Partial Inner Product Spaces: Some Categorical Aspects

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.

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, a 𝖯𝖨𝖯-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 single 𝖯𝖨𝖯-space 𝑉_𝐼 as a category by itself, called 𝐕_𝐈, with natural embeddings as arrows. For this category 𝐕_𝐈, 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 𝑉_𝐼, 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 Section 7) 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 indexed 𝖯𝖨𝖯-space 𝑉_𝐼={𝑉_𝑟,𝑟∈𝐼}, corresponding to the linear compatibility #. The assaying subspaces are denoted by 𝑉_𝑝,𝑉_𝑟,…,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, 𝑓#𝑔⟺∃𝑟∈𝐼suchthat𝑓∈𝑉_𝑟,𝑔∈𝑉_𝑟. (2.1) The lattice properties are the following: (i)involution: 𝑉_𝑟↔𝑉_𝑟∶=(𝑉_𝑟)^#,(ii)infimum: 𝑉_𝑝∧𝑞∶=𝑉_𝑝∧𝑉_𝑞=𝑉_𝑝∩𝑉_𝑞,(𝑝,𝑞,𝑟∈𝐼),(iii)supremum: 𝑉_𝑝∨𝑞∶=𝑉_𝑝∨𝑉_𝑞=(𝑉_𝑝+𝑉_𝑞)^##.

The smallest element of ℱ(𝑉,#) is 𝑉^#=⋂_𝑟𝑉_𝑟, and the greatest element is 𝑉=⋃_𝑟𝑉_𝑟, but often they do not belong to 𝑉_𝐼.

Each assaying subspace 𝑉_𝑟 carries its Mackey topology 𝜏(𝑉_𝑟,𝑉_𝑟), and 𝑉^# is dense in every 𝑉_𝑟, since the indexed 𝖯𝖨𝖯-space 𝑉_𝐼 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 indexed 𝖯𝖨𝖯-spaces 𝑉_𝐼,𝑌_𝐾, an operator 𝐴∶𝑉_𝐼→𝑌_𝐾 may be identified with the coherent collection of its representatives 𝐴≃{𝐴_𝑢𝑟}, where each 𝐴_𝑢𝑟∶𝑉_𝑟→𝑌_𝑢 is a continuous operator from 𝑉_𝑟 into 𝑌_𝑢. We will also need the set 𝖽(𝐴)={𝑟∈𝐼∶thereisa𝑢∈𝐾suchthat𝐴_𝑢𝑟exists}. 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 from 𝑉_𝐼 into 𝑌_𝐾 is denoted by Op(𝑉_𝐼,𝑌_𝐾) and the set of all homomorphisms by Hom(𝑉_𝐼,𝑌_𝐾).

For more details and references to related work, see Appendix A 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 1_𝑋∶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∘1_𝑋=𝛼 and 1_𝑌∘𝛼=𝛼.

In a category, an object 𝑆 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 categories C and D, a covariant functor 𝖥∶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 𝖥1_𝑋=1_𝖥(𝑋),𝖥𝛽∘𝛼=𝖥𝛽∘𝖥(𝛼), (2.2) whenever the arrow 𝛽∘𝛼 is defined in C.

Given a category C, the opposite category C^op has the same objects as C and all arrows reversed: to each arrow 𝛼∶𝑋→𝑌, there is an arrow 𝛼^op∶𝑌→𝑋, so that 𝛼^op∘𝛽^op=(𝛽∘𝛼)^op.

A contravariant functor 𝖥∶C→D may be defined as a functor 𝖥∶C^op→D, or directly on C, by writing 𝖥(𝛼)=𝖥(𝛼^op). Thus, we have 𝖥1_𝑋=1_𝖥(𝑋),𝖥𝛽∘𝛼=𝖥(𝛼)∘𝖥𝛽. (2.3) 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 of 𝖯𝖨𝖯-Spaces

3.1. A Single 𝖯𝖨𝖯-Space as Category

We begin by a trivial example.

A single-indexed 𝖯𝖨𝖯-space 𝑉_𝐼={𝑉_𝑟,𝑟∈𝐼} may be considered as a category 𝐕_𝐈, where (i)the objects are the assaying subspaces {𝑉_𝑟,𝑟∈𝐼,𝑉^#,𝑉}; (ii)the arrows are the natural embeddings {𝐸_𝑟𝑠∶𝑉_𝑟→𝑉_𝑠,𝑟⩽𝑠}, that is, the representatives of the identity operator on 𝑉_𝐼.

The axioms of categories are readily checked as follows. (i)For every 𝑉_𝑟, there exists an identity, 𝐸_𝑟𝑟∶𝑉_𝑟→𝑉_𝑟, the identity map. (ii)For every 𝑉_𝑟,𝑉_𝑠 with 𝑟⩽𝑠, one has 𝐸_𝑠𝑠∘𝐸_𝑠𝑟=𝐸_𝑠𝑟 and 𝐸_𝑠𝑟∘𝐸_𝑟𝑟=𝐸_𝑠𝑟. (iii)For every 𝑉_𝑟,𝑉_𝑠,𝑉_𝑡 with 𝑟⩽𝑠⩽𝑡, one has 𝐸_𝑡𝑠∘𝐸_𝑠𝑟=𝐸_𝑡𝑟. (iv)For every 𝑉_𝑟,𝑉_𝑠,𝑉_𝑡,𝑉_𝑢 with 𝑟⩽𝑠⩽𝑡⩽𝑢, one has (𝐸_𝑢𝑡∘𝐸_𝑡𝑠)∘𝐸_𝑠𝑟=𝐸_𝑢𝑡∘(𝐸_𝑡𝑠∘𝐸_𝑠𝑟).

In the category 𝐕_𝐈, we have: (i)𝑉_∞∶=𝑉^#=⋂_𝑟∈𝐼 is an initial object: for every 𝑉_𝑟∈𝑉_𝐼, there is a unique arrow 𝐸_𝑟∞∶𝑉^#→𝑉_𝑟. (ii)𝑉_∞∶=𝑉=∑_𝑟∈𝐼 is a terminal object: for every 𝑉_𝑟∈𝑉_𝐼, there is a unique arrow 𝐸_∞𝑟∶𝑉_𝑟→𝑉. (iii)The compatibility #∶𝑉_𝑟↦(𝑉_𝑟)^#=𝑉_𝑟 defines a contravariant functor 𝐕_𝐈→𝐕_𝐈.

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 indexed 𝖯𝖨𝖯-space 𝑉_𝐼={𝑉_𝑟,𝑟∈𝐼}, take a single totally regular operator, that is, an operator 𝐴 that leaves every 𝑉_𝑟 invariant. Hence so does each power 𝐴^𝑛,𝑛∈ℕ. Then this operator induces a category A(𝐕_𝐈), as follows. (i)The objects are the assaying subspaces {𝑉_𝑟,𝑟∈𝐼}. (ii)The arrows are the operators 𝐴^𝑛_𝑝𝑞∶𝑉_𝑞→𝑉_𝑝,𝑞⩽𝑝,𝑛∈ℕ.

The axioms of categories are readily checked as follows. (i)For every 𝑉_𝑟, there exists an identity, 𝐴_𝑟𝑟∶𝑉_𝑟→𝑉_𝑟, since 𝐴 is totally regular. (ii)For every 𝑉_𝑟,𝑉_𝑠 with 𝑟⩽𝑠, one has 𝐴_𝑠𝑠∘𝐴^𝑛_𝑠𝑟=𝐴^𝑛+1_𝑠𝑟 and 𝐴^𝑛_𝑠𝑟∘𝐴_𝑟𝑟=𝐴^𝑛+1_𝑠𝑟,forall𝑛∈ℕ. (iii)For every 𝑉_𝑟,𝑉_𝑠,𝑉_𝑡 with 𝑟⩽𝑠⩽𝑡, one has 𝐴^𝑚_𝑡𝑠∘𝐴^𝑛_𝑠𝑟=𝐴^𝑚+𝑛_𝑡𝑟,forall𝑚,𝑛∈ℕ. (iv)For every 𝑉_𝑟,𝑉_𝑠,𝑉_𝑡,𝑉_𝑢 with 𝑟⩽𝑠⩽𝑡⩽𝑢, one has (𝐴^𝑚_𝑢𝑡∘𝐴^𝑛_𝑡𝑠)∘𝐴^𝑘_𝑠𝑟=𝐴^𝑚_𝑢𝑡∘(𝐴^𝑛_𝑡𝑠∘𝐴^𝑘_𝑠𝑟),forall𝑚,𝑛,𝑘∈ℕ.

As for 𝐕_𝐈, the space 𝑉_∞∶=𝑉^# is an initial object in A(𝐕_𝐈) and 𝑉_∞∶=𝑉 is a terminal object.

The adjunction 𝐴↦𝐴^# defines a contravariant functor from A(𝐕_𝐈) into 𝐀(𝐕_𝐈)^#, where the latter is the category induced by 𝐴^#. The proof is immediate.

3.3. The Category PIP of Indexed 𝖯𝖨𝖯-Spaces

The collection of all indexed 𝖯𝖨𝖯-spaces constitutes a category, which we call PIP, where (i)objects are indexed 𝖯𝖨𝖯-spaces {𝑉_𝐼}; (ii)arrows are homomorphisms 𝐴∶𝑉_𝐼→𝑌_𝐾, where an operator 𝐴∈Op(𝑉_𝐼,𝑌_𝐾) is called a homomorphism if (a)for every 𝑟∈𝐼 there exists 𝑢∈𝐾 such that both 𝐴_𝑢𝑟 and 𝐴_𝑢𝑟 exist; (b)for every 𝑢∈𝐾 there exists 𝑟∈𝐼 such that both 𝐴_𝑢𝑟 and 𝐴_𝑢𝑟 exist.

For making the notation less cumbersome (and more automatic), we will, henceforth, denote by 𝐴_𝐾𝐼 an element 𝐴∈Hom(𝑉_𝐼,𝑌_𝐾). Then the axioms of a category are obviously satisfied as follows. (i)For every 𝑉_𝐼, there exists an identity, 1_𝐼∈Hom(𝑉_𝐼,𝑉_𝐼), the identity operator on 𝑉_𝐼. (ii)For every 𝑉_𝐼,𝑌_𝐾, one has 1_𝐾∘𝐴_𝐾𝐼=𝐴_𝐾𝐼 and 𝐴_𝐾𝐼∘1_𝐼=𝐴_𝐾𝐼. (iii)For every 𝑉_𝐼,𝑌_𝐾,𝑊_𝐿, one has 𝐵_𝐿𝐾∘𝐴_𝐾𝐼=𝐶_𝐿𝐼∈Hom(𝑉_𝐼,𝑊_𝐿). (iv)For every 𝑉_𝐼,𝑌_𝐾,𝑊_𝐿,𝑍_𝑀, one has (𝐶_𝑀𝐿∘𝐵_𝐿𝐾)∘𝐴_𝐾𝐼=𝐶_𝑀𝐿∘(𝐵_𝐿𝐾∘𝐴_𝐾𝐼)∈Hom(𝑉_𝐼,𝑍_𝑀).

The category PIP has no initial object and no terminal object; hence, it is not a topos.

One can define in the same way smaller categories LBS and LHS, 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 homomorphism 𝑀_𝐾𝐼∈Hom(𝑉_𝐼,𝑌_𝐾) is a monomorphism if 𝑀_𝐾𝐿𝐴_𝐿𝐼=𝑀_𝐾𝐿𝐵_𝐿𝐼 implies 𝐴_𝐿𝐼=𝐵_𝐿𝐼, for any pair 𝐴_𝐿𝐼,𝐵_𝐿𝐼 and any indexed 𝖯𝖨𝖯-space 𝑊_𝐿 (a typical example is given in Appendix A). Two monomorphisms 𝑀_𝐿𝐼,𝑁_𝐿𝐾 with the same codomain 𝑊_𝐿 are equivalent if there exists an isomorphism 𝑈_𝐾𝐼 such that 𝑁_𝐿𝐾𝑈_𝐾𝐼=𝑀_𝐿𝐼. Then a subobject of 𝑉_𝐼 is an equivalence class of monomorphisms into 𝑉_𝐼. 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, 𝑊_𝐼=𝑃𝑉_𝐼. Now the embedding 𝑀∶𝑊_𝐼=𝑃𝑉_𝐼↦𝑉_𝐼 is a monomorphism; thus, orthocomplemented subspaces are subobjects of PIP.

However, the converse is not true, at least for a general indexed 𝖯𝖨𝖯-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/LHS 𝑉_𝐼={𝑉_𝑟,𝑟∈𝐼} and a vector subspace 𝑊. In order that 𝑊 becomes a LBS/LHS 𝑊_𝐼 in its own right, we must require that, for every 𝑟∈𝐼, 𝑊_𝑟=𝑊∩𝑉_𝑟 and 𝑊_𝑟=𝑊∩𝑉_𝑟 are a dual pair with respect to their respective Mackey topologies and that the intrinsic Mackey topology 𝜏(𝑊_𝑟,𝑊_𝑟) coincides with the norm topology induced by 𝑉_𝑟. In other words, 𝑊 must be topologically regular, which is equivalent that it be orthocomplemented [5, Section 3.4.2]. Now the injection 𝑀_𝐼∶𝑊_𝐼→𝑉_𝐼 is clearly a monomorphism, and 𝑊_𝐼 is a subobject of 𝑉_𝐼. 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 subspace 𝑊 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 arbitrary 𝖯𝖨𝖯-spaces. However, when it comes to indexed 𝖯𝖨𝖯-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 homomorphism 𝑁_𝐾𝐿∈Hom(𝑉_𝐼,𝑊_𝐾) is an epimorphism if 𝐴_𝐼𝐾𝑁_𝐾𝐿=𝐵_𝐼𝐾𝑁_𝐾𝐿 implies 𝐴_𝐼𝐾=𝐵_𝐼𝐾, for any pair 𝐴_𝐼𝐾,𝐵_𝐼𝐾 and any indexed 𝖯𝖨𝖯-space 𝑌_𝐿. 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 on 𝖯𝖨𝖯-Spaces

4.1. Presheaves and Sheaves

Let 𝑋 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. A presheaf 𝖥 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 set 𝑈 of 𝑋, there is an object 𝖥(𝑈) in 𝐂; (PS₂)for each inclusion of open sets 𝑇⊆𝑈, there is given a restriction morphism 𝜌^𝑇_𝑉∶𝖥(𝑈)→𝖥(𝑇) in the category 𝐂, such that 𝜌^𝑈_𝑈 is the identity for every open set 𝑈 and 𝜌^𝑇_𝑆∘𝜌^𝑈_𝑇=𝜌^𝑈_𝑆 whenever 𝑆⊆𝑇⊆𝑈.

Definition 4.2. Let 𝖥 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. Let 𝑋 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, 𝜌^𝑈_𝑇 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. Let 𝖥 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 {𝑈_𝑖}_𝑖∈𝐼 of 𝑈, the following conditions are fulfilled: (S₁)given 𝑠,𝑠′∈𝖥(𝑈) such that 𝜌^𝑈_𝑈𝑖(𝑠)=𝜌^𝑈_𝑈𝑖(𝑠′), for every 𝑖∈𝐼, then 𝑠=𝑠′ (local identity); (S₂)given 𝑠_𝑖∈𝖥(𝑈_𝑖) such that 𝜌^𝑈𝑖_{𝑈𝑖∩𝑈𝑗}(𝑠_𝑖)=𝜌^𝑈𝑗_{𝑈𝑖∩𝑈𝑗}(𝑠_𝑗), for every 𝑖,𝑗∈𝐼, then there exists a section 𝑠∈𝖥(𝑈) such that 𝜌^𝑈_𝑈𝑖(𝑠)=𝑠_𝑖, for every 𝑖∈𝐼 (gluing).

The section 𝑠 whose existence is guaranteed by axiom (S₂) is called the gluing, concatenation, or collation of the sections 𝑠_𝑖. 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(𝑈_𝑖)},{𝜌^𝑈𝑖_𝑈𝑗}).

4.2. A Sheaf of Operators on an Indexed 𝖯𝖨𝖯-Space

Let 𝑉_𝐼={𝑉_𝑟,𝑟∈𝐼} be an indexed 𝖯𝖨𝖯-space, and let 𝐕_𝐼 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 𝑉_𝑟 carries its Mackey’s topology 𝜏(𝑉_𝑟,𝑉_𝑟).

We define a sheaf on 𝑉_𝐼 by the contravariant functor 𝖥∶𝐕_𝐼→Set given by 𝖥∶𝑉_𝑟⟼Op_𝑟∶=𝐴↾𝑉_𝑟∶𝐴∈Op𝑉_𝐼,𝑟∈𝖽(𝐴). (4.1) This means that an element of Op_𝑟 is a representative 𝐴_𝑤𝑟 from 𝑉_𝑟 into some 𝑉_𝑤. 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 Op_𝑟 may be called germs of operators. Op_𝑟 is the restriction to 𝑉_𝑟 of the set 𝐿_𝑟 of left multipliers [5, Section 6.2.3]: 𝐿_𝑟∶=𝐴∈Op𝑉_𝐼∶∃𝑤suchthat𝐴_𝑤𝑟exists=𝐴∈Op𝑉_𝐼∶𝑟∈𝖽(𝐴). (4.2)

Then we have:(i)When 𝑉_𝑞⊆𝑉_𝑝, define 𝜌^𝑝_𝑞∶Op_𝑝→Op_𝑞 by 𝜌^𝑝_𝑞(𝐴_∗𝑝)∶=𝐴_∗𝑞, for 𝐴_∗𝑝∈Op_𝑝. Clearly, 𝜌^𝑝_𝑝=𝑖𝑑_Op𝑝 and 𝜌^𝑞_𝑟∘𝜌^𝑝_𝑞=𝜌^𝑝_𝑟 if 𝑉_𝑟⊂𝑉_𝑞⊂𝑉_𝑝. Hence, 𝖥 is a presheaf.(ii) (S₁) is clearly satisfied. As for (S₂), if 𝐴_∗𝑟∈Op_𝑟 and 𝐴^′_∗𝑠∈Op_𝑠 are such that 𝜌^𝑟_𝑟∧𝑠(𝐴_∗𝑟)=𝜌^𝑠_𝑟∧𝑠(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(𝑉_𝐼). It remains to prove that 𝐴 extends to 𝑉_𝑟∨𝑠 and that its representative 𝐴_{∗𝑟∨𝑠} extends both 𝐴_∗𝑟 and 𝐴^′_∗𝑠.

Proposition 4.5. Let the indexed 𝖯𝖨𝖯-space 𝑉_𝐼 be additive; that is, 𝑉_𝑟∨𝑠=𝑉_𝑟+𝑉_𝑠,forall𝑟,𝑠∈𝐼. Then the map 𝖥 given in (4.1) is a sheaf of operators on 𝑉_𝐼.

Proof. By linearity, 𝐴_∗𝑟 and 𝐴_∗𝑠 can be extended to an operator 𝐴^{(𝑟∨𝑠)} on 𝑉_𝑟+𝑉_𝑠 as follows: 𝐴^{(𝑟∨𝑠)}𝑓_𝑟+𝑓_𝑠∶=𝐴_∗𝑟𝑓_𝑟+𝐴_∗𝑠𝑓_𝑠. (4.3) This operator is well defined. Let indeed 𝑓=𝑓_𝑟+𝑓_𝑠=𝑓^′_𝑟+𝑓^′_𝑠 be two decompositions of 𝑓∈𝑉_𝑟+𝑉_𝑠, so that 𝑓_𝑟−𝑓^′_𝑟=𝑓^′_𝑠−𝑓_𝑠∈𝑉_𝑟∩𝑉_𝑠. Then 𝐴^{(𝑟∨𝑠)}𝑓=𝐴_∗𝑟𝑓_𝑟+𝐴_∗𝑠𝑓_𝑠=𝐴_∗𝑟𝑓^′_𝑟+𝐴_∗𝑠𝑓^′_𝑠. Hence, 𝐴_∗𝑟(𝑓_𝑟−𝑓^′_𝑟)=𝐴_∗𝑠(𝑓^′_𝑠−𝑓_𝑠). Taking the restriction to 𝑉_𝑟∩𝑉_𝑠=𝑉_𝑟∧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, 𝑉_𝑟+𝑉_𝑠, with its inductive topology, coincides with 𝑉_𝑟∨𝑠, and, thus, 𝐴_{∗𝑟∨𝑠} is the (𝑟∨𝑠,∗)-representative of the operator 𝐴∈Op(𝑉_𝐼). Therefore, 𝖥 is a sheaf.

We recall that the most interesting classes of indexed 𝖯𝖨𝖯-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 on 𝖯𝖨𝖯-Spaces

5.1. Pre-Cosheaves and Cosheaves

Pre-cosheaves and cosheaves are the dual notions of presheaves and sheaves, respectively. Let again 𝑋 be a topological space, with closed sets 𝑊_𝑖 so that 𝑋=⋃_𝑖∈𝐼𝑊_𝑖, and let 𝐂 be a (concrete) category.

Definition 5.1. A pre-cosheaf 𝖦 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 set 𝑊 of 𝑋, there is an object 𝖦(𝑊) in 𝐂; (PC₂)for each inclusion of closed sets 𝑍⊇𝑊, there is given a extension morphism 𝛿^𝑍_𝑊∶𝖦(𝑊)→𝖦(𝑍) in the category 𝐂, such that 𝛿^𝑊_𝑊 is the identity for every closed set 𝑊 and 𝛿^𝑇_𝑍∘𝛿^𝑍_𝑊=𝛿^𝑇_𝑊 whenever 𝑇⊇𝑍⊇𝑊.

Definition 5.2. Let 𝖦 be a pre-cosheaf on the topological space 𝑋. We say that 𝖦 is a cosheaf if, for every nonempty closed set 𝑊=⋂_𝑗∈𝐽𝑊_𝑗,𝐽⊆𝐼 and for every family of (local) sections {𝑡_𝑗∈𝖦(𝑊_𝑗)}_𝑗∈𝐽, the following conditions are fulfilled: (CS₁)given 𝑡,𝑡^′∈𝖦(𝑊) such that 𝛿^𝑊𝑗_𝑊(𝑡)=𝛿^𝑊𝑗_𝑊(𝑡^′), for every 𝑗∈𝐼 such that 𝑊⊆𝑊_𝑗, then 𝑡=𝑡′;(CS₂)if 𝛿^{𝑊𝑗∪𝑊𝑘}_𝑊𝑗(𝑡_𝑗)=𝛿^{𝑊𝑗∪𝑊𝑘}_𝑊𝑘(𝑡_𝑘), for every 𝑗,𝑘∈𝐽, then there exists a unique section 𝑡∈𝖦(𝑊) such that 𝛿^𝑊𝑗_𝑊(𝑠)=𝑡_𝑗, for every 𝑗∈𝐽.

The cosheaf 𝖦 may be seen as a covariant functor from the category of closed sets of 𝑋 into 𝐂∶=({𝖦(𝑊_𝑖)},{𝛿^𝑊𝑖_𝑊𝑗}).

Now, there are situations where extensions do not exist for all inclusions 𝑍⊇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. Let 𝑋 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-cosheaf 𝖦 with values in 𝐂 is a map 𝖦 defined on 𝒞𝑙(𝑋), satisfying conditions (PC₁) and (pPC₂)if 𝑍≽𝑊, there is given an extension morphism 𝛿^𝑍_𝑊∶𝖦(𝑊)→𝖦(𝑍) in the category 𝐂, such that 𝛿^𝑊_𝑊 is the identity for every closed set 𝑊 and 𝛿^𝑇_𝑍∘𝛿^𝑍_𝑊=𝛿^𝑇_𝑊 whenever 𝑇≽𝑍≽𝑊.

We may use the term “partial,” since here not all pairs 𝑤⩽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. Let 𝖦 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∈𝐽𝑊_𝑗,𝐽⊆𝐼 and for every family of sections {𝑡_𝑗∈𝖦(𝑊_𝑗)}_𝑗∈𝐽, the following conditions are fulfilled: (pCS₁)given 𝑡,𝑡^′∈𝖦(𝑊) such that 𝛿^𝑊𝑗_𝑊(𝑡) and 𝛿^𝑊𝑗_𝑊(𝑡′) exist and are equal, for every 𝑗∈𝐽, then 𝑡=𝑡′;(pCS₂)if 𝛿^{𝑊𝑗∪𝑊𝑘}_𝑊𝑗(𝑡_𝑗) and 𝛿^{𝑊𝑗∪𝑊𝑘}_𝑊𝑘(𝑡_𝑘) exist and are equal, for every 𝑗,𝑘∈𝐽, then 𝛿^𝑊𝑗_𝑊 exists for every 𝑗∈𝐽 and there exists a unique section 𝑡∈𝖦(𝑊) such that 𝛿^𝑊𝑗_𝑊(𝑡)=𝑡_𝑗, for every 𝑗∈𝐽.

5.2. Cosheaves of Operators on Indexed 𝖯𝖨𝖯-Spaces

Let again 𝑉_𝐼={𝑉_𝑟,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): 𝖥∶𝑉_𝑟⟼Op_𝑟∶=𝐴_𝑤𝑟∶𝑉_𝑟⟶𝑉_𝑤,forsome𝑤∈𝐼,𝐴∈Op𝑉_𝐼. (5.1) Then we immediately face a problem. Given 𝑉_𝑞, there is no 𝑉_𝑝, with 𝑉_𝑞⊆𝑉_𝑝, such that 𝛿^𝑝_𝑞∶Op_𝑝→Op_𝑞 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 𝑉_𝑝. 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 operators Op(𝑉_𝐼). Take two assaying subspaces 𝑉_𝑟,𝑉_𝑠 and two operators 𝐴^(𝑟)∈Op_𝑟,𝐵^(𝑠)∈Op_𝑠, and assume that they have a common extension 𝐶^{(𝑟∨𝑠)} to Op_𝑟∨𝑠. This means that, for any suitable 𝑤, 𝐶^{(𝑟∨𝑠)}∶𝑉_𝑟∨𝑠→𝑉_𝑤 is the (𝑟∨𝑠,𝑤)-representative 𝐶_{𝑤,𝑟∨𝑠} of a unique operator 𝐶∈Op(𝑉_𝐼), and 𝐶=𝐴=𝐵. A fortiori, 𝐴, and 𝐵 coincide on 𝑉_𝑟∧𝑠=𝑉_𝑟∩𝑉_𝑠; that is, condition (pCS₂) is satisfied.

Assume now that 𝑉_𝐼 is additive; that is, 𝑉_𝑟∨𝑠 coincides with 𝑉_𝑟+𝑉_𝑠, 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)}𝑓_𝑟=𝐴_∗𝑟𝑓_𝑟 and 𝐶^{(𝑟∨𝑠)}𝑓_𝑠=𝐵_∗𝑠𝑓_𝑠, we may write, for any 𝑓=𝑓_𝑟+𝑓_𝑠∈𝑉_𝑟+𝑉_𝑠, 𝐶^{(𝑟∨𝑠)}𝑓=𝐶_{∗𝑟∨𝑠}𝑓_𝑟∨𝑠=𝐴_∗𝑟𝑓_𝑟+𝐵_∗𝑠𝑓_𝑠. (5.2) Here too, this operator is well defined. Let indeed 𝑓=𝑓_𝑟+𝑓_𝑠=𝑓^′_𝑟+𝑓^′_𝑠 be two decompositions of 𝑓∈𝑉_𝑟+𝑉_𝑠, so that 𝑓_𝑟−𝑓^′_𝑟=𝑓^′_𝑠−𝑓_𝑠∈𝑉_𝑟∩𝑉_𝑠. Then necessarily 𝐶^{(𝑟∨𝑠)}𝑓=𝐴_∗𝑟𝑓_𝑟+𝐵_∗𝑠𝑓_𝑠=𝐴_∗𝑟𝑓^′_𝑟+𝐵_∗𝑠𝑓^′_𝑠. Hence, 𝐴_∗𝑟(𝑓_𝑟−𝑓^′_𝑟)=𝐵_∗𝑠(𝑓^′_𝑠−𝑓_𝑠). Taking the restriction to 𝑉_𝑟∩𝑉_𝑠=𝑉_𝑟∧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 𝑉_𝑟∧𝑠, by the condition (pcS₂).

The conclusion is that, for any pair of assaying subspaces 𝑉_𝑟,𝑉_𝑠, the extension maps 𝛿_𝑟^𝑟∨𝑠 and 𝛿_𝑠^𝑟∨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 𝑉_𝑟,𝑉_𝑠,𝑉_𝑡 (and using the associativity of ∨). Thus, me may state the following.

Proposition 5.5. Let 𝑉_𝐼={𝑉_𝑟,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: 𝑟≼𝑠⟺∃𝑤∈𝐼suchthat𝑠=𝑟∨𝑤. (5.3)

5.2.2. Universal Left Multipliers

We consider only the operators 𝐴 which are everywhere defined; that is, 𝖽(𝐴)=𝐼. This is precisely the set of universal left multipliers: 𝐿Op𝑉_𝐼=𝐶∈Op𝑉_𝐼∶∀𝑟,∃𝑤suchthat𝐶_𝑤𝑟=𝑉exists𝐶∈Op_𝐼∶𝖽(𝐶)=𝐼=_𝑟∈𝐼𝐿_𝑟, (5.4) where 𝐿_𝑟∶={𝐶∈Op(𝑉_𝐼)∶𝑟∈𝖽(𝐶)}. Correspondingly, we consider the map 𝖦∶𝑉_𝑟⟼𝐿Op_𝑟∶=𝐴↾𝑉_𝑟∶𝐴∈𝐿Op𝑉_𝐼. (5.5) Here again, elements of 𝐿Op_𝑟 are representatives 𝐴_∗𝑟. In that case, extensions always exist. When 𝑉_𝑞⊆𝑉_𝑝, define 𝛿^𝑝_𝑞∶𝐿Op_𝑞→𝐿Op_𝑝 by 𝛿^𝑝_𝑞(𝐴_∗𝑞)∶=𝐴_∗𝑝 for 𝐴_∗𝑞∈𝐿Op_𝑞. Clearly, 𝛿^𝑝_𝑝=𝑖d_𝐿Op𝑝 and 𝛿^𝑝_𝑞∘𝛿^𝑞_𝑟=𝛿^𝑝_𝑟 if 𝑉_𝑟⊂𝑉_𝑞⊂𝑉_𝑝. Hence, 𝖦 is a pre-cosheaf. Moreover, if 𝛿_𝑞^𝑝∨𝑞(𝐴_∗𝑞)=𝛿_𝑝^𝑝∨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(𝑉_𝐼). Then, of course, one has 𝛿^𝑞_𝑝∧𝑞(𝐴_℘∧𝑞)=𝐴_∗𝑞 and 𝛿^𝑝_𝑝∧𝑞(𝐴_{∗𝑝∧𝑞})=𝐴_∗𝑝; thus, 𝖦 is a cosheaf. Therefore,

Proposition 5.6. Let 𝑉_𝐼={𝑉_𝑟,𝑟∈𝐼} be an arbitrary indexed 𝖯𝖨𝖯-space. Then the map given in (5.5) is a cosheaf on 𝑉_𝐼 with values in 𝐿Op(𝑉_𝐼), the set of universal left multipliers, with extensions given by 𝛿^𝑝_𝑞(𝐴_∗𝑞)=𝐴_∗𝑝,𝑞⩽𝑝,𝐴_∗𝑞∈𝐿Op_𝑞.

5.2.3. General Operators: The Projective Case

We consider again all operators in Op(𝑉_𝐼). The fact is that the set ⋃_𝑟∈𝐼Op_𝑟 is an initial set in Op(𝑉_𝐼), 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: 𝖦∶𝑉_𝑟⟼Op_𝑟∶=𝐴_𝑟∗∶𝐴∈Op𝑉_𝐼,𝑟∈𝗂(𝐴). (5.6) Now indeed ⋁_𝑟∈𝐼Op_𝑟 is a final set. Elements of Op_𝑟 have been denoted as 𝐴_𝑟∗, but, for definiteness, we could also replace 𝐴_𝑟∗ by 𝐴_𝑟∞ where 𝑉_∞∶=𝑉^#.

We claim that the map 𝖦 defines a cosheaf, with extensions 𝛿^𝑝_𝑞(𝐴_𝑞∗)∶=𝐴_𝑝∗, for 𝐴_𝑞∗∈Op_𝑞, which exist whenever 𝑞⩽𝑝.

As in the case of Section 5.2.1, assume that 𝐴_𝑟∗∈Op_𝑟,𝐵_𝑠∗∈Op_𝑠 have a common extension belonging to Op_𝑟∨𝑠; that is, 𝐴_{𝑟∨𝑠∗}∶=𝐸_{𝑟∨𝑠,𝑟}𝐴_𝑟∗=𝐵_{𝑟∨𝑠∗}∶=𝐸_{𝑟∨𝑠,𝑠}𝐵_𝑠∗. Call this operator 𝐶_{𝑟∨𝑠∗}. Thus, for any suitable 𝑤, 𝐶_{𝑟∨𝑠,𝑤} is the (𝑤,𝑟∨𝑠) representative of a unique operator 𝐶∈Op(𝑉_𝐼), which extends 𝐴 and 𝐵. Thus 𝐶=𝐴=𝐵∈Op(𝑉_𝐼). Assume now that 𝑉_𝐼 is projective; that is, 𝑉_𝑟∧𝑠=𝑉_𝑟∩𝑉_𝑠,forall𝑟,𝑠∈𝐼, with the projective topology. Then, 𝐶_{𝑟∧𝑠∗}=𝐴_{𝑟∧𝑠∗}=𝐵_{𝑟∧𝑠∗}; that is, condition (CS₂) is satisfied with extensions 𝛿^𝑟_𝑟∧𝑠(𝐶_{𝑟∧𝑠∗})=𝐴_𝑟∗ and 𝛿^𝑠_𝑟∧𝑠(𝐶_{𝑟∧𝑠∗})=𝐵_𝑠∗. The rest is obvious. As in the case of sheaves, extensions from 𝑉_𝑟,𝑉_𝑠 to 𝑉_𝑟∨𝑠 exist by linearity, since 𝑉_𝐼 is projective, hence, additive. Thus, we may state the following.

Proposition 5.7. Let the indexed 𝖯𝖨𝖯-space 𝑉_𝐼 be projective that is, 𝑉_𝑟∧𝑠=𝑉_𝑟∩𝑉_𝑠,forall𝑟,𝑠∈𝐼, with the projective topology. Then the map 𝖦 given in (5.6) is a cosheaf of operators on 𝑉_𝐼.

Corollary 5.8. Let 𝑉_𝐼={𝑉_𝑟,𝑟∈𝐼} be a projective indexed 𝖯𝖨𝖯-space. Then 𝑉_𝐼 generates in a natural fashion a sheaf and a cosheaf of operators. If 𝑉_𝐼 is additive, but not projective; it generates a sheaf and a partial cosheaf of operators.

6. Cohomology of an Indexed 𝖯𝖨𝖯-Space

6.1. Cohomology of Operator Sheaves

It is possible to introduce a concept of sheaf cohomology group defined on an indexed 𝖯𝖨𝖯-space according to the usual definition of sheaf cohomology [10]. Let 𝑉_𝐼 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(𝑉_𝑟,𝑉_𝑟), each 𝑉_𝑟 is a Hausdorff vector subspace of 𝑉.

Definition 6.1. Given the indexed 𝖯𝖨𝖯-space 𝑉_𝐼, let {𝑉_𝑗,𝑗∈𝐼} be the open covering of 𝑉 by its assaying subspaces, and let 𝖥∶𝑉_𝑟↦Op_𝑟={𝐴↾𝑉_𝑟∶𝐴∈Op(𝑉_𝐼),𝑟∈𝖽(𝐴)} be the sheaf of operators defined in (4.1) above. We call 𝑝-cochain with values in 𝖥 the map which associates, to each intersection of open sets 𝑉_𝑗0∩𝑉_𝑗1∩⋯∩𝑉_𝑗𝑝,𝑗_𝑘∈𝐼,𝑗₀<𝑗₁<⋯<𝑗_𝑝, an operator 𝐴_{𝑗0,𝑗1,…,𝑗𝑝} belonging to 𝖥(𝑉_𝑗0∩𝑉_𝑗1∩⋯∩𝑉_𝑗𝑝).
Thus, a 𝑝-cochain with value in 𝖥 is a set 𝒜={𝐴_{𝑗0,𝑗1,…,𝑗𝑝},𝑗₀<𝑗₁<⋯<𝑗_𝑝}. The set of 𝑝-cochains is denoted by 𝐶^𝑝(𝐼,𝖥). For example, a 0-cochain is a set {𝐴_𝑗,𝑗∈𝐼}. A 1-cochain is a set {𝐴_𝑗0,𝑗1,𝑗₀<𝑗₁},𝐴_𝑗0,𝑗1∈𝖥(𝑉_𝑗0∩𝑉_𝑗1),….

Definition 6.2. Given the notion of 𝑝-cochain, the corresponding coboundary operator 𝐷 is defined by 𝐷∶𝐶^𝑝(𝐼,𝖥)⟶𝐶^𝑝+1(𝐼,𝖥)∶𝐴_{𝑗0,𝑗1,…,𝑗𝑝}⟼(𝐷𝒜)_{𝑗0,𝑗1,…,𝑗𝑝,𝑗𝑝+1},(𝐷𝒜)_{𝑗0,𝑗1,…,𝑗𝑝,𝑗𝑝+1}=^𝑝+1_𝑘=0(−1)^𝑘𝜌^{𝑉𝑗0∩𝑉𝑗1∩⋯∩}_{𝑗𝑘∩⋯∩𝑉𝑗𝑝+1𝑉}𝑉_{𝑗0∩𝑉𝑗1∩⋯∩𝑉𝑗𝑝+1}𝐴_{𝑗0,𝑗1,…,}̂𝑗_{𝑘,…,𝑗𝑝+1}, (6.1) 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 (𝐷𝒜)_𝑗0,𝑗1=𝜌^𝑉𝑗1_{𝑉𝑗0∩𝑉𝑗1}𝐴_𝑗1−𝜌^𝑉𝑗0_{𝑉𝑗0∩𝑉𝑗1}𝐴_𝑗0. (6.2)
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 (𝐷𝒜)_{𝑗0,𝑗1,𝑗2}=𝜌^{𝑉𝑗1∩𝑉𝑗2}_{𝑉𝑗0∩𝑉𝑗1∩𝑉𝑗2}𝐴_𝑗1,𝑗2−𝜌^{𝑉𝑗0∩𝑉𝑗2}_{𝑉𝑗0∩𝑉𝑗1∩𝑉𝑗2}𝐴_𝑗0,𝑗2+𝜌^{𝑉𝑗0∩𝑉𝑗1}_{𝑉𝑗0∩𝑉𝑗1∩𝑉𝑗2}𝐴_𝑗0,𝑗1. (6.3) 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: (𝐷𝒜)_{𝑗0,𝑗1,𝑗2}=𝜌^{𝑉𝑗1∩𝑉𝑗2}_{𝑉𝑗0∩𝑉𝑗1∩𝑉𝑗2}𝜌^𝑉𝑗2_{𝑉𝑗1∩𝑉𝑗2}𝐵_𝑗2−𝜌^𝑉𝑗1_{𝑉𝑗1∩𝑉𝑗2}𝐵_𝑗1−𝜌^{𝑉𝑗0∩𝑉𝑗2}_{𝑉𝑗0∩𝑉𝑗1∩𝑉𝑗2}𝜌^𝑉𝑗2_{𝑉𝑗0∩𝑉𝑗2}𝐵_𝑗2−𝜌^𝑉𝑗0_{𝑉𝑗0∩𝑉𝑗2}𝐵_𝑗0+𝜌^{𝑉𝑗0∩𝑉𝑗1}_{𝑉𝑗0∩𝑉𝑗1∩𝑉𝑗2}𝜌^𝑉𝑗1_{𝑉𝑗0∩𝑉𝑗1}𝐵_𝑗1−𝜌^𝑉𝑗0_{𝑉𝑗0∩𝑉𝑗1}𝐵_𝑗0. (6.4) Using the properties of restrictions we get (𝐷𝒜)_{𝑗0,𝑗1,𝑗2}=𝜌^𝑉𝑗2_{𝑉𝑗0∩𝑉𝑗1∩𝑉𝑗2}𝐵_𝑗2−𝜌^𝑉𝑗1_{𝑉𝑗0∩𝑉𝑗1∩𝑉𝑗2}𝐵_𝑗1−𝜌^𝑉𝑗2_{𝑉𝑗0∩𝑉𝑗1∩𝑉𝑗2}𝐵_𝑗2+𝜌^𝑉𝑗0_{𝑉𝑗0∩𝑉𝑗1∩𝑉𝑗2}𝐵_𝑗0+𝜌^𝑉𝑗1_{𝑉𝑗0∩𝑉𝑗1∩𝑉𝑗2}𝐵_𝑗1−𝜌^𝑉𝑗0_{𝑉𝑗0∩𝑉𝑗1∩𝑉𝑗2}𝐵_𝑗0=0, (6.5) which shows that indeed 𝐷𝐷ℬ=0.

Now it is possible to define cohomology groups of the sheaf 𝖥 on a indexed 𝖯𝖨𝖯-space 𝑉.

Definition 6.3. Let 𝑉_𝐼 be an indexed 𝖯𝖨𝖯-space, endowed with the open covering of its assaying spaces, {𝑉_𝑗,𝑗∈𝐼}, and let 𝖥 be the corresponding sheaf of operators on 𝑉_𝐼. Then the 𝑝th cohomology group is defined as 𝐻^𝑝(𝐼,𝖥)∶=𝑍^𝑝(𝐼,𝖥)/𝐵^𝑝(𝐼,𝖥), where 𝑍^𝑝(𝐼,𝖥) is the set of 𝑝-cocycles, that is, 𝑝-cochains 𝒜 such that 𝐷𝒜=0, and 𝐵^𝑝(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 group 𝐻^𝑝(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 indexed 𝖯𝖨𝖯-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 {𝑉_𝑗,𝑗∈𝐽} of an indexed 𝖯𝖨𝖯-space 𝑉_𝐼 is finer than another one, {𝑈_𝑘,𝑘∈𝐾}, if there exists an application 𝑡∶𝐽→𝐾 such that 𝑉_𝑗⊂𝑈_𝑡(𝑗),forall𝑗∈𝐽. For instance, 𝑈_𝑡(𝑗) could be a union ∪_𝑙𝑉_𝑙 containing 𝑉_𝑗. According to the general theory of sheaf cohomology [9], this induces group homomorphisms 𝑡^𝐾_𝐽∶𝐻^𝑝(𝐾,𝖥)→𝐻^𝑝(𝐽,𝖥),forall𝑝⩾0. It is then possible to introduce cohomology groups that do not depend on the particular open covering, namely, by defining 𝐻^𝑝(𝑉,𝖥) as the inductive limit of the groups 𝐻^𝑝(𝐽,𝖥) with respect to the homomorphisms 𝑡^𝐾_𝐽.

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 indexed 𝖯𝖨𝖯-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 indexed 𝖯𝖨𝖯-space 𝑉_𝐼, define the sheaf 𝖥∶𝑉_𝑟→F(𝑉_𝑟)∶=Op_𝑟={𝐴_𝑟∶𝑉_𝑟→𝑉}. Then, if 𝑉 is metrizable for its Mackey topology, the sheaf 𝖥 is acyclic; that is, the cohomology groups 𝐻^𝑝(𝑉,𝖥) are trivial for all 𝑝⩾1; that is, 𝐻^𝑝(𝑉,𝖥)=0.

Proof. First of all, 𝑉 is a paracompact space, since it is metrizable.
Next, we check that the sheaf 𝖥∶𝑉_𝑟→𝖥(𝑉_𝑟)=Op_𝑟={𝐴_𝑟∶𝑉_𝑟→𝑉} on the paracompact space 𝑉 is fine. Indeed, let {𝑉_𝑗,𝑗∈𝐽} be a locally finite open covering of 𝑉. We can associate to the latter a partition of unity {𝜑_𝑗,𝑗∈𝐽}. Then we can use {𝜑_𝑗,𝑗∈𝐽} to define the homomorphisms ℎ_𝑗∶𝖥→𝖥. Let 𝑉_𝑟⊂∪_𝑠∈𝐾𝑉_𝑠 for some index set 𝐾. For each 𝑠∈𝐾, we consider the set of operators Op_𝑠={𝐴_∗𝑠∶𝑉_𝑠→𝑉}, and we define ℎ_𝑗∶𝐴_𝑠↦ℎ_𝑗(𝐴_𝑠)∶=𝜑_𝑗𝐴_𝑠. This defines a homomorphism Op_𝑠→Op_𝑠 and then a homomorphism 𝖥→𝖥. One can check that ℎ_𝑗∶𝖥→𝖥 satisfies conditions (1) and (2) in Definition B.3, and, thus, 𝖥 is a fine sheaf.
Then the result follows from Theorem B.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=(𝐶_𝑗0,𝑗1,𝑗₀<𝑗₁)∈𝐼 of 𝐶¹(𝐼,𝖥), such that 𝐷𝒞=0; that is, (𝐷𝒞)_{𝑗0,𝑗1,𝑗2}=𝜌^{𝑉𝑗1∩𝑉𝑗2}_{𝑉𝑗0∩𝑉𝑗1∩𝑉𝑗2}𝐶_𝑗1,𝑗2−𝜌^{𝑉𝑗0∩𝑉𝑗2}_{𝑉𝑗0∩𝑉𝑗1∩𝑉𝑗2}𝐶_𝑗0,𝑗2+𝜌^{𝑉𝑗0∩𝑉𝑗1}_{𝑉𝑗0∩𝑉𝑗1∩𝑉𝑗2}𝐶_𝑗0,𝑗1=0. (6.6) Using the fact that 𝖥 is a fine sheaf, we can choose the partition of unity {𝜑_𝑖,𝑖∈𝐼} to define the 0-cochain ℰ={𝐸_𝑗}, where 𝐸_𝑗=−∑_𝑘∈𝐼𝜑_𝑘(𝐶_𝑗𝑘). This sum is well defined since the covering is locally finite. Applying the coboundary operator, we get (𝐷ℰ)_𝑗0,𝑗1=𝜌^𝑉𝑗1_{𝑉𝑗0∩𝑉𝑗1}𝐸_𝑗1−𝜌^𝑉𝑗0_{𝑉𝑗0∩𝑉𝑗1}𝐸_𝑗0=−_𝑘∈𝐼𝜑_𝑘𝜌^𝑉𝑗1_{𝑉𝑗0∩𝑉𝑗1}𝐶_𝑗1𝑘+_𝑘∈𝐼𝜑_𝑘𝜌^𝑉𝑗0_{𝑉𝑗0∩𝑉𝑗1}𝐶_𝑗0𝑘=_𝑘∈𝐼𝜑_𝑘𝜌^𝑉𝑗0_{𝑉𝑗0∩𝑉𝑗1}𝐶_𝑗0𝑘−𝜌^𝑉𝑗1_{𝑉𝑗0∩𝑉𝑗1}𝐶_𝑗1𝑘=_𝑘∈𝐼𝜑_𝑘𝜌^{𝑉𝑗0∩𝑉𝑘}_{𝑉𝑗0𝑗1𝑘∩𝑉∩𝑉}𝐶_𝑗0𝑘−𝜌^{𝑉𝑗1∩𝑉𝑘}_{𝑉𝑗0𝑗1𝑘∩𝑉∩𝑉}𝐶_𝑗1𝑘. (6.7) Putting 𝑘=𝑗₂ and using the equation (𝐷𝒞)_{𝑗0,𝑗1,𝑗2}=0, we find (𝐷ℰ)_𝑗0,𝑗1=_𝑗2∈𝐼𝜑_𝑗2𝜌^{𝑉𝑗0∩𝑉𝑗1}_{𝑉𝑗0𝑗1𝑗2∩𝑉∩𝑉}𝐶_𝑗0𝑗1. (6.8) But since {𝜑_𝑖,𝑖∈𝐼} is a partition of unity, we get (𝐷ℰ)_𝑗0,𝑗1=𝐶_𝑗0𝑗1. This means that 𝐷ℰ=𝒞, and; thus, [𝒞]=0.

Of course, the restriction that 𝑉 is to be metrizable is quite strong, but still the result applies to a significant number of interesting situations, for instance, (1)a finite 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 indexed 𝖯𝖨𝖯-space 𝑉_𝐼 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 algebra 𝒮₀(ℝ^𝑑), which generates the triplet 𝒮₀ℝ^𝑑↪𝐿²ℝ^𝑑↪𝒮^×₀ℝ^𝑑. (6.9) 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 that 𝑉 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 indexed 𝖯𝖨𝖯-spaces. The assaying spaces can also be considered as closed sets. Let, thus, {𝑊_𝑗,𝑗∈𝐼} be such a closed covering of 𝑉, and let 𝖦 be a cosheaf of operators defined in (5.5), namely, 𝖦∶𝑊_𝑗↦𝐿Op_𝑗={𝐴↾𝑊_𝑗∶𝐴∈𝐿Op(𝑉_𝐼)} together with maps 𝛿^𝑊𝑗_𝑊𝑖∶𝖦(𝑊_𝑖)→𝖦(𝑊_𝑗),𝑊_𝑖⊂𝑊_𝑗 as above. In the sequel, 𝖦(𝑊_𝑗0∪𝑊_𝑗1∪⋯∪𝑊_𝑗𝑝) denotes the set of operators 𝑊_𝑗0∪𝑊_𝑗1∪⋯∪𝑊_𝑗𝑝→𝑉. Alternatively, one could consider the cosheaf defined in (5.6), 𝖦∶𝑊_𝑗↦Op_𝑗∶={𝐴_𝑗∗∶𝐴∈Op(𝑉_𝐼),𝑗∈𝗂(𝐴)} and proceed in the same way.

In this setup, we may introduce the necessary cohomological concepts, as in Definitions 6.1 and 6.2.

Definition 6.5. A 𝑝-cochain with values in the cosheaf 𝖦 is a map which associates to each union of closed sets 𝑊_𝑗0∪𝑊_𝑗1∪⋯∪𝑊_𝑗𝑝,𝑗_𝑘∈𝐼,𝑗₀<𝑗₁<⋯<𝑗_𝑝, an operator 𝐴_{𝑗0,𝑗1,…,𝑗𝑝} of 𝖦(𝑊_𝑗0∪𝑊_𝑗1∪⋯∪𝑊_𝑗𝑝). A 𝑝-cochain is, thus, a set 𝔄={𝐴_{𝑗0,𝑗1,…,𝑗𝑝},𝑗₀<𝑗₁<⋯<𝑗_𝑝}. The set of such 𝑝-cochains will be denoted by 𝐶^𝑝(𝐼,𝖦).

Definition 6.6. One can then introduce the coboundary operator 𝐷 as follows: 𝐷∶𝐶^𝑝(𝐼,𝖦)⟶𝐶^𝑝+1(𝐼,𝖦)∶𝔄=𝐴_{𝑗0,𝑗1,…,𝑗𝑝}⟼𝐷𝔄_{𝑗0,𝑗1,…,𝑗𝑝,𝑗𝑝+1},𝐷𝔄_{𝑗0,𝑗1,…,𝑗𝑝,𝑗𝑝+1}=^𝑝+1_𝑘=0(−1)^𝑘𝛿^{𝑊𝑗0∪𝑊𝑗1∪⋯∪𝑊𝑗𝑝+1}_{𝑊𝑗0∪𝑊𝑗1∪⋯∪}𝑊_{𝑗𝑘∪⋯∩𝑊𝑗𝑝+1}𝐴_{𝑗0,𝑗1,…,}̂𝑗_{𝑘,…,𝑗𝑝+1}. (6.10) In view of the properties of the maps 𝛿^∗_∗, we check that 𝐷𝐷=0.

In the case of 0-cochains, a straightforward application of the formula leads to 𝐷𝔄_𝑗0,𝑗1=𝛿^{𝑊𝑗0∪𝑊𝑗1}_𝑊𝑗1𝐴_𝑗1−𝛿^{𝑊𝑗0∪𝑊𝑗1}_𝑊𝑗0𝐴_𝑗0. (6.11) And if we put this to zero, we get the constraint 𝛿^{𝑊𝑗0∪𝑊𝑗1}_𝑊𝑗1𝐴_𝑗1=𝛿^{𝑊𝑗0∪𝑊𝑗1}_𝑊𝑗0𝐴_𝑗0, which has to be satisfied in order to build a cosheaf, according to the condition (CS₂). On 1-cochains, the coboundary action gives 𝐷𝔄_{𝑗0,𝑗1,𝑗2}=𝛿^{𝑊𝑗0∪𝑊𝑗1∪𝑊𝑗2}_{𝑊𝑗1∪𝑊𝑗2}𝐴_𝑗1,𝑗2−𝛿^{𝑊𝑗0∪𝑊𝑗1∪𝑊𝑗2}_{𝑊𝑗0∪𝑊𝑗2}𝐴_𝑗0,𝑗2+𝛿^{𝑊𝑗0∪𝑊𝑗1∪𝑊𝑗2}_{𝑊𝑗0∪𝑊𝑗1}𝐴_𝑗0,𝑗1. (6.12) The cohomology groups of the cosheaf 𝐻^𝑝(𝐼,𝖦)∶=𝑍^𝑝(𝐼,𝖦)/𝐵^𝑝(𝐼,𝖦), with obvious notations, can then be defined in a natural way. Similarly for 𝐻^𝑝(𝑉,𝖦) and 𝐻^𝑝(𝐼,𝖦),𝐻^𝑝(𝑉,𝖦).

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 indexed 𝖯𝖨𝖯-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 PIP^op? Could it be somehow linked to the category of partial *algebras, in the same way as 𝐒𝐞𝐭^op 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 indexed 𝖯𝖨𝖯-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, 𝐻^𝑝(𝑉,𝖦)=0, for all 𝑝⩾1, or, similarly, 𝐻^𝑝(𝑉,𝖦)=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 principle 𝖭𝖢𝖯¬(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!).

Appendices

A. Partial Inner Product Spaces

A.1. 𝖯𝖨𝖯-Spaces and Indexed 𝖯𝖨𝖯-Spaces

For the convenience of the reader, we have collected here the main features of partial inner product spaces and operators on them, keeping only what is needed for reading the paper. Further information may be found in our review paper [6] or our monograph [5].

The general framework is that of a 𝖯𝖨𝖯-space 𝑉, corresponding to the linear compatibility #, that is, a symmetric binary relation 𝑓#𝑔 which preserves linearity. We call assaying subspace of 𝑉 a subspace 𝑆 such that 𝑆^##=𝑆, and we denote by ℱ(𝑉,#) the family of all assaying subspaces of 𝑉, ordered by inclusion. The assaying subspaces are denoted by 𝑉_𝑟,𝑉_𝑞,…, and the index set is 𝐹. By definition, 𝑞⩽𝑟 if and only if 𝑉_𝑞⊆𝑉_𝑟. Thus, we may write 𝑓#𝑔⟺∃𝑟∈𝐹suchthat𝑓∈𝑉_𝑟,𝑔∈𝑉_𝑟. (A.1)

General considerations imply that the family ℱ(𝑉,#)∶={𝑉_𝑟,𝑟∈𝐹}, ordered by inclusion, is a complete involutive lattice; that is, it is stable under the following operations, arbitrarily iterated: (i)involution:𝑉_𝑟↔𝑉_𝑟=(𝑉_𝑟)^#,(ii)infimum: 𝑉_𝑝∧𝑞∶=𝑉_𝑝∧𝑉_𝑞=𝑉_𝑝∩𝑉_𝑞,(𝑝,𝑞,𝑟∈𝐹),(iii)supremum:𝑉_𝑝∨𝑞∶=𝑉_𝑝∨𝑉_𝑞=(𝑉_𝑝+𝑉_𝑞)^##.

The smallest element of ℱ(𝑉,#) is 𝑉^#=⋂_𝑟𝑉_𝑟, and the greatest element is 𝑉=⋃_𝑟𝑉_𝑟.

By definition, the index set 𝐹 is also a complete involutive lattice; for instance, 𝑉_𝑝∧𝑞^#=𝑉_𝑝∧𝑞=𝑉_𝑝∨𝑞=𝑉_𝑝∨𝑉_𝑞. (A.2)

Given a vector space 𝑉 equipped with a linear compatibility #, a partial inner product on (𝑉,#) is a Hermitian form ⟨⋅∣⋅⟩ defined exactly on compatible pairs of vectors. A partial inner product space (𝖯𝖨𝖯-space) is a vector space 𝑉 equipped with a linear compatibility and a partial inner product.

From now on, we will assume that our 𝖯𝖨𝖯-space (𝑉,#,⟨⋅∣⋅⟩) is nondegenerate; that is, ⟨𝑓∣𝑔⟩=0 for all 𝑓∈𝑉^# implies 𝑔=0. As a consequence, (𝑉^#,𝑉) and every couple (𝑉_𝑟,𝑉_𝑟),𝑟∈𝐹, are a dual pair in the sense of topological vector spaces [18]. Next, we assume that every 𝑉_𝑟 carries its Mackey topology 𝜏(𝑉_𝑟,𝑉_𝑟), so that its conjugate dual is (𝑉_𝑟)^×=𝑉_𝑟,forall𝑟∈𝐹. Then, 𝑟<𝑠 implies 𝑉_𝑟⊂𝑉_𝑠, and the embedding operator 𝐸_𝑠𝑟∶𝑉_𝑟→𝑉_𝑠 is continuous and has dense range. In particular, 𝑉^# is dense in every 𝑉_𝑟.

As a matter of fact, the whole structure can be reconstructed from a fairly small subset of ℱ, namely, a generating involutive sublattice ℐ of ℱ(𝑉,#), indexed by 𝐼, which means that 𝑓#𝑔⟺∃𝑟∈𝐼suchthat𝑓∈𝑉_𝑟,𝑔∈𝑉_𝑟. (A.3) The resulting structure is called an indexed 𝖯𝖨𝖯-space and denoted simply by 𝑉_𝐼∶=(𝑉,ℐ,⟨⋅∣⋅⟩).

Then an indexed 𝖯𝖨𝖯-space 𝑉_𝐼 is said to be (i)additive if 𝑉_𝑝∨𝑞=𝑉_𝑝+𝑉_𝑞,forall𝑝,𝑞∈𝐼,(ii)projective if 𝑉_𝑝∧𝑞|_𝜏≃(𝑉_𝑝∩𝑉_𝑞)_proj,forall𝑝,𝑞∈𝐼; here 𝑉_𝑝∧𝑞|_𝜏 denotes 𝑉_𝑝∧𝑞 equipped with the Mackey topology 𝜏(𝑉_𝑝∧𝑞,𝑉_𝑝∨𝑞), the right-hand side denotes 𝑉_𝑝∩𝑉_𝑞 with the topology of the projective limit from 𝑉_𝑝 and 𝑉_𝑞, and ≃ denotes an isomorphism of locally convex topological spaces.

For practical applications, it is essentially sufficient to restrict oneself to the case of an indexed 𝖯𝖨𝖯-space satisfying the following conditions: (i)every 𝑉_𝑟,𝑟∈𝐼, is a Hilbert space or a reflexive Banach space, so that the Mackey topology 𝜏(𝑉_𝑟,𝑉_𝑟) coincides with the norm topology;(ii)there is a unique self-dual, Hilbert, assaying subspace 𝑉_𝑜=𝑉_𝑜.

In that case, the structure 𝑉_𝐼∶=(𝑉,ℐ,⟨⋅∣⋅⟩) is called, respectively, a lattice of Hilbert spaces (LHS) or a lattice of Banach spaces (LBS) (see [5] for more precise definitions, including explicit requirements on norms). The important facts here are that(i)every projective indexed 𝖯𝖨𝖯-space is additive;(ii)an LBS or an LHS is projective if and only if it is additive.

Note that 𝑉^#,𝑉 themselves usually do not belong to the family {𝑉_𝑟,𝑟∈𝐼}, but they can be recovered as 𝑉^#=_𝑟∈𝐼𝑉_𝑟,𝑉=_𝑟∈𝐼𝑉_𝑟. (A.4) A standard, albeit trivial, example is that of a Rigged Hilbert space (RHS) Φ⊂ℋ⊂Φ^# (it is trivial because the lattice ℱ contains only three elements). One should note that the construction of an RHS from a directed family of Hilbert spaces, via projective and inductive limits, has been investigated recently by Bellomonte and Trapani [19]. Similar constructions, in the language of categories, may be found in the work of Mityagin and Shvarts [20] and that of Semadeni and Zidenberg [21].

Let us give some concrete examples.

(i) Sequence Spaces
Let 𝑉 be the space 𝜔 of all complex sequences 𝑥=(𝑥_𝑛) and define on it (i) a compatibility relation by 𝑥#𝑦⇔∑^∞_𝑛=1|𝑥_𝑛𝑦_𝑛|<∞, (ii) a partial inner product ⟨𝑥∣𝑦⟩=∑^∞_𝑛=1𝑥_𝑛𝑦_𝑛. Then 𝜔^#=𝜑, the space of finite sequences, and the complete lattice ℱ(𝜔,#) consists of Köthe's perfect sequence spaces [18, § 30]. Among these, a nice example is the lattice of the so-called ℓ_𝜙 spaces associated to symmetric norming functions or, more generally, the Banach sequence ideals discussed in [5, Section 4.3.2] and previously in [20, § 6] (in this example, the extreme spaces are, resp., ℓ¹ and ℓ^∞).

(ii) Spaces of Locally Integrable Functions
Let 𝑉 be 𝐿¹_loc(ℝ,d𝑥), the space of Lebesgue measurable functions, integrable over compact subsets, and define a compatibility relation on it by 𝑓#𝑔⇔∫_ℝ|𝑓(𝑥)𝑔(𝑥)|d𝑥<∞ and a partial inner product ⟨𝑓∣𝑔⟩=∫_ℝ𝑓(𝑥)𝑔(𝑥)d𝑥. Then 𝑉^#=𝐿^∞_𝑐(ℝ,d𝑥), the space of bounded measurable functions of compact support. The complete lattice ℱ(𝐿¹_loc,#) consists of the so-called Köthe function spaces. Here again, normed ideals of measurable functions in 𝐿¹([0,1],d𝑥) are described in [20, § 8].

A.2. Operators on Indexed 𝖯𝖨𝖯-Spaces

Let 𝑉_𝐼 and 𝑌_𝐾 be two nondegenerate indexed 𝖯𝖨𝖯-spaces (in particular, two LHSs or LBSs). Then an operator from 𝑉_𝐼 to 𝑌_𝐾 is a map 𝐴 from a subset 𝒟(𝐴)⊂𝑉 into 𝑌, such that (i)𝒟(𝐴)=⋃_{𝑞∈𝖽(𝐴)}𝑉_𝑞, where 𝖽(𝐴) is a nonempty subset of 𝐼.(ii)For every 𝑟∈𝖽(𝐴), there exists 𝑢∈𝐾 such that the restriction of 𝐴 to 𝑉_𝑟 is a continuous linear map into 𝑌_𝑢 (we denote this restriction by 𝐴_𝑢𝑟). (iii)𝐴 has no proper extension satisfying (i) and (ii).

We denote by Op(𝑉_𝐼,𝑌_𝐾) the set of all operators from 𝑉_𝐼 to 𝑌_𝐾 and, in particular, Op(𝑉_𝐼)∶=Op(𝑉_𝐼,𝑉_𝐼). The continuous linear operator 𝐴_𝑢𝑟∶𝑉_𝑟→𝑌_𝑢 is called a representative of 𝐴. Thus, the operator 𝐴 may be identified with the collection of its representatives, 𝐴≃{𝐴_𝑢𝑟∶𝑉_𝑟→𝑌_𝑢existsasaboundedoperator}. We will also need the following sets: 𝖽(𝐴)=𝑟∈𝐼∶thereisa𝑢suchthat𝐴_𝑢𝑟,𝗂(𝐴)=exists𝑢∈𝐾∶thereisa𝑟suchthat𝐴_𝑢𝑟exists. (A.5) The following properties are immediate: (i)𝖽(𝐴) is an initial subset of 𝐼: if 𝑟∈𝖽(𝐴) and 𝑟′<𝑟, then 𝑟∈𝖽(𝐴), and 𝐴_𝑢𝑟′=𝐴_𝑢𝑟𝐸_𝑟𝑟′, where 𝐸_𝑟𝑟′ is a representative of the unit operator.(ii)𝗂(𝐴) is a final subset of 𝐾: if 𝑢∈𝗂(𝐴) and 𝑢′>𝑢, then 𝑢^′∈𝗂(𝐴), and 𝐴_𝑢′𝑟=𝐸_𝑢′𝑢𝐴_𝑢𝑟.

Although an operator may be identified with a separately continuous sesquilinear form on 𝑉^#×𝑉^#, it is more useful to keep also the algebraic operations on operators, namely, (i)adjoint: every 𝐴∈Op(𝑉_𝐼,𝑌_𝐾) has a unique adjoint 𝐴^×∈Op(𝑌_𝐾,𝑉_𝐼) and one has 𝐴^××=𝐴, for every 𝐴∈Op(𝑉_𝐼,𝑌_𝐾): no extension is allowed, by the maximality condition (iii) of the definition. (ii)partial multiplication: let 𝑉_𝐼, 𝑊_𝐿, and 𝑌_𝐾 be nondegenerate indexed 𝖯𝖨𝖯-spaces (some, or all, may coincide). Let 𝐴∈Op(𝑉_𝐼,𝑊_𝐿), and, 𝐵∈Op(𝑊_𝐿,𝑌_𝐾). We say that the product 𝐵𝐴 is defined if and only if there is a 𝑡∈𝗂(𝐴)∩𝖽(𝐵); that is, if and only if there is continuous factorization through some 𝑊_𝑡: 𝑉⟶_𝑟^𝐴𝑊⟶_𝑡^𝐵𝑌_𝑢,thatis,(𝐵𝐴)_𝑢𝑟=𝐵_𝑢𝑡𝐴_𝑡𝑟,forsome𝑟∈𝖽(𝐴),𝑢∈𝗂(𝐵). (A.6) Among operators on indexed 𝖯𝖨𝖯-spaces, a special role is played by morphisms.

An operator 𝐴∈Op(𝑉_𝐼,𝑌_𝐾) is called a homomorphism if (i)for every 𝑟∈𝐼, there exists 𝑢∈𝐾 such that both 𝐴_𝑢𝑟 and 𝐴_𝑢𝑟 exist;(ii)for every 𝑢∈𝐾, there exists 𝑟∈𝐼 such that both 𝐴_𝑢𝑟 and 𝐴_𝑢𝑟 exist.

We denote by Hom(𝑉_𝐼,𝑌_𝐾) the set of all homomorphisms from 𝑉_𝐼 into 𝑌_𝐾 and by Hom(𝑉_𝐼) those from 𝑉_𝐼 into itself. The following properties are immediate.

Proposition A.1. Let 𝑉_𝐼,𝑌_𝐾,… be indexed 𝖯𝖨𝖯-spaces. Then, (i)𝐴∈Hom(𝑉_𝐼,𝑌_𝐾) if and only if 𝐴^×∈Hom(𝑌_𝐾,𝑉_𝐼); (ii)the product of any number of homomorphisms (between successive 𝖯𝖨𝖯-spaces) is defined and is a homomorphism; (iii)if 𝐴∈Hom(𝑉_𝐼,𝑌_𝐾), then 𝑓#𝑔 implies 𝐴𝑓#𝐴𝑔.

The definition of homomorphisms just given is tailored in such a way that one may consider the category PIP of all indexed 𝖯𝖨𝖯-spaces, with the homomorphisms as morphisms (arrows), as we have done in Section 3.3 above. In the same language, we may define particular classes of morphisms, such as monomorphisms, epimorphisms, and isomorphisms. (i)Let 𝑀∈Hom(𝑊_𝐿,𝑌_𝐾). Then 𝑀 is called a monomorphism if 𝑀𝐴=𝑀𝐵 implies 𝐴=𝐵, for any two elements of 𝐴,𝐵∈Hom(𝑉_𝐼,𝑊_𝐿), where 𝑉_𝐼 is any indexed 𝖯𝖨𝖯-space.(ii)Let 𝑁∈Hom(𝑊_𝐿,𝑌_𝐾). Then 𝑁 is called an epimorphism if 𝐴𝑁=𝐵𝑁 implies 𝐴=𝐵, for any two elements 𝐴,𝐵∈Hom(𝑌_𝐾,𝑉_𝐼), where 𝑉_𝐼 is any indexed 𝖯𝖨𝖯-space.(iii)An operator 𝐴∈Op(𝑉_𝐼,𝑌_𝐾) is an isomorphism if 𝐴∈Hom(𝑉_𝐼,𝑌_𝐾), and there is a homomorphism 𝐵∈Hom(𝑌_𝐾,𝑉_𝐼) such that 𝐵𝐴=1_𝑉,𝐴𝐵=1_𝑌, the identity operators on 𝑉,𝑌, respectively.

Typical examples of monomorphisms are the inclusion maps resulting from the restriction of a support, for instance, the natural injection 𝑀^(Ω)∶𝐿¹_loc(Ω,𝖽𝑥)→𝐿¹_loc(ℝ,𝖽𝑥), where ℝ=Ω∪Ω′ is the partition of ℝ in two measurable subsets of nonzero measure. More examples and further properties of morphisms may be found in [5, Section 3.3] and in [7].

Finally, an orthogonal projection on a nondegenerate indexed 𝖯𝖨𝖯-space 𝑉_𝐼, in particular, an LBS or an LHS, is a homomorphism 𝑃∈Hom(𝑉_𝐼) such that 𝑃²=𝑃^×=𝑃.

A 𝖯𝖨𝖯-subspace 𝑊 of a 𝖯𝖨𝖯-space 𝑉 is defined in [5, Section 3.4.2] as an orthocomplemented subspace of 𝑉, that is, a vector subspace 𝑊 for which there exists a vector subspace 𝑍⊆𝑉 such that 𝑉=𝑊⊕𝑍 and (i){𝑓}^#={𝑓_𝑊}^#∩{𝑓_𝑍}^# for every 𝑓∈𝑉, where 𝑓=𝑓_𝑊+𝑓_𝑍,𝑓_𝑊∈𝑊,𝑓_𝑍∈𝑍; (ii)if 𝑓∈𝑊,𝑔∈𝑍 and 𝑓#𝑔, then ⟨𝑓∣𝑔⟩=0.

In the same Section 3.4.2 of [5], it is shown that a vector subspace 𝑊 of a nondegenerate 𝖯𝖨𝖯-space is orthocomplemented if and only if it is topologically regular, which means that it satisfies the following two conditions: (i)for every assaying subset 𝑉_𝑟⊆𝑉, the intersections 𝑊_𝑟=𝑊∩𝑉_𝑟 and 𝑊_𝑟=𝑊∩𝑉_𝑟 are a dual pair in 𝑉;(ii)the intrinsic Mackey topology 𝜏(𝑊_𝑟,𝑊_𝑟) coincides with the Mackey topology 𝜏(𝑉_𝑟,𝑉_𝑟)|_𝑊𝑟 induced by 𝑉_𝑟.

Then the fundamental result, which is the analogue to the similar statement for a Hilbert space, says that a vector subspace 𝑊 of the nondegenerate 𝖯𝖨𝖯-space 𝑉 is orthocomplemented if and only if it is the range of an orthogonal projection: 𝑊=𝑃𝑉,𝑉=𝑊⊕𝑊^⟂=𝑃𝑉⊕(1−𝑃)𝑉. (A.7) Clearly, this raises the question, discussed in Section 3.3.1, of identifying the subobjects of any category consisting of 𝖯𝖨𝖯-spaces.

B. Fine Sheaves

We collect here some classical notions and results used in Section 6. We recall first that the support of the continuous function 𝜑∶𝑋→ℝ on the topological space 𝑋 is the closed set supp 𝜑∶=closure{𝑥∈𝑋∶𝜑≠0}, which is the smallest closed set outside which 𝜑 is zero. The same definition applies to a distribution. Then we recall the standard notion of a partition of unity.

Definition B.1. Let 𝑈={𝑈_𝑖,𝑖∈𝐼} be an open covering of the topological space 𝑋. A set of real and continuous functions {𝜑_𝑖,𝑖∈𝐼} defined on 𝑋 is called a partition of unity with respect to 𝑈 if one has the following. (i)𝜑_𝑖(𝑥)⩾0,forall𝑥∈𝑋. (ii)supp 𝜑_𝑖⊂𝑈_𝑖,forall𝑖∈𝐼. (iii)Each point 𝑥∈𝑋 has an open neighborhood that meets supp 𝜑 for a finite number of 𝑖∈𝐼 only. (iv)∑_𝑖∈𝐼𝜑_𝑖(𝑥)=1,forall𝑖∈𝐼. This sum is well defined by (iii). We recall that a topological space is paracompact if it is separated (Hausdorff), and every open covering admits a locally finite open covering that is finer [22, § 6]. Every metrizable locally convex space is paracompact, but there are nonmetrizable paracompact spaces as well. The following result is standard.

Theorem B.2. 𝑋 is paracompact if and only if 𝑋 is a separated topological space and each open covering of 𝑋 admits a partition of unity.

The main use of paracompact spaces is for the definition of a fine sheaf, to which the Cartan-Leray theorem applies (see below).

Definition B.3. Let 𝖥 be a sheaf on a paracompact topological space 𝑋. One says that 𝖥 is fine if, for every locally finite open covering 𝑈={𝑈_𝑖,𝑖∈𝐼} of 𝑋, there exists a set of homomorphisms {ℎ_𝑖∶𝖥→𝖥,𝑖∈𝐼} such that (1)for each 𝑖∈𝐼, there exists a closed set 𝑀_𝑖 of 𝑋 such that 𝑀_𝑖⊂𝑈_𝑖 and ℎ_𝑖(𝖥_𝑥)=0 for 𝑥∉𝑀_𝑖, where 𝖥_𝑥 is the stalk of the sheaf 𝖥 at the point 𝑥 (the stalk is defined by the inductive limit li^→m_{𝑉𝑘∋𝑥}𝖥(𝑉_𝑘));(2)∑_𝑖∈𝐼ℎ_𝑖=1. This sum is well defined since the covering 𝑈 is locally finite.
Then the basic result is the following standard theorem. (A good introduction to the cohomology of sheaves and to the Cartan-Leray theorem can be found in [23, pages 166–197].)

Theorem B.4 (Cartan-Leray). Let 𝖥 be a fine sheaf on a paracompact topological space 𝑋. Then 𝖥 is acyclic; that is, the higher-order sheaf cohomology groups are trivial; 𝐻^𝑝(𝑋,𝖥)=0 for all 𝑝≥1.

Acknowledgments

J.-P. Antoine thanks Juma Shabani (UNESCO) for communicating his unpublished results. D. Lambert thanks Bertrand Hespel (FUNDP) for inspiring discussions. The authors all thank the anonymous referee for his constructive remarks that have definitely improved the paper.