Partial inner product spaces: Some categorical aspects

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.


Motivation
Partial inner product spaces (pip-spaces) were introduced some time ago by A. 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, etc. [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, that 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.

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 ∈ I such that f ∈ V r , g ∈ V r . (2.1) The lattice properties are the following: . involution: V r ↔ V r := (V r ) # , . infimum: The smallest element of F(V, #) is V # = r V r and the greatest element is V = r V 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 particular, we consider projective and additive indexed pip-spaces (see Appendix A) and, in particular, lattices of 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 : there is a u ∈ K such that A ur exists}. 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].

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 α −→ Y , satisfying the following axioms.
• Identity: for any object X, there is a unique arrow 1 X : X → X.
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 and Given a category C, the opposite category C op has the same objects as C and all arrows reversed: to each arrow α : 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 and Some standard examples of categories are . Set, the category of sets with functions as arrows.
. Top, the category of topological spaces with continuous functions as arrows.
. Grp, the category of groups with group homomorphisms as arrows. For more details, we refer to standard texts, such as Mac Lane [8].

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 • The objects are the assaying subspaces {V r , r ∈ I, V # , V }.
• 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 • For every V r , there exists an identity, E rr : V r → V r , the identity map.
• For every V r , V s with r s, one has E ss • E sr = E sr and E sr • E rr = E sr .
• For every V r , V s , V t with r s t, one has E ts • E sr = E tr .
In the category V I , . 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.

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: • The objects are the assaying subspaces {V r , r ∈ I}.
• The arrows are the operators A n pq : V q → V p , q p, n ∈ N.
The axioms of categories are readily checked: • For every V r , there exists an identity, A rr : V r → V r , since A is totally regular.
• For every V r , V s with r s, one has A ss • A n sr = A n+1 sr and A n sr As for V I , the space where the latter is the category induced by A # . The proof is immediate.

The category PIP of indexed pip-spaces
The collection of all indexed pip-spaces constitutes a category, that we call PIP, where • Objects are indexed pip-spaces {V I }.
• Arrows are homomorphisms A : (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.
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: • For every V I , there exists an identity, 1 I ∈ Hom(V I , V I ), the identity operator on V I .
• For every V I , Y K , one has 1 K • A KI = A KI and A KI • 1 I = A KI .
• For every V I , Y K , W L , one has B LK • A KI = C LI ∈ Hom(V I , W L ).
• For every V I , Y K , W L , Z M , 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 Banach spaces (LBS) and lattices of Hilbert spaces (LHS), the arrows being still the corresponding homomorphisms.

Subobjects
We recall that a homomorphism M KI ∈ Hom(V I , Y K ) is a monomorphism if M KL A LI = M KL B 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 Section A.2). Two monomorphisms M LI , N LK with the same codomain W L are equivalent if there exists an isomorphism U KI such that N LK U 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 = P V I . Now the embedding M : W I = P V 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 non-complete prehilbert space (i.e., V = V # ). Then every subspace is a subobject, but need not be the range of a projection. To give a concrete example [5,Sec. 3.4.5], take V = S(R), the Schwartz space of test functions. Let W = S + := {ϕ ∈ S : ϕ(x) = 0 for x 0}. Then W ⊥ = S − := {ψ ∈ S : ψ(x) = 0 for x 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,Sec.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 precidely the orthocomplemented subspaces.
Coming back to the previous example of a non-complete 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,Sec. 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.

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 IK N KL = B IK N 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, i.e., cosheaves.

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.
family of open subsets of X such that: 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. We say 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: , for every i ∈ I, then s = s ′ (local identity); , for every i, j ∈ I, then there exists a section s ∈ F(U ) such that ρ U U i (s) = s i , for every i ∈ I (gluing). The section s whose existence is guaranteed by axiom S 2 is called the gluing, concatenation, or collation of the sections s i . By axiom S 1 it is unique. The sheaf F may be seen as a contravariant functor from the category of open sets of X into C :

A sheaf of operators on an indexed pip-space
Let V I = {V r , r ∈ I} be an indexed pip-space and V I 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 topology τ (V r , V r ). We define a sheaf on V I by the contravariant functor F : V I → Set given by 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 of left multipliers [5, Sec.6.2.3] Then we have • (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 .
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: 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 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.

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∈I W 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: 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∈J W j , J ⊆ I and for every family of (local) sections {t j ∈ G(W j )} j∈J , the following conditions are fulfilled: , for every j, k ∈ J, then there exists a unique section t ∈ G(W ) The cosheaf G may be seen as a covariant functor from the category of closed sets of X into 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, Cl(X) the family of closed subsets of X and a coarsening of the set inclusion in Cl(X), that is, W Z implies W ⊆ Z, but not necessarily the opposite. A partial 4 pre-cosheaf G with values in C is a map G defined on Cl(X), satisfying condition (PC 1 ) and 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∈J W j , J ⊆ I and for every family of sections {t j ∈ G(W j )} j∈J , the following conditions are fulfilled: (t k ) exist and are equal, for every j, k ∈ J, then δ W j W exists for every j ∈ J and there exists a unique section t ∈ G(W ) such that δ W j W (t) = t j , for every j ∈ J .

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): 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.

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 Assume now that V I is additive, i.e., 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 * r f r and C (r∨s) f s = B * s f s , we may write, for any Here too, this operator is well-defined. The conclusion is that, for any pair of assaying subspaces V r , V s , the extension maps δ r∨s r and δ r∨s s 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 V r , V s , V t (and using the associativity of ∨). Thus me may state: 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 ∈ I such that s = r ∨ w.

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 Here again, elements of LOp r are representatives A * r . In that case, extensions exist always. When * ,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 (5.1) 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 .

General operators: the projective case
We consider again all operators in Op(V I ). The fact is that the set r∈I Op 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: Now indeed r∈I Op 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 , ∀ r, s ∈ I, with the projective topology. Then C r∧s * = A r∧s * = B r∧s * , that is, condition (CS 2 ) 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: 6 Cohomology of an indexed pip-space

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 .
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, i.e., 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
The previous formula becomes: Using the properties of restrictions we get: which shows that indeed DDB = 0. Now it is possible to define cohomology groups of the sheaf F on a indexed pip-space V . 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) , ∀ j ∈ J. For instance, U t(j) could be a union ∪ l V 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), ∀ p 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 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. Proof. First of all, V is a paracompact space, since it is metrizable.
Next, we check that the sheaf F : 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∈I ϕ k (C jk ). This sum is well-defined since the covering is locally finite. Applying the coboundary operator we get: Putting k = j 2 and using the equation (DC) j 0 ,j 1 ,j 2 = 0, we find But since {ϕ i , i ∈ I} is a partition of unity, we get (DE) j 0 ,j 1 = C j 0 j 1 . This means that DE = C and thus [C] = 0.
Of course, the restriction that V 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 H 1 ֒→ H 0 ֒→ H 1 or any of its refinements, as discussed in [5,Sec.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,Sec.4.6.3].
(3) A Banach Gel'fand triple, in the sense of Feichtinger [11], that is, a 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 S 0 (R d ), 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,Sec.5.3 and 5.4].
In fact, what is really needed for the Cartan-Leray theorem is not that V be metrizable, but that it be paracompact. Indeed, without that condition the situation becomes totally unmanageable. However, except for metrizable spaces, we could not find interesting examples of indexed pipspaces with V paracompact.

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 (5.1), namely, G : In the sequel, G(W j 0 ∪ W j 1 ∪ . . . ∪ W jp ) denotes the set of operators W j 0 ∪ W j 1 ∪ . . . ∪ W jp → V . Alternatively, one could consider the cosheaf defined in (5.2), 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 j 0 ∪ W j 1 ∪ . . . ∪ W jp , j k ∈ I, j 0 < j 1 < . . . < j p , an operator A j 0 ,j 1 ,...,jp of G(W j 0 ∪ W j 1 ∪ . . . ∪ W jp ). A p-cochain is thus a set A = {A j 0 ,j 1 ,...,jp , j 0 < j 1 < . . . < 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: A j 0 ,j 1 ,..., j k ,...,j p+1 .
In view of the properties of the maps δ * * , we check that D D = 0. In the case of 0-cochains, a straighforward application of the formula leads to And if we put this to zero, we get the constraint δ A j 0 , which has to be satisfied in order to build a cosheaf, according to condition (CS2). On 1-cochains, the coboundary action gives 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 won't venture into it.

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,Sec. 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, Sec. I.1.10].
More precisely, classical logic satisfies the noncontradiction principle NCP ¬(p ∧ ¬p)) and the excluded middle principle EMP (p ∨ ¬p). 5 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 [16,17] 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!).
Appendix 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 General considerations imply that the family F(V, #) := {V r , r ∈ F }, ordered by inclusion, is a complete involutive lattice, i.e., it is stable under the following operations, arbitrarily iterated: . involution: The smallest element of F(V, #) is V # = r V r and the greatest element is V = r V r .
By definition, the index set F is also a complete involutive lattice; for instance, 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 [21]. 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 , ∀ r ∈ 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 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: here V p∧q | τ denotes V p∧q equipped with the Mackey topology τ (V p∧q , V p∨q ), the r.h.s. 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 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) A LBS or a 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 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 a RHS from a directed family of Hilbert spaces, via projective and inductive limits, has been investigated recently by Bellomonte and Trapani [18]. Similar constructions, in the language of categories, may be found in the work of Mityagin and Shvarts [19] and that of Semadeni and Zidenberg [20].
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 n y n | < ∞; (ii) a partial inner product x|y = ∞ n=1 x n y n . Then ω # = ϕ , the space of finite sequences, and the complete lattice F(ω, #) consists of Köthe's perfect sequence spaces [21, § 30]. Among these, a nice example is the lattice of the so-called ℓ φ spaces associated to symmetric norming functions or, more generally, Banach sequence ideals discussed in [5,Sec.4.3.2] and previously in [19, § 6] (in this example, the extreme spaces are, respectively, ℓ 1 and ℓ ∞ ).

(ii) Spaces of locally integrable functions
Let V be L 1 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 = R f (x)g(x) dx. Then V # = L ∞ c (R, dx), the space of bounded measurable functions of compact support. The complete lattice F(L 1 loc , #) consists of the so-called Köthe function spaces. Here again, normed ideals of measurable functions in L 1 ([0, 1], dx) are described in [19, § 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 from a subset D(A) ⊂ V into Y , such that 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 u exists as a bounded operator}. We will also need the following sets: there is a r such that A ur exists}.
The following properties are immediate: . d(A) is an initial subset of I: if r ∈ d(A) and r ′ < r, then r ∈ d(A), and A ur ′ = A ur E rr ′ , where E rr ′ is a representative of the unit operator.
Although an operator may be identified with a separately continous 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 : 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. (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 M A = M B 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) : L 1 loc (Ω, dx) → L 1 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,Sec.3.3] and in [7].
Finally, an orthogonal projection on a nondegenerate indexed pip-space V I , in particular, a LBS or a LHS, is a homomorphism P ∈ Hom(V I ) such that P 2 = P × = P .
A pip-subspace W of a pip-space V is defined in [5,Sec.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 (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 : Clearly, this raises the question, discussed in Section 3.3.1, of identifying the subobjects of any category consisting of pip-spaces.
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 (i) ϕ i (x) 0, ∀ x ∈ X ; (ii) supp ϕ i ⊂ U i , ∀ i ∈ I ; (iii) each point x ∈ X has an open neighborhood that meets supp ϕ for a finite number of i ∈ I only; (iv) i∈I ϕ i (x) = 1, ∀ i ∈ 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 non-metrizable paracompact spaces as well. The following result is standard. (2) i∈I h i = 1. This sum is well-defined since the covering U is locally finite. Then the basic result is the following standard theorem. 6 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.