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Advances in Mathematical Physics

Volume 2011 (2011), Article ID 957592, 22 pages

http://dx.doi.org/10.1155/2011/957592

## Partial Inner Product Spaces: Some Categorical Aspects

^{1}Institut de Recherche en Mathématique et Physique (IRMP), Université Catholique de Louvain, 1348 Louvain-la-Neuve, Belgium^{2}Dipartimento di Matematica e Informatica, Università di Palermo, 90123 Palermo, Italy^{3}Namur Center for Complex Systems (NAXYS), Facultés Universitaires Notre-Dame de la Paix, B-5000 Namur, Belgium

Received 16 March 2011; Revised 20 July 2011; Accepted 10 August 2011

Academic Editor: Partha Guha

Copyright © 2011 Jean-Pierre Antoine et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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

#### 1. Motivation

Partial inner product spaces (-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 -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) -spaces, with homomorphisms as arrows (they are defined precisely to play that role), as well as several other categories of -spaces.

In a second part, we consider a single -space as a category by itself, called , with natural embeddings as arrows. For this category , we show, in Sections 4 and 5, respectively, that one can construct sheaves and cosheaves of operators. There are some restrictions on the -space , 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 -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 -space , corresponding to the linear compatibility . The assaying subspaces are denoted by . The set indexes a generating involutive sublattice of the complete lattice of all assaying subspaces defined by ; that is, The lattice properties are the following: (i)involution: ,(ii)infimum: ,(iii)supremum: .

The smallest element of is , and the greatest element is , but often they do not belong to .

Each assaying subspace carries its Mackey topology , and is dense in every , since the indexed -space is assumed to be nondegenerate. In the sequel, we consider projective and additive indexed -spaces (see Appendix A) and, in particular, lattices of the Banach or Hilbert spaces (LBS/LHS).

Given two indexed -spaces , an operator may be identified with the coherent collection of its representatives , where each is a continuous operator from into . We will also need the set . Every operator has an adjoint , 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 into is denoted by and the set of all homomorphisms by .

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 and arrows or morphisms , where we note or , satisfying the following axioms. (i)*Identity:* for any object , there is a unique arrow .(ii)*Composition:* whenever , there is a unique arrow such that .(iii)*Associativity:* whenever , one has .(iv)*Unit law:* whenever , one has and .

In a category, an object is *initial* if, for each object , there is exactly one arrow . An object is *final* or *terminal* if, for each object , there is exactly one arrow . Two terminal objects are necessarily isomorphic (isomorphisms in categories are defined exactly as for indexed -spaces, see Appendix A).

Given two categories **C** and **D**, a *covariant functor * is a morphism between the two categories. To each object of **C**, it associates an object of **D**, and, to each arrow in **C**, it associates an arrow of **D** in such a way that
whenever the arrow is defined in **C**.

Given a category **C**, the *opposite* category has the same objects as **C** and all arrows reversed: to each arrow , there is an arrow , so that .

A *contravariant functor * may be defined as a functor , or directly on **C**, by writing . Thus, we have
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 -Spaces

##### 3.1. A Single -Space as Category

We begin by a trivial example.

A single-indexed -space may be considered as a category , where (i)the objects are the assaying subspaces ; (ii)the arrows are the natural embeddings , that is, the representatives of the identity operator on .

The axioms of categories are readily checked as follows. (i)For every , there exists an identity, , the identity map. (ii)For every with , one has and . (iii)For every with , one has . (iv)For every with , one has .

In the category , we have: (i) is an initial object: for every , there is a unique arrow . (ii) is a terminal object: for every , there is a unique arrow . (iii)The compatibility defines a contravariant functor .

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 -space , take a single totally regular operator, that is, an operator that leaves every invariant. Hence so does each power . Then this operator induces a category **A(****)**, as follows. (i)The objects are the assaying subspaces . (ii)The arrows are the operators .

The axioms of categories are readily checked as follows. (i)For every , there exists an identity, , since is totally regular. (ii)For every with , one has and . (iii)For every with , one has . (iv)For every with , one has .

As for , the space is an initial object in **A(****)** and is a terminal object.

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

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

The collection of all indexed -spaces constitutes a category, which we call** PIP**, where (i)objects are indexed -spaces ; (ii)arrows are homomorphisms , where an operator is called a *homomorphism* if (a)for every there exists such that both and exist; (b)for every there exists such that both and exist.

For making the notation less cumbersome (and more automatic), we will, henceforth, denote by an element . Then the axioms of a category are obviously satisfied as follows. (i)For every , there exists an identity, , the identity operator on . (ii)For every , one has and . (iii)For every , one has . (iv)For every , one has .

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 is a monomorphism if implies , for any pair and any indexed -space (a typical example is given in Appendix A). Two monomorphisms with the same codomain are *equivalent* if there exists an isomorphism such that . Then a *subobject* of is an equivalence class of monomorphisms into . A -subspace of an indexed -space is defined as an orthocomplemented subspace of , and this holds if and only if is the range of an orthogonal projection, . Now the embedding is a monomorphism; thus, orthocomplemented subspaces are subobjects of **PIP**.

However, the converse is not true, at least for a general indexed -space. Take the case where is a noncomplete prehilbert space (i.e., ). 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 , the Schwartz space of test functions. Let . Then ; hence, . However, is not orthocomplemented, since every satisfies , so that . Yet is the range of a monomorphism (the injection), hence, a subobject. However, this example addresses an indexed -space which is *not* a LBS/LHS.

Take now a LBS/LHS and a vector subspace . In order that becomes a LBS/LHS in its own right, we must require that, for every , and are a dual pair with respect to their respective Mackey topologies and that the intrinsic Mackey topology coincides with the norm topology induced by . In other words, must be topologically regular, which is equivalent that it be orthocomplemented [5, Section 3.4.2]. Now the injection is clearly a monomorphism, and is a subobject of . 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 need *not* be orthocomplemented, because it may fail to be topologically regular. Indeed the intrinsic topology does not coincide with the norm topology, unless is orthocomplemented (see the discussion in [5, Section 3.4.5]). In the Schwartz example above, one has , which means that is closed, hence, norm closed, but it is not orthocomplemented.

*Remark 3.1. *Homomorphisms are defined between arbitrary -spaces. However, when it comes to *indexed *-spaces, the discussion above shows that the notion of homomorphism is more natural between two indexed -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 is an epimorphism if implies , for any pair and any indexed -space . 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 -Spaces

##### 4.1. Presheaves and Sheaves

Let be a topological space, and let be a (concrete) category. Usually 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 * on with values in is a map defined on the family of open subsets of such that (PS_{1})for each open set of , there is an object in ; (PS_{2})for each inclusion of open sets , there is given a *restriction morphism * in the category , such that is the identity for every open set and whenever .

*Definition 4.2. *Let be a presheaf on , and let be an open set of . Every element is called a *section* of over . A section over is called a *global section*.

*Example 4.3. *Let be a topological space; let be the category of vector spaces. Let associate to each open set the vector space of continuous functions on with values in . If , associates to each continuous function on its restriction to . This is a presheaf. Any continuous function on is a section of on .

*Definition 4.4. *Let be a presheaf on the topological space . One says that is a *sheaf* if, for every open set and for every open covering of , the following conditions are fulfilled: (S_{1})given such that , for every , then (local identity); (S_{2})given such that , for every , then there exists a section such that , for every (gluing).

The section whose existence is guaranteed by axiom (S_{2}) is called the * gluing*, * concatenation*, or *collation* of the sections . By axiom (S_{1}), it is unique. The sheaf may be seen as a contravariant functor from the category of open sets of into .

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

Let be an indexed -space, and let be the corresponding category defined in Section 3.1. If we put on the discrete topology, then defines an open covering of . Each carries its Mackey’s topology .

We define a sheaf on by the contravariant functor given by This means that an element of is a representative from into some . In the sequel, we will use the notation whenever the dependence on the first index may be neglected without creating ambiguities. By analogy with functions, the elements of may be called germs of operators. is the restriction to of the set of left multipliers [5, Section 6.2.3]:

Then we have:(i)When , define by , for . Clearly, and if . Hence, is a presheaf.(ii) (S_{1}) is clearly satisfied. As for (S_{2}), if and are such that ; that is, , then these two operators are the representative of a unique operator . It remains to prove that extends to and that its representative extends both and .

Proposition 4.5. *Let the indexed -space be additive; that is, . Then the map given in (4.1) is a sheaf of operators on .*

*Proof. *By linearity, and can be extended to an operator on as follows:
This operator is well defined. Let indeed be two decompositions of , so that . Then . Hence, . Taking the restriction to , this relation becomes , so that the vector is uniquely defined.

Next, by additivity, , with its inductive topology, coincides with , and, thus, is the -representative of the operator . Therefore, is a sheaf.

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

##### 5.1. Pre-Cosheaves and Cosheaves

Pre-cosheaves and cosheaves are the dual notions of presheaves and sheaves, respectively. Let again be a topological space, with closed sets so that , and let be a (concrete) category.

*Definition 5.1. * A *pre-cosheaf * on with values in is a map defined on the family of closed subsets of such that (PC_{1})for each closed set of , there is an object in ; (PC_{2})for each inclusion of closed sets , there is given a *extension morphism * in the category , such that is the identity for every closed set and whenever .

*Definition 5.2. * Let be a pre-cosheaf on the topological space . We say that is a *cosheaf* if, for every nonempty closed set and for every family of (local) sections , the following conditions are fulfilled: (CS_{1})given such that , for every such that , then ;(CS_{2})if , for every , then there exists a unique section such that , for every .

The cosheaf may be seen as a covariant functor from the category of closed sets of into .

Now, there are situations where extensions do not exist for all inclusions but only for certain pairs. This will be the case for operators on a -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 be a topological space, let be the family of closed subsets of , and let a be coarsening of the set inclusion in ; that is, implies , but not necessarily the opposite. A *partial pre-cosheaf * with values in is a map defined on , satisfying conditions (PC_{1}) and (pPC_{2})if , there is given an *extension morphism * in the category , such that is the identity for every closed set and whenever .

We may use the term “partial,” since here not all pairs admit extensions, but only certain pairs, namely, those that satisfy . This is analogous to the familiar situation of a compatibility.

*Definition 5.4. *Let be a partial pre-cosheaf on the topological space . We say that is a *partial cosheaf* if, for every nonempty closed set and for every family of sections , the following conditions are fulfilled: (pCS_{1})given such that and exist and are equal, for every , then ;(pCS_{2})if and exist and are equal, for every , then exists for every and there exists a unique section such that , for every .

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

Let again be an indexed -space. If we put on the discrete topology, then the assaying subspaces may be taken as closed sets in . In order to build a cosheaf, we consider the same map as before in (4.1): Then we immediately face a problem. Given , there is no , with , such that exists. One can always find an operator such that and ; in other words, exists, but it cannot be extended to . This happens, for instance, each time 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 . Take two assaying subspaces and two operators , and assume that they have a common extension to . This means that, for any suitable , is the -representative of a unique operator , and . *A fortiori*, , and coincide on ; that is, condition (pCS_{2}) is satisfied.

Assume now that is additive; that is, coincides with , with its inductive topology. Then we can proceed as in the case of a sheaf, by defining the extension of and by linearity. In other words, since and , we may write, for any ,
Here too, this operator is well defined. Let indeed be two decompositions of , so that . Then necessarily . Hence, . Taking the restriction to , this relation becomes , and this equals 0, since on , by the condition (pcS_{2}).

The conclusion is that, for any pair of assaying subspaces , the extension maps and always exist, and the condition (pCS_{2}) 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 (pPC_{2}) is also satisfied, as one can see by taking supremums (= sums) of successive pairs within three assaying subspaces (and using the associativity of ). Thus, me may state the following.

Proposition 5.5. *Let be an additive indexed -space. Then the map given in (4.1) is a partial cosheaf with respect to the partial order on :
*

###### 5.2.2. Universal Left Multipliers

We consider only the operators which are everywhere defined; that is, . This is precisely the set of universal left multipliers: where . Correspondingly, we consider the map Here again, elements of are representatives . In that case, extensions always exist. When , define by for . Clearly, and if . Hence, is a pre-cosheaf. Moreover, if ; that is, , these two operators are the representative of a unique operator . Then, of course, one has and ; thus, is a cosheaf. Therefore,

Proposition 5.6. *Let be an arbitrary indexed -space. Then the map given in (5.5) is a cosheaf on with values in , the set of universal left multipliers, with extensions given by .*

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

We consider again all operators in . The fact is that the set is an initial set in , like every ; 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 . Hence, we define the map:
Now indeed is a final set. Elements of have been denoted as , but, for definiteness, we could also replace by where .

We claim that the map defines a cosheaf, with extensions , for , which exist whenever .

As in the case of Section 5.2.1, assume that have a common extension belonging to ; that is, . Call this operator . Thus, for any suitable , is the representative of a unique operator , which extends and . Thus . Assume now that is projective; that is, , with the projective topology. Then, ; that is, condition (CS_{2}) is satisfied with extensions and . The rest is obvious. As in the case of sheaves, extensions from to exist by linearity, since is projective, hence, additive. Thus, we may state the following.

Proposition 5.7. *Let the indexed -space be projective that is, , with the projective topology. Then the map given in (5.6) is a cosheaf of operators on .*

Corollary 5.8. *Let be a projective indexed -space. Then generates in a natural fashion a sheaf and a cosheaf of operators. If is additive, but not projective; it generates a sheaf and a partial cosheaf of operators.*

#### 6. Cohomology of an Indexed -Space

##### 6.1. Cohomology of Operator Sheaves

It is possible to introduce a concept of sheaf cohomology group defined on an indexed -space according to the usual definition of sheaf cohomology [10]. Let be an indexed -space. The set of its assaying spaces, endowed with the discrete topology, defines an open covering of . Endowed with the Mackey topology, , each is a Hausdorff vector subspace of .

*Definition 6.1. *Given the indexed -space , let be the open covering of by its assaying subspaces, and let be the sheaf of operators defined in (4.1) above. We call *-cochain with values in * the map which associates, to each intersection of open sets , an operator belonging to .

Thus, a -cochain with value in is a set . The set of -cochains is denoted by . For example, a 0-cochain is a set . A 1-cochain is a set .

*Definition 6.2. *Given the notion of -cochain, the corresponding *coboundary operator * is defined by
where the hat corresponds to the omission of the corresponding symbol. One checks that is a coboundary operator; that is, .

If we compute the action of the coboundary operator on 0-cochains, we get

This equation is nothing but the condition (S_{1}) 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 Now, let us suppose that a 1-cochain is defined by , where is a 0-cochain on . The previous formula becomes: Using the properties of restrictions we get which shows that indeed .

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

*Definition 6.3. *Let be an indexed -space, endowed with the open covering of its assaying spaces, , and let be the corresponding sheaf of operators on . Then the th cohomology group is defined as , where is the set of -cocycles, that is, -cochains such that , and is the set of -coboundaries, that is, -cochains for which there exists a -cochain with .

The definition of the group is motivated by the fact that the -cocycle is given only up to a coboundary .

Our definition of cohomology groups of sheaves on indexed -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 of an indexed -space is finer than another one, , if there exists an application such that . For instance, could be a union containing . According to the general theory of sheaf cohomology [9], this induces group homomorphisms . It is then possible to introduce cohomology groups that do not depend on the particular open covering, namely, by defining as the inductive limit of the groups with respect to the homomorphisms .

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 -spaces. We collect in Appendix B the definitions and the results needed for this discussion. We know that an indexed -space endowed with the Mackey topology is a separated (Hausdorff) locally convex space, but it is not necessarily paracompact, unless 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 and apply the Cartan-Leray Theorem B.4. This justifies the restriction of metrizability in the following theorem.

Theorem 6.4. *Given an indexed -space , define the sheaf . Then, if is metrizable for its Mackey topology, the sheaf is acyclic; that is, the cohomology groups are trivial for all ; that is, .*

*Proof. *First of all, is a paracompact space, since it is metrizable.

Next, we check that the sheaf on the paracompact space is fine. Indeed, let be a locally finite open covering of . We can associate to the latter a partition of unity . Then we can use to define the homomorphisms . Let for some index set . For each , we consider the set of operators , and we define . This defines a homomorphism and then a homomorphism . One can check that satisfies conditions (1) and (2) in Definition B.3, and, thus, 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 be an element of . We want to show that . As an equivalence class, can be represented by an element of , such that ; that is, Using the fact that is a fine sheaf, we can choose the partition of unity to define the 0-cochain , where . This sum is well defined since the covering is locally finite. Applying the coboundary operator, we get Putting and using the equation , we find But since is a partition of unity, we get . This means that , and; thus, .

Of course, the restriction that 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 or any of its refinements, as discussed in [5, Section 5.2.2],(2)an indexed -space whose extreme space 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 *, which generates the triplet
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 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 -spaces with paracompact.

##### 6.2. Cohomology of Operator Cosheaves

It is also possible to get cohomological concepts on cosheaves defined from indexed -spaces. The assaying spaces can also be considered as closed sets. Let, thus, be such a closed covering of , and let be a cosheaf of operators defined in (5.5), namely, together with maps as above. In the sequel, denotes the set of operators . Alternatively, one could consider the cosheaf defined in (5.6), 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 *-cochain* with values in the cosheaf is a map which associates to each union of closed sets , an operator of . A -cochain is, thus, a set . The set of such -cochains will be denoted by .

*Definition 6.6. *One can then introduce the coboundary operator as follows:
In view of the properties of the maps , we check that .

In the case of 0-cochains, a straightforward application of the formula leads to
And if we put this to zero, we get the constraint , which has to be satisfied in order to build a cosheaf, according to the condition (CS_{2}). On 1-cochains, the coboundary action gives
The cohomology groups of the cosheaf , with obvious notations, can then be defined in a natural way. Similarly for and .

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 -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 ? Could it be somehow linked to the category of partial *algebras, in the same way as 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 -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, , for all , or, similarly, , for all ?

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 and the excluded middle principle () (i.e., : = Not-( And Not-) and : = Or Not-.). Intuitionistic logic satisfies but not . Finally, we know that Paraconsistent logic, satisfying but not , 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 -spaces are connected to quantum physics. Is there any hope to connect cosheaves of operators on some -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 into !).

#### Appendices

#### A. Partial Inner Product Spaces

##### A.1. -Spaces and Indexed -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 -space , corresponding to the linear compatibility , that is, a symmetric binary relation which preserves linearity. We call *assaying subspace* of a subspace such that , and we denote by the family of all assaying subspaces of , ordered by inclusion. The assaying subspaces are denoted by , and the index set is . By definition, if and only if . Thus, we may write

General considerations imply that the family , ordered by inclusion, is a complete involutive lattice; that is, it is stable under the following operations, arbitrarily iterated: (i)involution:,(ii)infimum: ,(iii)supremum:.

The smallest element of is , and the greatest element is .

By definition, the index set is also a complete involutive lattice; for instance,

Given a vector space equipped with a linear compatibility , a *partial inner product* on is a Hermitian form defined exactly on compatible pairs of vectors. A *partial inner product space* (-space) is a vector space equipped with a linear compatibility and a partial inner product.

From now on, we will assume that our -space is *nondegenerate*; that is, for all implies . As a consequence, and every couple , are a dual pair in the sense of topological vector spaces [18]. Next, we assume that every carries its Mackey topology , so that its conjugate dual is . Then, implies , and the embedding operator is continuous and has dense range. In particular, is dense in every .

As a matter of fact, the whole structure can be reconstructed from a fairly small subset of , namely, a *generating* involutive sublattice of , indexed by , which means that
The resulting structure is called an* indexed **-space *and denoted simply by .

Then an indexed -space is said to be (i)*additive* if ,(ii)*projective* if ; here denotes equipped with the Mackey topology , the right-hand side denotes with the topology of the projective limit from and , 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 -space satisfying the following conditions: (i)every , is a Hilbert space or a reflexive Banach space, so that the Mackey topology coincides with the norm topology;(ii)there is a unique self-dual, Hilbert, assaying subspace .

In that case, the structure 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 -space is additive;(ii)an LBS or an LHS is projective if and only if it is additive.

Note that themselves usually do *not* belong to the family , but they can be recovered as
A standard, albeit trivial, example is that of a Rigged Hilbert space (RHS) (it is trivial because the lattice 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 be the space of

*all*complex sequences and define on it (i) a compatibility relation by , (ii) a partial inner product . Then , the space of finite sequences, and the complete lattice consists of Köthe's perfect sequence spaces [18, § 30]. Among these, a nice example is the lattice of the so-called 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., and ).

*(ii) Spaces of Locally Integrable Functions *

Let be , the space of Lebesgue measurable functions, integrable over compact subsets, and define a compatibility relation on it by and a partial inner product . Then , the space of bounded measurable functions of compact support. The complete lattice consists of the so-called Köthe function spaces. Here again, normed ideals of measurable functions in are described in [20, § 8].

##### A.2. Operators on Indexed -Spaces

Let and be two nondegenerate indexed -spaces (in particular, two LHSs or LBSs). Then an *operator* from to is a map from a subset into , such that (i), where is a nonempty subset of .(ii)For every , there exists such that the restriction of to is a continuous linear map into (we denote this restriction by ). (iii) has no proper extension satisfying (i) and (ii).

We denote by the set of all operators from to and, in particular, . The continuous linear operator is called a *representative* of . Thus, the operator may be identified with the collection of its representatives, . We will also need the following sets:
The following properties are immediate: (i) is an initial subset of : if and , then , and , where is a representative of the unit operator.(ii) is a final subset of : if and , then , and .

Although an operator may be identified with a separately continuous sesquilinear form on , it is more useful to keep also the *algebraic operations* on operators, namely, (i)*adjoint:* every has a unique adjoint and one has , for every : no extension is allowed, by the maximality condition (iii) of the definition. (ii)*partial multiplication:* let , , and be nondegenerate indexed -spaces (some, or all, may coincide). Let , and, . We say that the product is defined if and only if there is a ; that is, if and only if there is continuous factorization through some :
Among operators on indexed -spaces, a special role is played by morphisms.

An operator is called a *homomorphism* if (i)for every , there exists such that both and exist;(ii)for every , there exists such that both and exist.

We denote by the set of all homomorphisms from into and by those from into itself. The following properties are immediate.

Proposition A.1. *Let be indexed -spaces. Then, *(i)* if and only if ; *(ii)*the product of any number of homomorphisms (between successive -spaces) is defined and is a homomorphism; *(iii)*if , then implies . *

The definition of homomorphisms just given is tailored in such a way that one may consider the category ** PIP** of all indexed -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 . Then is called a *monomorphism* if implies , for any two elements of , where is any indexed -space.(ii)Let . Then is called an *epimorphism* if implies , for any two elements , where is any indexed -space.(iii)An operator is an *isomorphism* if , and there is a homomorphism such that , the identity operators on , respectively.

Typical examples of monomorphisms are the inclusion maps resulting from the restriction of a support, for instance, the natural injection , where is the partition of 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 -space , in particular, an LBS or an LHS, is a homomorphism such that .

A -subspace of a -space is defined in [5, Section 3.4.2] as an *orthocomplemented* subspace of , that is, a vector subspace for which there exists a vector subspace such that and (i) for every , where ; (ii)if and , then .

In the same Section 3.4.2 of [5], it is shown that a vector subspace of a nondegenerate -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 , the intersections and are a dual pair in ;(ii)the intrinsic Mackey topology coincides with the Mackey topology induced by .

Then the fundamental result, which is the analogue to the similar statement for a Hilbert space, says that a vector subspace of the nondegenerate -space is orthocomplemented if and only if it is the range of an orthogonal projection: Clearly, this raises the question, discussed in Section 3.3.1, of identifying the subobjects of any category consisting of -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 on the topological space is the closed set supp , which is the smallest closed set outside which is zero. The same definition applies to a distribution. Then we recall the standard notion of a partition of unity.

*Definition B.1. *Let be an open covering of the topological space . A set of real and continuous functions defined on is called a *partition of unity* with respect to <