is a cosheaf. Therefore,Proposition 5.6.
LetVI={Vr,r∈I} be an arbitrary  indexed  PIP-space. Then the map given in (5.5) is a cosheaf on  VI with values in  LOp(VI), the set of universal left multipliers, with extensions given by  δqp(A*q)=A*p,q⩽p,A*q∈LOpq.


## 5.2.3. General Operators: The Projective Case

We consider again all operators inOp(VI). The fact is that the set ⋃r∈IOpr is an initial set in Op(VI), like every  d(A); that is, it contains all predecessors of any of its elements, and this is natural for constructing sheaves into it. But it is not when considering cosheaves. By duality, one should rather take a final set, that is, containing all successors of any of its elements, just like any  i(A). Hence, we define the map: (5.6)Ĝ:Vr⟼Op̂r:={Ar*:A∈Op(VI),r∈i(A)}.
Now indeed ⋁r∈IOp̂r is a final set. Elements of Op̂r have been denoted as Ar*, but, for definiteness, we could also replace Ar* by Ar∞¯ where V∞¯:=V#.We claim that the mapĜ defines a cosheaf, with extensions δqp(Aq*):=Ap*, for Aq*∈Op̂q, which exist whenever q⩽p.As in the case of Section5.2.1, assume that Ar*∈Op̂r,Bs*∈Op̂s have a common extension belonging to Op̂r∨s; that is, Ar∨s*:=Er∨s,rAr*=Br∨s*:=Er∨s,sBs*. Call this operator Cr∨s*. Thus, for any suitable w, Cr∨s,w is the (w,r∨s) representative of a unique operator C∈Op(VI), which extends A and B. Thus C=A=B∈Op(VI). Assume now that VI is projective; that is, Vr∧s=Vr∩Vs,forallr,s∈I, with the projective topology. Then, Cr∧s*=Ar∧s*=Br∧s*; that is, condition (CS2) is satisfied with extensions δr∧sr(Cr∧s*)=Ar* and δr∧ss(Cr∧s*)=Bs*. The rest is obvious. As in the case of sheaves, extensions from Vr,Vs to Vr∨s exist by linearity, since VI is projective, hence, additive. Thus, we may state the following.Proposition 5.7.
Let the  indexedPIP-space   VI be projective that is,  Vr∧s=Vr∩Vs,forallr,s∈I, with the projective topology. Then the map  Ĝ given in (5.6) is a cosheaf of operators on  VI.Corollary 5.8.
LetVI={Vr,r∈I} be a projective  indexed  PIP-space. Then  VI generates in a natural fashion a sheaf and a cosheaf of operators. If  VI is additive, but not projective; it generates a sheaf and a partial cosheaf of operators.


## 6. Cohomology of an  IndexedPIP-Space

### 6.1. Cohomology of Operator Sheaves

It is possible to introduce a concept of sheaf cohomology group defined on an  indexedPIP-space  according to the usual definition of sheaf cohomology [10]. Let VI be an  indexed  PIP-space. The set of its assaying spaces, endowed with the discrete topology, defines an open covering of V. Endowed with the Mackey topology, τ(Vr,Vr¯), each Vr is a Hausdorff vector subspace of V.Definition 6.1.
Given the  indexedPIP-space  VI, let {Vj,j∈I} be the open covering of V by its assaying subspaces, and let F:Vr↦Opr={A↾Vr:A∈Op(VI),r∈d(A)} be the sheaf of operators defined in (4.1) above. We call p-cochain with values inF the map which associates, to each intersection of open sets Vj0∩Vj1∩⋯∩Vjp,jk∈I,j0<j1<⋯<jp, an operator Aj0,j1,…,jp belonging to F(Vj0∩Vj1∩⋯∩Vjp).
Thus, ap-cochain with value in F is a set 𝒜={Aj0,j1,…,jp,j0<j1<⋯<jp}. The set of p-cochains is denoted by Cp(I,F). For example, a 0-cochain is a set {Aj,j∈I}. A 1-cochain is a set {Aj0,j1,j0<j1},Aj0,j1∈F(Vj0∩Vj1),….Definition 6.2.
Given the notion ofp-cochain, the corresponding coboundary operatorD is defined by
(6.1)D:Cp(I,F)⟶Cp+1(I,F):{Aj0,j1,…,jp}⟼{(DA)j0,j1,…,jp,jp+1},(DA)j0,j1,…,jp,jp+1=∑k=0p+1(-1)kρVj0∩Vj1∩⋯∩Vjp+1Vj0∩Vj1∩⋯∩V̂jk∩⋯∩Vjp+1Aj0,j1,…,ĵk,…,jp+1,
where the hat corresponds to the omission of the corresponding symbol. One checks that D is a coboundary operator; that is, DD=0.
If we compute the action of the coboundary operator on 0-cochains, we get(6.2)(DA)j0,j1=ρVj0∩Vj1Vj1Aj1-ρVj0∩Vj1Vj0Aj0.
This equation is nothing but the condition (S1) in Definition 4.4, that is, the necessary condition for getting a sheaf.We can also compute the action of the coboundary operator on 1-cochains. We get(6.3)(DA)j0,j1,j2=ρVj0∩Vj1∩Vj2Vj1∩Vj2Aj1,j2-ρVj0∩Vj1∩Vj2Vj0∩Vj2Aj0,j2+ρVj0∩Vj1∩Vj2Vj0∩Vj1Aj0,j1.
Now, let us suppose that a 1-cochain is defined by 𝒜=Dℬ, where ℬ is a 0-cochain on V. The previous formula becomes: (6.4)(DA)j0,j1,j2=ρVj0∩Vj1∩Vj2Vj1∩Vj2(ρVj1∩Vj2Vj2Bj2-ρVj1∩Vj2Vj1Bj1)-ρVj0∩Vj1∩Vj2Vj0∩Vj2(ρVj0∩Vj2Vj2Bj2-ρVj0∩Vj2Vj0Bj0)+ρVj0∩Vj1∩Vj2Vj0∩Vj1(ρVj0∩Vj1Vj1Bj1-ρVj0∩Vj1Vj0Bj0).
Using the properties of restrictions we get (6.5)(DA)j0,j1,j2=ρVj0∩Vj1∩Vj2Vj2Bj2-ρVj0∩Vj1∩Vj2Vj1Bj1-ρVj0∩Vj1∩Vj2Vj2Bj2+ρVj0∩Vj1∩Vj2Vj0Bj0+ρVj0∩Vj1∩Vj2Vj1Bj1-ρVj0∩Vj1∩Vj2Vj0Bj0=0,
which shows that indeed DDℬ=0.Now it is possible to define cohomology groups of the sheafF on a  indexed  PIP-space  V.Definition 6.3.
LetVI be an  indexed  PIP-space, endowed with the open covering of its assaying spaces, {Vj,j∈I}, and let F be the corresponding sheaf of operators on VI. Then the pth cohomology group is defined as Hp(I,F):=Zp(I,F)/Bp(I,F), where Zp(I,F) is the set of p-cocycles, that is, p-cochains 𝒜 such that D𝒜=0, and Bp(I,F) is the set of p-coboundaries, that is, p-cochains ℬ for which there exists a (p-1)-cochain 𝒞 with ℬ=D𝒞.The definition of the groupHp(I,F) is motivated by the fact that the p-cocycle 𝒜 is given only up to a coboundary ℬ=D𝒞:D𝒜=0=D(𝒜+ℬ)=D(𝒜+D𝒞).Our definition of cohomology groups of sheaves on  indexedPIP-spaces depends on the particular open covering we are choosing. So far we have used the open covering given by all assaying subspaces, but there might be other ones, typically consisting of unions of assaying subspaces. We say that an open covering {Vj,j∈J} of an  indexed  PIP-space VI is finer than another one, {Uk,k∈K}, if there exists an application t:J→K such that Vj⊂Ut(j),forallj∈J. For instance, Ut(j) could be a union ∪lVl containing Vj. According to the general theory of sheaf cohomology [9], this induces group homomorphisms tJK:Hp(K,F)→Hp(J,F),forallp⩾0. It is then possible to introduce cohomology groups that do not depend on the particular open covering, namely, by defining Hp(V,F) as the inductive limit of the groups Hp(J,F) with respect to the homomorphisms tJK.Using a famous result of Cartan and Leray and an unpublished work of J. Shabani, we give now a theorem which characterizes the cohomology of sheaves on  indexedPIP-spaces. We collect in Appendix B the definitions and the results needed for this discussion. We know that an  indexed  PIP-space  V endowed with the Mackey topology is a separated (Hausdorff) locally convex space, but it is not necessarily paracompact, unless V is metrizable, in particular, a Banach or a  Hilbert space. In that case, it is possible to define a fine sheaf (see Definition B.3) of operators on V and apply the Cartan-Leray Theorem B.4. This justifies the restriction of metrizability in the following theorem.Theorem 6.4.
Given an  indexedPIP-space   VI, define the sheaf  F:Vr→F(Vr):=Opr={Ar:Vr→V}. Then, if  V is metrizable for its Mackey topology, the sheaf  F is acyclic; that is, the cohomology groups  Hp(V,F) are trivial for all  p⩾1;  that is, Hp(V,F)=0.Proof.
First of all,V is a paracompact space, since it is metrizable.
Next, we check that the sheafF:Vr→F(Vr)=Opr={Ar:Vr→V} on the paracompact space V is fine. Indeed, let {Vj,j∈J} be a locally finite open covering of V. We can associate to the latter a partition of unity {φj,j∈J}. Then we can use {φj,j∈J} to define the homomorphisms hj:F→F. Let Vr⊂∪s∈KVs for some index set K. For each s∈K, we consider the set of operators Ops={A*s:Vs→V}, and we define hj:As↦hj(As):=φjAs. This defines a homomorphism Ops→Ops and then a homomorphism F→F. One can check that hj:F→F satisfies conditions (1) and (2) in Definition B.3, and, thus, F is a fine sheaf.
Then the result follows from TheoremB.4.The crucial point of the proof here is the Cartan-Leray theorem. Let us give a flavor of the proof of a particular case of this classical result. Let[𝒞] be an element of H1(V,F). We want to show that [𝒞]=0. As an equivalence class, [𝒞] can be represented by an element 𝒞=(Cj0,j1,j0<j1)∈I of C1(I,F), such that D𝒞=0; that is,
(6.6)(DC)j0,j1,j2=ρVj0∩Vj1∩Vj2Vj1∩Vj2Cj1,j2-ρVj0∩Vj1∩Vj2Vj0∩Vj2Cj0,j2+ρVj0∩Vj1∩Vj2Vj0∩Vj1Cj0,j1=0.
Using the fact that F is a fine sheaf, we can choose the partition of unity {φi,i∈I} to define the 0-cochain ℰ={Ej}, where Ej=-∑k∈Iφk(Cjk). This sum is well defined since the covering is locally finite. Applying the coboundary operator, we get (6.7)(DE)j0,j1=ρVj0∩Vj1Vj1Ej1-ρVj0∩Vj1Vj0Ej0=-∑k∈Iφk(ρVj0∩Vj1Vj1Cj1k)+∑k∈Iφk(ρVj0∩Vj1Vj0Cj0k)=∑k∈Iφk(ρVj0∩Vj1Vj0Cj0k-ρVj0∩Vj1Vj1Cj1k)=∑k∈Iφk(ρVj0∩Vj1∩VkVj0∩VkCj0k-ρVj0∩Vj1∩VkVj1∩VkCj1k).
Putting k=j2 and using the equation (D𝒞)j0,j1,j2=0, we find (6.8)(DE)j0,j1=∑j2∈Iφj2(ρVj0∩Vj1∩Vj2Vj0∩Vj1Cj0j1).
But since {φi,i∈I} is a partition of unity, we get (Dℰ)j0,j1=Cj0j1. This means that Dℰ=𝒞, and; thus, [𝒞]=0.Of course, the restriction thatV is to be metrizable is quite strong, but still the result applies to a significant number of interesting situations, for instance,   (1)
afinite chain of reflexive Banach spaces or  Hilbert spaces, for instance, a triplet of  Hilbert spaces ℋ1↪ℋ0↪ℋ1¯ or any of its refinements, as discussed in [5, Section 5.2.2],(2)
an  indexedPIP-space  VI whose extreme space V is itself a  Hilbert space, like the LHS of functions analytic in a sector described in [5, Section 4.6.3],(3)
a Banach Gel'fand triple, in the sense of Feichtinger (see [11]), that is, an RHS (or LBS) in which the extreme spaces are (nonreflexive) Banach spaces. A nice example, extremely useful in Gabor analysis, is the so-called Feichtinger algebra𝒮0(ℝd), which generates the triplet
(6.9)S0(Rd)↪L2(Rd)↪S0×(Rd).
The latter can often replace the familiar Schwartz triplet of tempered distributions. Of course, one can design all sorts of LHSs of LBSs interpolating between the extreme spaces, as explained in [5, Sections 5.3 and 5.4].In fact, what is really needed for the Cartan-Leray theorem is not thatV becomes metrizable, but that it becomes paracompact. Indeed, without that condition, the situation becomes totally unmanageable. However, except for metrizable spaces, we could not find interesting examples of  indexed  PIP-spaces with V paracompact.


### 6.2. Cohomology of Operator Cosheaves

It is also possible to get cohomological concepts on cosheaves defined from  indexedPIP-spaces. The assaying spaces can also be considered as closed sets. Let, thus, {Wj,j∈I} be such a closed covering of V, and let G be a cosheaf of operators defined in (5.5), namely, G:Wj↦LOpj={A↾Wj:A∈LOp(VI)} together with maps δWiWj:G(Wi)→G(Wj),Wi⊂Wj as above. In the sequel, G(Wj0∪Wj1∪⋯∪Wjp) denotes the set of operators Wj0∪Wj1∪⋯∪Wjp→V. Alternatively, one could consider the cosheaf defined in (5.6), Ĝ:Wj↦Op̂j:={Aj*:A∈Op(VI),j∈i(A)} and proceed in the same way.In this setup, we may introduce the necessary cohomological concepts, as in Definitions6.1 and 6.2.Definition 6.5.
Ap-cochain with values in the cosheaf G is a map which associates to each union of closed sets Wj0∪Wj1∪⋯∪Wjp,jk∈I,j0<j1<⋯<jp, an operator Aj0,j1,…,jp of G(Wj0∪Wj1∪⋯∪Wjp). A p-cochain is, thus, a set 𝔄={Aj0,j1,…,jp,j0<j1<⋯<jp}. The set of such p-cochains will be denoted by Ĉp(I,G).Definition 6.6.
One can then introduce the coboundary operatorD̂ as follows:
(6.10)D̂:Ĉp(I,G)⟶Ĉp+1(I,G):A={Aj0,j1,…,jp}⟼{(D̂A)j0,j1,…,jp,jp+1},(D̂A)j0,j1,…,jp,jp+1=∑k=0p+1(-1)kδWj0∪Wj1∪⋯∪Ŵjk∪⋯∩Wjp+1Wj0∪Wj1∪⋯∪Wjp+1Aj0,j1,…,ĵk,…,jp+1.
In view of the properties of the maps δ**, we check that D̂D̂=0.In the case of 0-cochains, a straightforward application of the formula leads to(6.11)(D̂A)j0,j1=δWj1Wj0∪Wj1Aj1-δWj0Wj0∪Wj1Aj0.
And if we put this to zero, we get the constraint δWj1Wj0∪Wj1Aj1=δWj0Wj0∪Wj1Aj0, which has to be satisfied in order to build a cosheaf, according to the condition (CS2). On 1-cochains, the coboundary action gives (6.12)(D̂A)j0,j1,j2=δWj1∪Wj2Wj0∪Wj1∪Wj2Aj1,j2-δWj0∪Wj2Wj0∪Wj1∪Wj2Aj0,j2+δWj0∪Wj1Wj0∪Wj1∪Wj2Aj0,j1.
The cohomology groups of the cosheaf Ĥp(I,G):=Ẑp(I,G)/B̂p(I,G), with obvious notations, can then be defined in a natural way. Similarly for Ĥp(V,G) and Ĥp(I,Ĝ),Ĥp(V,Ĝ).Now it is tempting to proceed as in the case of sheaves and define the analog of a fine cosheaf, in such a way that one can apply a result similar to the Cartan-Leray theorem. But this is largely unexplored territory, so we will not venture into it.


## 6.1. Cohomology of Operator Sheaves

It is possible to introduce a concept of sheaf cohomology group defined on an  indexedPIP-space  according to the usual definition of sheaf cohomology [10]. Let VI be an  indexed  PIP-space. The set of its assaying spaces, endowed with the discrete topology, defines an open covering of V. Endowed with the Mackey topology, τ(Vr,Vr¯), each Vr is a Hausdorff vector subspace of V.Definition 6.1.
Given the  indexedPIP-space  VI, let {Vj,j∈I} be the open covering of V by its assaying subspaces, and let F:Vr↦Opr={A↾Vr:A∈Op(VI),r∈d(A)} be the sheaf of operators defined in (4.1) above. We call p-cochain with values inF the map which associates, to each intersection of open sets Vj0∩Vj1∩⋯∩Vjp,jk∈I,j0<j1<⋯<jp, an operator Aj0,j1,…,jp belonging to F(Vj0∩Vj1∩⋯∩Vjp).
Thus, ap-cochain with value in F is a set 𝒜={Aj0,j1,…,jp,j0<j1<⋯<jp}. The set of p-cochains is denoted by Cp(I,F). For example, a 0-cochain is a set {Aj,j∈I}. A 1-cochain is a set {Aj0,j1,j0<j1},Aj0,j1∈F(Vj0∩Vj1),….Definition 6.2.
Given the notion ofp-cochain, the corresponding coboundary operatorD is defined by
(6.1)D:Cp(I,F)⟶Cp+1(I,F):{Aj0,j1,…,jp}⟼{(DA)j0,j1,…,jp,jp+1},(DA)j0,j1,…,jp,jp+1=∑k=0p+1(-1)kρVj0∩Vj1∩⋯∩Vjp+1Vj0∩Vj1∩⋯∩V̂jk∩⋯∩Vjp+1Aj0,j1,…,ĵk,…,jp+1,
where the hat corresponds to the omission of the corresponding symbol. One checks that D is a coboundary operator; that is, DD=0.
If we compute the action of the coboundary operator on 0-cochains, we get(6.2)(DA)j0,j1=ρVj0∩Vj1Vj1Aj1-ρVj0∩Vj1Vj0Aj0.
This equation is nothing but the condition (S1) in Definition 4.4, that is, the necessary condition for getting a sheaf.We can also compute the action of the coboundary operator on 1-cochains. We get(6.3)(DA)j0,j1,j2=ρVj0∩Vj1∩Vj2Vj1∩Vj2Aj1,j2-ρVj0∩Vj1∩Vj2Vj0∩Vj2Aj0,j2+ρVj0∩Vj1∩Vj2Vj0∩Vj1Aj0,j1.
Now, let us suppose that a 1-cochain is defined by 𝒜=Dℬ, where ℬ is a 0-cochain on V. The previous formula becomes: (6.4)(DA)j0,j1,j2=ρVj0∩Vj1∩Vj2Vj1∩Vj2(ρVj1∩Vj2Vj2Bj2-ρVj1∩Vj2Vj1Bj1)-ρVj0∩Vj1∩Vj2Vj0∩Vj2(ρVj0∩Vj2Vj2Bj2-ρVj0∩Vj2Vj0Bj0)+ρVj0∩Vj1∩Vj2Vj0∩Vj1(ρVj0∩Vj1Vj1Bj1-ρVj0∩Vj1Vj0Bj0).
Using the properties of restrictions we get (6.5)(DA)j0,j1,j2=ρVj0∩Vj1∩Vj2Vj2Bj2-ρVj0∩Vj1∩Vj2Vj1Bj1-ρVj0∩Vj1∩Vj2Vj2Bj2+ρVj0∩Vj1∩Vj2Vj0Bj0+ρVj0∩Vj1∩Vj2Vj1Bj1-ρVj0∩Vj1∩Vj2Vj0Bj0=0,
which shows that indeed DDℬ=0.Now it is possible to define cohomology groups of the sheafF on a  indexed  PIP-space  V.Definition 6.3.
LetVI be an  indexed  PIP-space, endowed with the open covering of its assaying spaces, {Vj,j∈I}, and let F be the corresponding sheaf of operators on VI. Then the pth cohomology group is defined as Hp(I,F):=Zp(I,F)/Bp(I,F), where Zp(I,F) is the set of p-cocycles, that is, p-cochains 𝒜 such that D𝒜=0, and Bp(I,F) is the set of p-coboundaries, that is, p-cochains ℬ for which there exists a (p-1)-cochain 𝒞 with ℬ=D𝒞.The definition of the groupHp(I,F) is motivated by the fact that the p-cocycle 𝒜 is given only up to a coboundary ℬ=D𝒞:D𝒜=0=D(𝒜+ℬ)=D(𝒜+D𝒞).Our definition of cohomology groups of sheaves on  indexedPIP-spaces depends on the particular open covering we are choosing. So far we have used the open covering given by all assaying subspaces, but there might be other ones, typically consisting of unions of assaying subspaces. We say that an open covering {Vj,j∈J} of an  indexed  PIP-space VI is finer than another one, {Uk,k∈K}, if there exists an application t:J→K such that Vj⊂Ut(j),forallj∈J. For instance, Ut(j) could be a union ∪lVl containing Vj. According to the general theory of sheaf cohomology [9], this induces group homomorphisms tJK:Hp(K,F)→Hp(J,F),forallp⩾0. It is then possible to introduce cohomology groups that do not depend on the particular open covering, namely, by defining Hp(V,F) as the inductive limit of the groups Hp(J,F) with respect to the homomorphisms tJK.Using a famous result of Cartan and Leray and an unpublished work of J. Shabani, we give now a theorem which characterizes the cohomology of sheaves on  indexedPIP-spaces. We collect in Appendix B the definitions and the results needed for this discussion. We know that an  indexed  PIP-space  V endowed with the Mackey topology is a separated (Hausdorff) locally convex space, but it is not necessarily paracompact, unless V is metrizable, in particular, a Banach or a  Hilbert space. In that case, it is possible to define a fine sheaf (see Definition B.3) of operators on V and apply the Cartan-Leray Theorem B.4. This justifies the restriction of metrizability in the following theorem.Theorem 6.4.
Given an  indexedPIP-space   VI, define the sheaf  F:Vr→F(Vr):=Opr={Ar:Vr→V}. Then, if  V is metrizable for its Mackey topology, the sheaf  F is acyclic; that is, the cohomology groups  Hp(V,F) are trivial for all  p⩾1;  that is, Hp(V,F)=0.Proof.
First of all,V is a paracompact space, since it is metrizable.
Next, we check that the sheafF:Vr→F(Vr)=Opr={Ar:Vr→V} on the paracompact space V is fine. Indeed, let {Vj,j∈J} be a locally finite open covering of V. We can associate to the latter a partition of unity {φj,j∈J}. Then we can use {φj,j∈J} to define the homomorphisms hj:F→F. Let Vr⊂∪s∈KVs for some index set K. For each s∈K, we consider the set of operators Ops={A*s:Vs→V}, and we define hj:As↦hj(As):=φjAs. This defines a homomorphism Ops→Ops and then a homomorphism F→F. One can check that hj:F→F satisfies conditions (1) and (2) in Definition B.3, and, thus, F is a fine sheaf.
Then the result follows from TheoremB.4.The crucial point of the proof here is the Cartan-Leray theorem. Let us give a flavor of the proof of a particular case of this classical result. Let[𝒞] be an element of H1(V,F). We want to show that [𝒞]=0. As an equivalence class, [𝒞] can be represented by an element 𝒞=(Cj0,j1,j0<j1)∈I of C1(I,F), such that D𝒞=0; that is,
(6.6)(DC)j0,j1,j2=ρVj0∩Vj1∩Vj2Vj1∩Vj2Cj1,j2-ρVj0∩Vj1∩Vj2Vj0∩Vj2Cj0,j2+ρVj0∩Vj1∩Vj2Vj0∩Vj1Cj0,j1=0.
Using the fact that F is a fine sheaf, we can choose the partition of unity {φi,i∈I} to define the 0-cochain ℰ={Ej}, where Ej=-∑k∈Iφk(Cjk). This sum is well defined since the covering is locally finite. Applying the coboundary operator, we get (6.7)(DE)j0,j1=ρVj0∩Vj1Vj1Ej1-ρVj0∩Vj1Vj0Ej0=-∑k∈Iφk(ρVj0∩Vj1Vj1Cj1k)+∑k∈Iφk(ρVj0∩Vj1Vj0Cj0k)=∑k∈Iφk(ρVj0∩Vj1Vj0Cj0k-ρVj0∩Vj1Vj1Cj1k)=∑k∈Iφk(ρVj0∩Vj1∩VkVj0∩VkCj0k-ρVj0∩Vj1∩VkVj1∩VkCj1k).
Putting k=j2 and using the equation (D𝒞)j0,j1,j2=0, we find (6.8)(DE)j0,j1=∑j2∈Iφj2(ρVj0∩Vj1∩Vj2Vj0∩Vj1Cj0j1).
But since {φi,i∈I} is a partition of unity, we get (Dℰ)j0,j1=Cj0j1. This means that Dℰ=𝒞, and; thus, [𝒞]=0.Of course, the restriction thatV is to be metrizable is quite strong, but still the result applies to a significant number of interesting situations, for instance,   (1)
afinite chain of reflexive Banach spaces or  Hilbert spaces, for instance, a triplet of  Hilbert spaces ℋ1↪ℋ0↪ℋ1¯ or any of its refinements, as discussed in [5, Section 5.2.2],(2)
an  indexedPIP-space  VI whose extreme space V is itself a  Hilbert space, like the LHS of functions analytic in a sector described in [5, Section 4.6.3],(3)
a Banach Gel'fand triple, in the sense of Feichtinger (see [11]), that is, an RHS (or LBS) in which the extreme spaces are (nonreflexive) Banach spaces. A nice example, extremely useful in Gabor analysis, is the so-called Feichtinger algebra𝒮0(ℝd), which generates the triplet
(6.9)S0(Rd)↪L2(Rd)↪S0×(Rd).
The latter can often replace the familiar Schwartz triplet of tempered distributions. Of course, one can design all sorts of LHSs of LBSs interpolating between the extreme spaces, as explained in [5, Sections 5.3 and 5.4].In fact, what is really needed for the Cartan-Leray theorem is not thatV becomes metrizable, but that it becomes paracompact. Indeed, without that condition, the situation becomes totally unmanageable. However, except for metrizable spaces, we could not find interesting examples of  indexed  PIP-spaces with V paracompact.


## 6.2. Cohomology of Operator Cosheaves

It is also possible to get cohomological concepts on cosheaves defined from  indexedPIP-spaces. The assaying spaces can also be considered as closed sets. Let, thus, {Wj,j∈I} be such a closed covering of V, and let G be a cosheaf of operators defined in (5.5), namely, G:Wj↦LOpj={A↾Wj:A∈LOp(VI)} together with maps δWiWj:G(Wi)→G(Wj),Wi⊂Wj as above. In the sequel, G(Wj0∪Wj1∪⋯∪Wjp) denotes the set of operators Wj0∪Wj1∪⋯∪Wjp→V. Alternatively, one could consider the cosheaf defined in (5.6), Ĝ:Wj↦Op̂j:={Aj*:A∈Op(VI),j∈i(A)} and proceed in the same way.In this setup, we may introduce the necessary cohomological concepts, as in Definitions6.1 and 6.2.Definition 6.5.
Ap-cochain with values in the cosheaf G is a map which associates to each union of closed sets Wj0∪Wj1∪⋯∪Wjp,jk∈I,j0<j1<⋯<jp, an operator Aj0,j1,…,jp of G(Wj0∪Wj1∪⋯∪Wjp). A p-cochain is, thus, a set 𝔄={Aj0,j1,…,jp,j0<j1<⋯<jp}. The set of such p-cochains will be denoted by Ĉp(I,G).Definition 6.6.
One can then introduce the coboundary operatorD̂ as follows:
(6.10)D̂:Ĉp(I,G)⟶Ĉp+1(I,G):A={Aj0,j1,…,jp}⟼{(D̂A)j0,j1,…,jp,jp+1},(D̂A)j0,j1,…,jp,jp+1=∑k=0p+1(-1)kδWj0∪Wj1∪⋯∪Ŵjk∪⋯∩Wjp+1Wj0∪Wj1∪⋯∪Wjp+1Aj0,j1,…,ĵk,…,jp+1.
In view of the properties of the maps δ**, we check that D̂D̂=0.In the case of 0-cochains, a straightforward application of the formula leads to(6.11)(D̂A)j0,j1=δWj1Wj0∪Wj1Aj1-δWj0Wj0∪Wj1Aj0.
And if we put this to zero, we get the constraint δWj1Wj0∪Wj1Aj1=δWj0Wj0∪Wj1Aj0, which has to be satisfied in order to build a cosheaf, according to the condition (CS2). On 1-cochains, the coboundary action gives (6.12)(D̂A)j0,j1,j2=δWj1∪Wj2Wj0∪Wj1∪Wj2Aj1,j2-δWj0∪Wj2Wj0∪Wj1∪Wj2Aj0,j2+δWj0∪Wj1Wj0∪Wj1∪Wj2Aj0,j1.
The cohomology groups of the cosheaf Ĥp(I,G):=Ẑp(I,G)/B̂p(I,G), with obvious notations, can then be defined in a natural way. Similarly for Ĥp(V,G) and Ĥp(I,Ĝ),Ĥp(V,Ĝ).Now it is tempting to proceed as in the case of sheaves and define the analog of a fine cosheaf, in such a way that one can apply a result similar to the Cartan-Leray theorem. But this is largely unexplored territory, so we will not venture into it.


## 7. Outcome

The analysis so far shows that several aspects of the theory of  indexedPIP-spaces and operators on them have a natural formulation in categorical terms. Of course, this is only a first step, many questions remain open. For instance, does there exist a simple characterisation of the dual category PIPop? Could it be somehow linked to the category of partial *algebras, in the same way as Setop is isomorphic to the category of complete atomic Boolean algebras (this is the so-called Lindenbaum-Tarski duality [12, Section VI.4.6]?In addition, our constructions yield new concrete examples of sheaves and cosheaves, namely, (co)sheaves of operators on an  indexedPIP-space, and this is probably the most important result of this paper. Then, another open question concerns the cosheaf cohomology groups. Can one find conditions under which the cosheaf is acyclic, that is, Ĥp(V,G)=0, for all p⩾1, or, similarly, Ĥp(V,Ĝ)=0, for all p⩾1?In guise of conclusion, let us note that cosheaf is a new concept which was introduced in a logical framework in order to dualize the sheaf concept [13]. In fact one knows that the category of sheaves (which is in fact a topos) is related to Intuitionistic logic and Heyting algebras, in the same way as the category of sets has deep relations with the classical proposition logic and Boolean algebras [12, Section I.1.10].More precisely, Classical logic satisfies the noncontradiction principleNCP¬(p∧¬p) and the excluded middle principle EMP (p∨¬p) (i.e.,  NCP: = Not-(p And Not-p) and  EMP: = p Or Not-p.). Intuitionistic logic satisfies NCP but not EMP. Finally, we know that Paraconsistent logic, satisfying  EMP but not  NCP, is related to Brouwer algebras, also called co-Heyting algebras [14, 15]. Then, it is natural to wonder what is the category, if any, (mimicking the category of sets for the classical case and the topos of sheaves for the intuitionistic case), corresponding to Paraconsistent logic? The category of cosheaves can be a natural candidate for this. And this is the reason why it was tentatively introduced in a formal logic context. The category of closed sets of a topological space happens to be a cosheaf. But up to now we did not know any other examples of cosheaves in other areas of mathematics. Therefore, it is interesting to find here additional concrete examples of cosheaves in the field of functional analysis.In a completely different field, the search for a quantum gravity theory, Shahn Majid [16, page 294] (see also the diagram in [17, page 122]) has proposed to unify quantum field theory and general relativity using a self-duality principle expressed in categorical terms. His approach shows deep connections, on the one hand, between quantum concepts and Heyting algebras (the relations between quantum physics and Intuitionistic logic are well known) and, on the other hand, following a suggestion of Lawvere [16], between general relativity (Riemannian geometry and uniform spaces) and co-Heyting algebras (Brouwer algebras). Therefore, it is very interesting to shed light on concepts arising naturally from Brouwer algebras, and this is precisely the case of cosheaves. Sheaves of operators on   PIP-spaces are connected to quantum physics. Is there any hope to connect cosheaves of operators on some PIP-spaces to (pseudo-)Riemannian geometry, uniform spaces, or to theories describing gravitation? This is an open question suggested by Majid's idea of a self-duality principle (let us note that classical logic is the prototype of a self-dual structure, self-duality being given by the de Morgan rule, which transforms  NCP into  EMP!).



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