-SEMI-SHEAVES, DISTRUCTURES AND
by Andrée C. EHRESMANN
Université de Picardie Jules Verne, Amiens
In the early 50's Charles Ehresmann is
interested in mathematical species of structures (a concept also discussed
during the Bourbaki meetings to which he participated); in particular he gives
the definition of a species of local
structures in connection with his work on locally trivial fibred spaces and
foliated manifolds (, Part I). To such a species he associates the pseudogroup
of its local automorphisms and, as soon as 1952, remarks that it is a groupoid
in the sense of Brandt, which plays the same role as an operator group on a
homogeneous space. But it is only in his important 1957 paper « Gattungen
von lokalen Strukturen »  that he did make the link with the categories
In this paper, he introduces the notion of a species of structures over a category C and of the associated of action of the category C on the set S of fibers of this species, and he shows that it is equivalent to a functor F from C to the category Set sending an object e of C on the fiber Se (hence, in modern terms, to a presheaf on Cop). He also gives the construction (often erroneously attributed to Grothendieck) of the associated discrete fibration p: C*S → C, which it calls "hypermorphism category", and characterizes the functors so obtained.
This very substantial 1957 paper has been at the root of many later works, all the more since Charles also 'internalizes" the above notions in the category of complete sup-lattices by defining what he calls local categories, their actions and the corresponding local species of structures. Moreover he introduces two main constructions: 1. given a species of structures on C and a category C' containing C, he extends the action of C to C' by the "enlargement process"; and, in the local case, he gives a "completion process" of a local species which, in modern terms, corresponds to the formation of the associated sheaf.
This paper was followed by his 1959 paper « Catégories topologiques, catégories différentiables »  where he introduces internal categories, and their actions, in the category Top of topological spaces and Diff or differentiable manifolds (well before the notion of an internal category had been considered). There he characterizes the locally trivial groupoids and proves that their theory is equivalent to the theory of principal fibred spaces, while their actions give the associated locally trivial fibred spaces. In the brief abstract of the lecture he gave at the Amiens Conference in 1975 7, Part I), he claims that differential geometry reduces to the study of the differentiable sub-categories of the category of jets, and of their actions, and this subsumes his approach to his foundation of differential geometry,
In the first times of our relation, Charles was correcting the proofs of his 1957 paper and writing the 1959 paper (which I typed for him). These papers were my initiation to category theory and, for a long time, remained all what I knew of this theory. I was then working in functional analysis (under the direction of Choquet) and was beginning to write my thesis (from 1959 to 1961)  where I propose a notion of distructure to unify different notions of "generalized functions" and, in particular, to extend the (vector-valued) Schwartz distributions to infinite dimensional vector spaces. Thinking of the local characterization of distributions as formal derivatives of continuous functions, which are then glued together, I was naturally led to compare this with the enlargement process of a category action, and then its completion.
However Charles' definitions were not well adapted for this. Indeed, in the enlargement process, there is initially an action of a category C on a set S which is extended to a larger category; but the monoid A of differential operators acts only partially on continuous functions, and it is this partial action which has to be extended; moreover the fibers are not sets but locally convex spaces whose structure has to be preserved. And it is the same for the completion process in which not only the locally convex structure of the fibers has to be preserved, but also the action of A on these fibers. So I had to modify and extend Charles' constructions to the case of a 'partial action' (here I'll use the term of 'semi-action') of a category, which is at the root of the theory of distructures given in the second part of my thesis.
The concept of a partial action is also used in my later works, first in my 1963-66 study of control and optimization problems  in the frame of control systems which use partial topological or differentiable actions; then in the hierarchical evolutive systems [8-9] which I introduced (with J.-P. Vanbremeersch) to model living systems which correspond to enriched partial actions, the fibers being hierarchical categories (a hierarchical category being a category in which the objects are partitioned into levels, with an object of a level being the colimit of at least one diagram with values in the strictly lower levels).
However In my former papers, the notion of an enriched or internalized partial action is applied in various particular cases, not in a general setting. Later on, to unify these various examples, Charles has introduced the notions of a system of structures and of a dominated system of structures , and in a Note of 1966  he gives existence theorems for its extension into a dominated species of structures (in other words, extension of the partial action into a global one), but under strong conditions on the category in which the 'domination' rests.
Here I propose a general setting to study and extend enriched partial actions of a category, and define distructures which are generated by presheaves of such partial actions. We know that an action of a category C is equivalent to a presheaf on Cop; likewise a semi-action (which is a partial action with a transitivity action stronger than in the Ehresmann's systems of structures) can be defined by what I call a semi-sheaf. The Associated Presheaf Theorem asserts that a (K, M)-semi-sheaf, where K is a category with colimits and M an adequate class of monomorphisms, can be extended into a presheaf with values in K, and this presheaf is explicitly constructed. A similar theorem and construction are valid for internal semi-actions and will be given in a later paper. Then a (K, M)-generator of distructures on H is defined as a presheaf on H with values on the category of (K, M)-semi-sheaves; it generates a K-distructures presheaf and, if H is equipped with a Grothendieck topology, a K-distructures sheaf. As an application, given two locally convex topological spaces E and E', I construct the sheaf of E'-valued distributions on E; if E is finite dimensional these distributions correspond to Schwartz E'-valued distributions.
1. Semi-actions of categories and semi-sheaves of sets
To simplify, I'll consider only semi-actions of categories and semi-sheaves on a category, though everything would extend if the categories are replaced by semi-categories in the sense of Schröder & Herrlich . (I recall that a semi-category satisfies the same axioms as a category, except that the composite of two consecutive arrows is not always defined, so that the associativity becomes: if the composites yx and zy are defined, then (zy)x is defined if and only if z(yx) is defined, and then both are equal.)
a) Semi-sheaves of sets
We know that the concept of an action of a category is in 1-1 correspondence with that of a presheaf. There will be an analog relation between semi-actions and semi-sheaves. Depending on what we want to do, one or the other notion will be more practical: semi-sheaves are easily 'enriched', while semi-actions are easily 'internalized'. To make the definition more concrete, we'll begin with the notion of a semi-sheaf of sets.
Definition. A semi-sheaf (of sets) on the category C is defined by a map F from C to Set satisfying the following conditions:
<![if !supportLists]>· <![endif]>For each object e of C, F(e) is the identity of a set Fe, called the fiber of F on e.
<![if !supportLists]>· <![endif]>For x: e' → e, F(x) is a map from a subset Fx of Fe to Fe'. The image F(x)(s) of an element s of Fx will be denoted by sx and called the composite of (x, s).
<![if !supportLists]>· <![endif]>(Transitivity) If the composites xx' in C and sx are defined, then the composite (sx)x' is defined if and only if s(xx') is defined, and then we have (sx)x' = s(xx').
Let S be the disjoint union of the fibers Fe and p0 the map from S to C sending an element of Fe to e. We denote by C*S the set of pairs (x, s) such that s is in Fx and by k' the map from C*S to S associating to (x, s) its composite sx. Then we say that (C, S, p0, k') defines a (right) semi-action of C on S. It is an action of C on S if the fibers are non void and if Fx = Fe for each x: e → e'.
Remark. This can be compared to the notion of a system of structures (C. Ehresmann ) on the opposite category Cop. However the transitivity axiom we impose is more restrictive and more symmetrical since, in a system of structures, the composite s(xx') is defined as soon as (sx)x' is defined and nothing is said in the other way (cf. Section 3)..
Example. The definition extends to the case where C is a semi-category. In this case, we may define a semi-sheaf F on C whose fiber on e is the set of arrows whose domain is e and Fx, for an x: e' → e in C is the set of z such that zx is defined in C, the composite being zx. The transitivity is then equivalent to the associativity axiom of a semi-category.
The next proposition will be useful to define the transitivity in the 'enriched' case where Set is replaced by another category K (cf. next section).
Proposition. The transitivity means that, if xx' is defined in C, the inverse image of Fx by F(x) is equal to the intersection Ixx' of Fx and Fxx'.
Proof. If s is in Ixx' the composites sx et s(xx') are defined, and the transitivity implies than (sx)x' must be defined, so that sx = F(x)(s) must be in Fx’ hence s is in the inverse image of Fx by F(x). Conversely, if (sx)x' is defined, s(xx') must be defined, so that s is in the intersection Ixx' of Fx and Fxx'.
b) Semi-fibration associated to a semi-sheaf
To a presheaf on C, or rather to the corresponding action of C on the set S of its fibers, is naturally associated a discrete fibration over C. This extends to the case of a semi-sheaf F and to the semi-action (C, S, p0, k') it defines. Indeed, we obtain a category by equipping the set C*S of the composition:
(x, s)(x', s') = (xx', s) if and only if xx' is defined in C, s’ = sx and s' is in Fx'.
The transitivity ensures that this law defines a category whose objects are the pairs (e, s) such that s is in Fe. The map sending (x, s) to x defines a functor p from this category to C, which satisfies the condition:
(SF) Each morphism is cartesian and there is at most one morphism with a given codomain and image by p.
Let us recall that h is cartesian if, for any morphism h" having the same codomain and such that p(h') = p(h)x', there exists one and only one h' such that h" = hh' and p(h') = x'. (In Charles' terminology , a cartesian morphism is called a p-injection, and SF implies the p is 'well faithful'.)
Definition. A functor satisfying the condition (SF) is called a discrete semi-fibration over C. The functor p: C*S → C is called the semi-fibration associated to F (or to the semi-action defined by F).
Proposition. There is a 1-1 correspondence between the semi-sheaves on C and the discrete semi-fibrations over C.
Proof. If p: H → C is a semi-fibration, it is associated to the semi-sheaf constructed as follows: the fiber Fe is the inverse image of e by p; for x: e'→ e, Fx is the set of objects s of Fe such that there exists a (necessarily unique) h with s as its codomain and x as its image by p; and F(x) sends s to the domain sx of h. For the transitivity, if sx and xx' are defined, then s(xx') is defined iff there exists h" with codomain s and image xx'; as h is cartesian it means that there exists h' with codomain sx and image x', hence that (sx)x' is defined.
c) The category of semi-sheaves
The semi-sheaves are the objects of the category SS whose morphisms are defined as follows: if F is a semi-sheaf on C and F* a semi-sheaf on C*, a morphism from F to F* is a pair (λ, µ) of a functor λ from C to C* and of a map µ from C to Set sending x to the map μ(x) from Fx to F*λ(x) such that:
µ(x)(s)λ(x) = µ(e')(sx) for x: e' → e and s in Fx.
In terms of the associated semi-fibrations p and p*, it corresponds to a pair of functors (λ: C → C', Λ: C*S → C**S*) such that p*Λ = λp.
The category SS admits the category PS of presheaves as a full sub-category. In the next section, we'll show that PS is a reflective sub-category of SS as a corollary of the same assertion for 'enriched' semi-sheaves and we'll give an explicit construction of the presheaf associated to a semi-sheaf. For the semi-sheaves of sets considered in this section, this result can also be deduced from the theorem of Street and Walters  which decomposes any functor p: H → C as the composite p'° j of a fibration p' on C and a final functor j. If p is the semi-fibration associated to a semi-sheaf F, then p' is the fibration corresponding to the presheaf F' associated to F. Let us recall that the union S' of the fibers of p' is constructed by these authors as the colimit of the presheaf A on H defined as follows:
A(s) = HomC(p(s), -) and A(h), for h: s → s', is the map y’ |→ y’p(h) from A(s’) to A(s).
It follows that the fiber F'e on e is the class of the connected components of the comma category p↓e.
2. (K, M)-semi-sheaves and their associated K-presheaves
In the applications to analysis or modeling that I have developed the fibers of the semi-sheaves are enriched (or 'dominated' in Charles' terminology) by supplementary structures: locally convex spaces, categories, or even (in the case of distructures) by enriched semi-sheaves. The concept of (K, M)-semi-sheaf will encompass these different cases. In the Set case studied above we had the choice between 3 equivalent definitions: semi-sheaves, semi-actions or semi-fibrations, However to enrich the fibers, we'll have to take the semi-sheaf approach.
If we try to transpose the definition of a Set-semi-sheaf F on C to a category K, by what will we replace the inclusion of Fx in the fiber Fe for a morphism x: e' → e of C? If K is a concrete category it seems natural to ask that F(x) be a morphism from a sub-structure Fx of Fe to Fe' (for instance a functor from a sub-category if K is Cat). However this will not be adapted for the application to distributions where the fibers will be a space of continuous maps equipped with the compact-open topology and the semi-action is that of the differential operators which are not continuous for the topology it induces on the differentiable functions. On the other hand the definition would be too lax if we took for Fx any sub-object of Fe. Thus we will select a particular class of sub-objects. Moreover to transpose the transitivity axiom, we need pullbacks (to define the analog of the intersection Ixx' of Fx and Fxx'). Thus we are led to the following setting.
a) The category of (K, M)-semi-sheaves
Let K be a category (generally it will admit pullbacks), and M an ss-admissible class of monomorphisms of K, meaning that it satisfies the following conditions:
• M contains the identities and there is at most one element of M between two objects.
• M is stable by pullbacks: if a pair (m, f) of an element m of M and a morphism f admits a pullback I, its projection m' opposite to m is in M:
Examples. If there is a faithful functor p: K → Set, then M could be the class of the cartesian morphisms which define a p-sub-structure (i.e., whose image by p is an inclusion), for instance K may be the category Cat of categories and M the class of inclusions of a sub-category. To define distributions [2,3], I took for K the category Lcs of locally convex topological vector spaces and for M the class of inclusion of a vector sub-space, but with a finer locally convex topology. In my study of Control Systems , K was the category Top of topological spaces, and M the class of inclusions of an open sub-space. These examples are particular cases of an ordered category K (i.e., internal to an adequate category of posets) which is completely regular, with M being the class of morphisms ('pseudoproducts') s's: s → s' such that s < s's < s'; this unifying setting is used in Charles' 1966 Note  to define dominated systems of structures (cf. Section 3).
Definition. A (K, M)-semi-sheaf on C is a map F from C to K satisfying the following conditions:
<![if !supportLists]>· <![endif]>F(e) for an object e of C is the identity of an object Fe of K, called the fiber of F on e.
<![if !supportLists]>· <![endif]>For each x: e' → e in C, the morphism F(x): Fx → Fe', called composition by x has for its domain Fx an object such that there exists a (necessarily unique) mx : Fx → Fe in M.
<![if !supportLists]>· <![endif]>(Transitivity) If xx' is defined in C, there exists a pullback Ixx' of (F(x), mx’) which is also the intersection of (mx, mxx') (hence their pullback); moreover we have :
F(x’)pxx' = F(xx')p'xx'
where pxx' and p'xx' are the projections from Ixx' respectively to Fx and Fxx'. (Since M is stable by pullbacks and has only one element between two objects, the other projection m' is in M and is the same for the two pullbacks.)
<![if !vml]><![endif]>A (K, M)-semi-sheaf F such that Fx = Fe for each x with codomain e is just a functor from Cop to K, hence a K-presheaf, and M has no more rôle. Indeed, Ixx' is equal to Fe since we have Fxx' = Fe = Fx so that pxx = F(x), and the transitivity condition becomes: F(x’)F(x) = F(xx').
The (K, M)-semi-sheaves are the objects of the category SS(K, M) whose morphisms are defined as follows. If F is (K, M)-semi-sheaf on C and F* a (K, M)-semi-sheaf on C*, we define a morphism (λ, µ) : F → F' by:
<![if !supportLists]>· <![endif]>A map µ from C to K which sends x: e' → e to a morphism µ(x): Fx → F*λ(x) so that:
m*λ(x) µ(x) = µ(e’) mx and F*(λ(x))µ(x) = µ(e’)F(x)
There is a functor Base: SS(K, M) → Cat mapping (λ, μ) : F → F* on λ: C → C*.
b) The Associated Presheaf Theorem
The category PS(K) of K-presheaves is identified to a full sub-category of the category SS(K, M) of (K, M)-semi-sheaves. We are going to prove that it is a reflective sub-category. It means that a (K, M)-semi-sheaf F on C can be 'universally' extended into a K-presheaf F' on C; equivalently, it will extend a semi-action of C into an action of C. This construction was essential in my thesis to extend differential operators to continuous functions.
Theorem (K-presheaf associated to a (K, M)-semi-sheaf). If K admits pullbacks and colimits, the category SS(K, M) admits the category PS(K) as a reflective sub-category, and the reflection F' of a (K, M)-semi-sheaf F is explicitly constructed.
Proof. Let F be a (K, M)-semi-sheaf on C.
• We first define the fiber F'ê on ê of the associated K-presheaf F'. It will be the colimit of a functor Gê: Sub(ê↓C) → K where Sub(ê↓C) is the subdivision category of the category ê↓C of objects under ê. The category ê↓C has for objects the morphisms y with domain ê and for morphisms the commutative triangles t = y,x,y'; its subdivision category has for objects all the elements of ê↓C, and it has one morphism it: t → y and one morphism ct: t → y'; roughly each morphism t of ê↓C is replaced by a span (it, ct). The functor Gê is defined by:
Gê(it) = mx: Fx → Fe and Gê(ct) = F(x).
• Let z: ê' → ê be a morphism in C. It defines a functor z↓C: ê↓C → ê'↓C sending t = y,x,y' onto tz = yz,x,y'z, hence a functor γz: Sub(ê↓C) → Sub(ê'↓C) which sends it and ct on itz and ctz. Since Gê'°γz = Gê there exists a morphism F'(z) from the colimit F'ê of Gê to the colimit F'ê'' of Gê'' such that F'(z)jy = jyz for each y: ê → e, where jy is the injection of Fe = Gê(y) into the colimit F'ê of Gê and jyz the injection of F(e) = Gê'(yz) into the colimit F'ê' of Gê'. We have thus constructed the K-presheaf F'.
• We define a morphism (IdC, ρ): F → F' by taking for ρ(ê) the injection jê of F(ê) into the colimit of Gê and for ρ(z) the composite ρ(ê)mz. Indeed, for z: ê' → ê we have
F'(z)ρ(z) = F'(z)ρ(ê)mz = F'(z)jêmz = jzmz = jê'F(z) = ρ(ê')F(z).
• It remains to prove that (IdC, ρ) defines F' as a reflection of F into the sub-category PS(K) of K-presheaves. Indeed, let (λ, μ): F → F* be a morphism from F to a K-presheaf F* on a category C*. We must find a morphism (λ', μ'): F' → F* whose composite with (IdC, ρ) gives (λ, μ). For λ' we take λ itself. For an object ê of C, we have a natural transformation n from Gê to F*ê such that:
n(y) = F*(λ(y))μ(e) and n(t) = F*(λ(y))μ(e)mx for y = xy' and t = y,x,y'
It is a natural transformation since
n(y)mx = n(t) = F*(λ(y))μ(e)mx = F*(λ(y'))F*(λ(x))μ(x) = F*(λ(y'))μ(e')F(x) = n(y')F(x)
This natural transformation glues into the morphism μ'(ê) from the colimit F'ê of Gê to F*λ(ê) such that n(y) = μ'(ê)jy. In particular, for y = ê, we have
μ(ê) = n(ê) = μ'(ê)jê = μ'(ê)ρ(ê).
It is a natural transformation since
n(y)mx = n(t) = F*(λ(y))μ(e)mx = F*(λ(y'))F*(λ(x))μ(x) = F*(λ(y'))μ(e')F(x) = n(y')F(x)
Let us prove that μ' defines a natural transformation from F' to F*. For z: ê' → ê and each y: ê → e, we have
μ'(ê')F'(z)jy = μ'(ê')jyz (by definition of F'(z))
= F*(λ(yz))μ(e) (by definition of μ'(ê') )
= F*(λ(z))F*(λ(y))μ(e) = F*(λ(z))μ'(ê)jy (by definition of μ'(ê) ).
whence μ'(ê')F'(z) = F*(λ(z))μ'(ê) since the jy are the injections toward the colimit F'ê. Thus (λ, μ') defines a morphism from F' to F* such that (λ, μ) = (λ, μ')(IdC, ρ). This morphism is unique because, if (λ, μ") was another one, for each ê the equality μ'(ê)ρ(ê) = μ"(ê)ρ(ê) implies:
μ'(ê)jy = μ'(ê)F'(y)ρ(e) = F*(λ(y))μ'(e)ρ(e) = F*(λ(y))μ"(e)ρ(e) = μ"(ê)F'(y)ρ(e) = μ"(ê)jy.
for each y, and thus μ'(ê) = μ"(ê). This completes the proof that (IdC, ρ): F → F' is a reflector into the sub-category of K-presheaves.
Corollary. The presheaf F' associated to a semi-sheaf of sets F has for fiber on ê the quotient of the disjoint union Σê of the family (Fe)y:ê→e by the equivalence relation Rê generated by the relations: (y', sx) ~ (xy', s) if s is in Fx.
Proof. A semi-sheaf of sets F is a (Set, M)-semi-sheaf where M is the class of inclusions of a subset into a set. By the above construction, the fiber F'ê of the associated presheaf of sets is the colimit of the functor Gê. In Set this colimit can be constructed as follows: we first take the set Σê consisting of all the pairs (y, s) where y: ê → e and s is in Fe. For each triangle t = y,x,y' we must have jymx = jy'F(x) so that, for s in Fx we must identify (y, s) and (y', sx), whence the relation Rê since y = xy'.
3. Comparison with dominated systems of structures
Since a semi-sheaf is a particular system of structures, the Associated presheaf Theorem of Section 2 can be compared to the "Dominated Expansion Theorem" for systems of structures dominated in a concrete ordered category (K, <) given by Charles Ehresmann . For that, we must transpose his notion of a dominated system of structures in our setting.
a) (K, M)-dominated systems of structures
Given a category admitting pullbacks and an ss-admissible class M of monomorphisms of K, we define a (K, M)-dominated system of structures in the same way as a (K, M)-semi-sheaf, except that the transitivity is replaced by:
If xx' is defined in C, there exists a pullback Ixx' of (F(x), mx’) and a p'xx': Ixx' → Fxx 'in M such that F(x’)pxx' = F(xx')p'xx' where pxx' is the projection from Ixx' to Fx and Fxx'.
The difference with a semi-sheaf is that here we do not impose that Ixx' be the intersection of Fx and Fxx'. As this stronger condition is not used in the proof of the Associated Presheaf Theorem, this theorem extends to (K, M)-dominated systems of structures: Whence
Theorem. If K admits colimits, PS(K) is also a reflective sub-category of the category Dss(K, M) of (K, M)-dominated systems of structures (the morphisms being defined as for semi-sheaves), and the associated K-presheaf is constructed as for semi-sheaves.
This theorem is much stronger than the "Dominated Expansion Theorem" of Charles which supposes that K is a concrete category whose forgetful functor to Set has strong enough properties. In a comment that I have written to supplement Charles' Note in "Charles Ehresmann : Oeuvres complètes et commentées"  (Comment 468-1, Part III), I prove the existence of the associated K-presheaf (but not its construction) under the above conditions. For this, I give another characterization of a (K, M)-dominated system of structures and of its associated K-presheaf, when the ss-admissible class M is moreover stable by composition (so that it is a sub-category of K), and I deduce the existence of the associated presheaf from an existence theorem on lax colimits. As it allows another view on these problems, I am going to recall this characterization.
To (K, M) I associate the following 2-sub-category KM of the bicategory of spans of K whose 1-morphisms are the spans (f, m) where m is in M, their composition being done via pullbacks. As we have imposed the stability of M by pullbacks and that there exists at most one element of M between 2 objects, the identities are the sole invertible elements in M (otherwise the inverse would also be in M), so that we obtain a usual category for the composition:
(f', m')(f, m) = (f'f", m"m) if m' and f have the same codomain,
where f", m" are the projections of the pullback of (m', f). A 2-cell m*: (f, m) → (f1, m1) of KM is defined by an m* in M such that m = m1m* and f = f1m*.
The 2-category KM is the 2-category of objects for a 3-fold category KM* with K as the category of objects for the 3rd composition; 3-blocks are degenerated squares of spans, and their composition is deduced from the 'ver1tical' composition of squares.
Proposition. A (K, M)-dominated system of structures can be defined as a unitary lax 2-functor F from the trivial 2-category Cop to KM, hence also as a 2-functor lF from an appropriate 2-category C2 to KM. And the associated K-presheaf is a KM*-wise colimit of the 2-functor lF.
The category C2 has been constructed by several authors (Gray , Vaugelade ). The notion of a KM*-wise colimit of a 2-functor is introduced in a 1974 paper on multiple categories I have written with Charles. In this paper we give an existence theorem for such lax colimits. In the comment mentioned above, I use this theorem to prove that, if K is cocomplete, then lF admits a KM*-wise colimit, so that F has an associated K-presheaf.
b) Completely regular ordered categories
In fact in Charles' Note  as well as in my comment on it , the setting seems somewhat different since the 'domination' depends on an appropriate order < on K rather than on an ss-admissible sub-category M of K. However, the difference is not essential as I am going to prove.
There it is supposed that (K, <) is a completely regular ordered category. It means that it is a category internal to the category Pos of posets which satisfies the following conditions:
(i) If f and f' have the same domain, the same codomain and are both less than f", then f = f'.
(ii) If k < gf, there exists g’ < g and f’ < f such that k = g’f’.
(iii) If s' is an object less than the codomain β(f) of f, the set of f' such that f' < f and β(f') = β(f) has a maximal element, denoted by s'f and called the pseudoproduct of (s', f).
(iv) For two objects s' < s, there exists an ss': s' → s, called the pseudoproduct of (s, s'), such that s' < ss' < s
In such a category it is said that g and f admit a pseudoproduct gf if the class of composites g'f' where g' < g and f' < f admits gf as a largest element. This pseudoproduct extends the composition of the category, but it does not always exist. Charles has proved  that, if the pseudoproducts (hg)f and h(gf) are both defined, they are equal.
Proposition. Let (K, <) be a completely regular ordered category. Then the class M< of the pseudoproducts s's of objects such that s' < s is ss-admissible. Conversely, if K is a category admitting pullbacks and M is an ss-admissible class of monomorphisms of K closed by composition, then there exists an order < on K for which M = M<.
Proof. 1. M< contains the identities and ss': s' → s is the unique element of M< between s and s'. To prove that it is stable by pullbacks, let us show that, for each f: s* → s there exists a pullback of (ss', f) whose projections are s'f and s*s'*: We have (ss')(s'f) = f(s*s'*) both composites having the same domain and codomain and being less than f. If (ss')g = fg', the pseudoproduct s'*g' factors both g and g'. Indeed, using the associativity of the pseudoproduct, we get:
(s'f)(s'*g') = ((s'f)s'*)g' = (s'f)g' = s'(fg') = s'((ss')g) = (s'ss')g = g;
it follows that s'*g' has the same domain as g, whence (s*s'*)(s'*g') = g', both having the same doman and codomain and being less than g'.
2. Let M be an ss-admissible sub-category of K. We define an order < on K as follows:
f' < f if there exist m and m* in M such that fm* = mf'.
The axioms of an ordered category are easily proved, as well as condition (i) above, using the fact that there exists at most one monomorphism in M between two objects. And for each m: s' → s in M, we have s' < m < s, hence m is the pseudoproduct ss'
To prove condition (ii) we suppose that k < gf, so that there exist m' and m* in M with m'k = gfm. We form the pullback of (g, m'), its projections being m in M and g'; since m'g' = gm, we have g' < g. Since m'k = gfm* by definition of a pullback, there exists an f' such that g'f' = k and mf' = fm*, hence f' < f.
It remains to prove that condition (iii) is satisfied. If f: s* → s and s' < s we form the pullback of (f, m), where m = ss'. Its projections are m* in M and f'. As fm* = mf', we see that f' is an element less than f and with codomain s'. It is the pseudoproduct s'f; indeed, for any h < f with codomain s' we have mh = fm' for some m' in M, so that there exists an n with m*n = m' and f'n = h; the first equality implies that n is in M, and the second that h < f'.
In my thesis, completely regular inductive categories were also used as a setting to construct the associated sheaf. The following proposition (which I gave in Comment 143.1, Part II of ) explains why it is possible.
A completely regular inductive category is a category internal to the category of complete join-lattices (the join of a family (si)i is noted Visi) which is also completely regular as an ordered category and moreover satisfies:
(v) If s = Visi for f: s' → s, then f = Vsif.
Proposition. If K is a completely regular inductive category, then there is a Grothendieck topology on K generated by the pretopology admitting as covering families of s the families (ssi)i of the pseudoproducts such that s = Vsi.
The axioms of a pretopology are easily verified, using the fact that sif is a projection of the pullback of (f, ssi) and the axiom (v).
4. Semi-sheaves with values in concrete categories
Here we consider the case where K is a concrete category, thus a category equipped with a faithful functor q to the category Set. We suppose that q preserves pullbacks and that the ss-admissible class M of monomorphisms of K is sent by q into the class Ins of inclusions of a subset into a set. Then a (K, M)-semi-sheaf F has an underlying semi-sheaf of sets qF whose fibers are the images by q of those of F and qF(x) is the image by q of F(x). The associated presheaf theorem can be applied both to F, giving the K-presheaf F', and to qF. However the underlying presheaf qF' to F' is generally not the presheaf (qF)' associated to qF, except if q also preserves colimits (for example if K is the category Top); there is only a non-invertible morphism from (qF)' to qF'.
In this section we consider two such situations which are extensively used in my former work: semi-sheaves of categories which I initially called category of categories  and semi-sheaves of locally convex spaces which are at the root of in my theory of distructures.
a) Locally convex semi-sheaves
To define locally convex semi-sheaves, we take for the category K the category Lcs of locally convex topological vector spaces, and for M the class MLcs of inclusions m: E' → E of a vector sub-space E' of E, but with a locally convex topology finer than the topology induced by E. The reason for this choice of M comes from the initial example which I met to define distributions, where the fibers are spaces of continuous functions with the compact-open topology, on which differential operators act on sub-spaces of differentiable functions, the operation being continuous only if these sub-spaces are equipped with a finer topology, namely the compact-open topology for the functions and their derivatives.
Definition. A locally convex semi-sheaf is defined as a (Lcs, MLcs)-semi-sheaf.
Thus a locally convex semi-sheaf F on C has for fiber Fe a locally convex space, and F(x), for an x: e' → e, is a continuous linear map to Fe' from a vector sub-space Fx of Fe equipped with a locally convex topology finer than that induced by Fe. If we note F(x)(s) = sx for s in Fx the transitivity becomes:
If sx is so defined and if xx' is defined in C, then s(xx') is defined if and only if (sx)x' is defined, in which case both are equal.
The Associated Presheaf Theorem associates to F a presheaf F' of locally convex spaces on C. As we have proved in Section 2 (of which we take back the notations), its fiber F'ê on the object ê of C is the colimit of the functor Gê from the subdivision category of ê↓C to Lcs which, for each commutative triangle t = y,x,y' sends it on the inclusion mx:: Fx → Fe and ct on F(x). To construct this colimit in Lcs, we first take the direct locally convex sum of the family (Fe)y:ê→e; its elements are linear combinations of pairs (y, s) where y: ê → e and s is in Fe. Then F'ê is the locally convex space quotient of this sum by the vector sub-space generated by the elements (y’x, s) – (y’, sx), where y' has ê for its domain and y'x is defined in C. The topology of F'ê is the final locally convex topology for the sink (jy), where jy: Fe → F'ê maps s on the equivalence class [y, s] of (y, s).
Thus the underlying (Set-)presheaf to F' is 'larger' than the presheaf P associated to the semi-sheaf underlying F; more specifically, the fibers of F' are generated (in the sense of vector spaces) by those of P.
b) Lcs-sheaf associated to a locally convex semi-sheaf
In the application to distributions on a locally convex space E, we want not only to construct a presheaf (which will correspond to finite order distributions), but also to associate a sheaf to this presheaf, a distribution gluing together finite order distributions. To define a sheaf, more structure must be given on the category C; in the classical case of distributions C is the category of open sets of E. In my thesis  I supposed that C was a completely regular inductive category, and the associated sheaf was defined in terms of completion of an inductive species of structures (in the terminology of C. Ehresmann ). More generally, C can be equipped with a Grothendieck topology J (why it is more general is explained at the end of Section 3). Then an Lcs-sheaf on C for J is an Lcs-presheaf F* on C satisfying the following condition, for each covering sieve (ci)i of e:
For each pair (i, j) of indices, let eij be the pullback of (ci, cj) with its projections prji and prij. There exist morphisms pr and pr', from the product of the fibers of F* on the different ei to the product of the fibers on the different eij, which glue together respectively the family (F*(prij))i and (F*(prij))i. Then F*e is the kernel of (pr, pr').
Theorem (Lcs-sheaf associated to a locally convex semi-sheaf). If J is a Grothendieck topology on C, the category of locally convex semi-sheaves on C admits the category Sh(C,J)(Lcs) of Lcs-sheaves as a reflective sub-category. The reflection F* of a locally convex semi-sheaf F is the Lcs-sheaf associated to the Lcs-presheaf F' associated to F.
Proof. To describe the Lcs-sheaf F*, we use a method similar to the construction of the complete enlargement of an inductive species of structure given by C. Ehresmann (to be compared to the construction given by Farina & Meloni  through "locally compatible families with covering support").
1. Let e an object of H. The set E underlying F*e will be the set of complete compatible families on e, where a complete compatible family on u is defined as a family σ = (σx)x satisfying the following conditions:
• its index set, denoted Iσ, is a covering sieve of e for J; for each morphism x: ex → e in Iσ, σx is an element of the fiber of F' on ex;
• if y has ex for codomain, the 'restriction' σxy = F'(y)( σx) of σx belongs to σ;
• two elements σx and σx' of σ are compatible, meaning that σxp = σx'p' where p and p' are the projections of the pullback of (x, x');
• σx belongs to σ if there exists a covering sieve W of ex such that its restrictions σxw to the elements w of W belong to σ.
2. For each x with codomain e, let Ax be the subset of E formed by the σ for which Iσ contains x, and fx the map from Ax to the fiber F'x of F' on ex sending σ on σx. If p has ex for its codomain, Ax is included in Axp since σ contains with σx all its restrictions.
We equip Ax with the locally convex structure inverse image by fx of the locally convex space F'x. For each p the above inclusion becomes a linear continuous map A(p) from Ax to Axp which satisfies the equality: F'(p)fx = fxpA(p). This defines a functor A from the category e↓Cop to Lcs. The fiber F*e is the colimit of this functor.
More explicitly: its vector structure is such that kσ = (kσx)x and σ + σ' is the complete compatible family generated by the family (σx' + σ'x')x' indexed by the intersection of Iσ and Iσ'. The topology is the finest topology for which the inclusions of Ax in F*e become continuous. A basis of neighborhoods of 0 is of the form N((Wy)yεY) where Y is a covering family of e and Wy is a convex neighborhood of 0 in the fiber of F' on ey for each y: ey → e in Y, and N((Wy)yεY) is the set of σ for which Iσ contains y and σy is in Wy. for each y in Y.
3. For z: ê → e, the continuous linear map F*(z): F*e → F*ê sends σ on the complete compatible family generated by the restrictions σwz of the elements of σ.
The natural transformation η': F' → F* defining the reflection into Sh(C,J)(Lcs) is defined as follows: η'(e): F'e → F*e is the continuous linear map sending s on the family (sx)x of all its restrictions.
Corollaire. The above construction also gives the Top-sheaf for J associated to a Top-presheaf, or to a (Top, Ins)-semi-sheaf.
Proposition. If the fibers Fe of F are complete locally convex spaces, then F' is a presheaf of complete locally convex spaces and F* a sheaf of complete locally convex spaces.
Indeed, the locally convex sum of a family of complete locally convex spaces is complete, the quotient of a complete locally convex space is complete as well as the inverse image and a colimit.
c) Semi-sheaves of categories
A semi-sheaf of categories is defined as a (Cat, MCat)-semi-sheaf F on C, where MCat is the class of inclusions of a sub-category in a category. Thus F associates to each object e of C a category Fe and to an x: e' → e a functor from a sub-category Fx of Fe to Fe' subject to the transitivity axiom: if xx' is defined in C and b is in Fx then b is in the intersection of Fx and Fxx' if and only if bx = F(x)(b) is in Fx' and then (bx)x' = b(xx').
The construction given in Section 2 and the form of colimits in Cat imply that the fiber F'ê of the associated presheaf of categories F' is the quasi-quotient category of the category L coproduct of the family (Fe)y:ê→e by the equivalence relation R generated by the relations:
(y, bx) ~ (yx, b) for each y: ê → e and b in Fe.
(For the definition and constructions of a quasi-quotient category, cf. C. Ehresmann (1965) who shows that the underlying set of the quasi-quotient category of L by R is generally larger than the quotient set L/R .).
If J is a Grothendieck topology on C, the construction given above of the associated sheaf via complete compatible families is readily transcribed to construct the sheaf of categories F*associated to F', hence also to the semi-sheaf F. In particular the objects of the category F*e are the complete compatible families e whose terms σx are objects in the corresponding fibers of F'.
The Evolutive Systems which we have introduced with Jean-Paul Vanbremeersch  to model the evolution of a living system are semi-sheaves of categories F on a category Time which is a category defining the order > on an interval or a finite part T of the space R+ of positive real numbers, so that its morphisms are pairs (t, t') with t < t'. The idea is that, for each instant t the category Ft models the configuration of the system around t: its objects are the state at t of the components of the system which then exist, and the morphisms between them represent the interactions around t. The functor F(t, t'), called transition, models the change of the system from t to t' and, to take account of the possible loss of some components and/or relations between t and t', it is defined only on a sub-category Ftt' of Ft. If at is an object of Ft, either it models a component that still exists at t', and then F(t, t')(at) is its new state at t'; or it does no more exist at t' and then it is not in Ftt'; and the same for the interactions represented by the morphisms. If S is the set consisting of all the objects of the different fibers Ft and p: S → T its projection map, a component of the system can be defined as a maximal partial section of p consisting of its various successive states during its 'life'.
Living systems have a whole hierarchy of components of various complexity levels. To model this hierarchy we have enriched the evolutive systems and defined the Hierarchical Evolutive Systems. The idea is that a complex component a of level n+1 can be modeled as the colimit of a diagram modeling its lower level internal structure, that is its components of levels less than, or equal to, n and their interactions concerning a. Thus we define a hierarchical category as a category whose objects are partitioned into a finite number of 'levels', say 0, 1, ..., m, so that an object of level n+1 be the colimit of at least one diagram with values in the full sub-category whose objects are of levels less than, or equal to, n. A hierarchical functor is a functor between hierarchical categories which respects the levels (but it may or may not preserve colimits). Let HCat be the category of hierarchical categories.
A Hierarchical Evolutive System is a (HCat, MCat)-semi-sheaf on a category Time. Otherwise, it is an evolutive system whose fibers are hierarchical categories, the transition functors respecting the levels .
The theory of Memory Evolutive Systems developed with Jean-Paul Vanbremeersch (cf. our book , summarized on our internet site ) is based on such hierarchical evolutive systems.
5. (K, M)-distructures
The definition of distributions will be made 'locally' on an open set U of a locally convex space E, by extending the differential operators to continuous functions defined on U (using the Associated Presheaf Theorem); and then we'll have to connect the distributions so defined on various U. Thus we have to consider not just one semi-sheaf on U, but a 'presheaf of semi-sheaves' on E. It is this situation which the notion of a distructure generalizes.
a) Generators of distructures
We suppose that K is a category admitting pullbacks and colimits and M an ss-admissible class of monomorphisms of K. Then the category SS(K, M) of (K, M)-semi-sheaves has a functor Base to Cat and an 'associated presheaf' functor Ps toward the category Ps(K) of K-presheaves.
Definition. An SS(K, M)-presheaf D on a category H is called a (K, M)-generator of distructures. The composite of D with Base is a presheaf of categories Γ on H called the base of D, and the composite D' of D with Psh: SS(K, M) → PS(K) is called the K-distructures presheaf generated by D.
Let D be a generator of (K, M)-distructures on H, For each object u of H, the fiber Du of D is a (K, M)-semi-sheaf on the category Γu which is the fiber on u of the presheaf of categories Γ base of D. For a morphism v: u' → u of H, D(v) is a morphism of semi-sheaves: D(v) = (Γ(v), μv): Du → Du'. The K-distructures presheaf D' generated by D is a Ps(K)-presheaf on H with the same base Γ, whose fiber on u is the K-presheaf associated to Du.
If H is reduced to a unique object, then a (K, M)-generator of distructures reduces to a (K, M)-semi-sheaf, its base being reduced to one category.
We denote by GDisH(K, M) the category of (K, M)-generators of distructures on H which is a full sub-category of PS(SS(K, M)). A morphism from D to E is a natural transformation. Let DisH(K) be its sub-category of K-distructures presheaves. Since PS(K) is reflective in SS(K, M), we have:
Proposition. The category DisH(K) is a reflective sub-category of GDisH(K, M). The fiber D'u of the reflection D' of D is the K-presheaf associated to Du.
b) Distructures as connecting two semi-sheaves
We are going to give another characterization of a (K, M)-generator of distructures with base Γ by considering the fibration Fib: F(Γ) → H associated to Γ. It will explain in which sense we can speak of 'di'structures. Let us recall that the class of objects of F(Γ) is the (disjoint) union of the classes of objects of the various Γu; a morphism from e' in Γu' to e in Γu is defined by a pair (v, x) where
v: u' → u is in H and x: e' → ev = Γ(v)(e) is in Γu';
and its composite with (v', x'): e" → e', where v': u" → u', is (vv', xv'.x'): e" → e (the dot denotes the composition in Γu"). The functor Fib maps (v, x) on v.
The category F(Γ) has 2 distinguished sub-categories:
• the category Vert of 'vertical morphisms' (those mapped by Fib on an object) which is the coproduct of the categories Γu; for y in Γu; we identify (u, y) with y if no confusion is possible;
• the sub-category Cart of cartesian morphisms which are of the form (u, e) where e is an object of Γu.
These sub-categories generate F(Γ) since we have: (v, x) = (v, ev)x for x in Γu'.
Theorem. There is an isomorphism φ from the category of (K, M)-generators of distructures with base Γ onto the category of (K, M)-semi-sheaves on F(Γ) whose restriction to Cart is a K-presheaf. It sends the presheaf of K-distructures generated by B to the K-presheaf associated to φB. If D' is a presheaf of K-distructures, then φD' is a presheaf;
Proof. 1. If D is a (K, M)-generator of distructures, we define a (K, M)-semi-sheaf φD on F(Γ) as follows: Its fiber φDe on the object e of Γu is the fiber (Du)e of the (K, M)-semi-sheaf Du on e. For a morphism (v, x): e' → e, in the semi-sheaf Du we have mx: (Du')x → (Du')ev in M and we take the pullback φDv,x of (mx, μv(e)), where D(v) = (Γ(v), μv); the projections in the corresponding pullback square PBx are denoted qv,x and mv,x in M (since M is stable by pullbacks), and we define:
φD(v, x) = Du'(x)qv,x: φDv,x → (Du')ev.
To prove that φD so defined is a (K, M)-semi-sheaf, we must prove the transitivity. For this we also consider (v', x'): e" → e', and its φD(v', x') = Du"(x')qv',x'.: φDv',x' → (Du")e". To construct the image by φD of the composite c = (vv', xv'.x') we first construct the image by Du" of the composite xv'.x'. Since Du" is a semi-sheaf, the domain of Du"(xv'.x') is the codomain of the projection p'xv'.x: I → (Du")xv'.x in the pullback square PBm (drawn in red in the diagram), and I is also the pullback of (mx', Du"(xv')), with corresponding pullback square PB1. Then we have φD(c) = Du"(xv'.x)qc where qc is the projection of the pullback of (mxv'.x, μv'(ev)μv(e)), whose corresponding pullback square is PBxv'.x.
The transitivity of φD means that the pullback I' of (mv',x', φD(v, x)) (with pullback square PB2) is also the pullback of (mv,x, mc). To show this, we consider the composite pullback square PBx''°PB2 and the adjacent pullback square PB1. Their bases form a commutative square with μv'(x)qv,x as its 4th side, so that there exists a morphism q from I' to I closing a pullback square PB3 (with I' and (Du')xv' as opposite vertices) which satisfies PB1'°PB3 = PBx''°PB2.
Similarly, considering the composite pullback square PBm°PB3 and the adjacent square PBxv'.x' connected by mv,x we complete by p' a pullback square PB4 (in red on the diagram) which defines I' as the pullback of (mv,x, mc).
For a cartesian morphism (v, e), the above construction implies φD(v, e) = μv(e) and the restriction of φD to Cart is reduced to a K-presheaf.
If D takes its values in the category of K-presheaves, so that it is already a presheaf of K-distructures, then mx for x in Γu' is an identity so that
mv,x = Id and φD(v, x) = Du'(x)μv(e).
Thus φD is a K-presheaf (and not only a (K, M)-semi-sheaf).
In this case, there is another construction of φD (which, for K = Set, I gave in Comment 490-1, p. 808, Part III ). It relies on the fact that the fibration F(Γ) is a lax colimit of Γ looked at as a 2-functor from Hop to the 2–category CAT, the canonical lax cocone with vertex F(Γ)op associating to v the 2-cell (Γ(v), (v, -v)) from the inclusion Insu: Γuop → F(Γ)op to Insu'. Now D can be seen as defining a lax cocone with basis Γ and vertex K associating to v the 2-cell from Du to Du' defined by D(v) = (Γ(v), μv). This lax cocone factors through φD.
2. Conversely, let B be a (K, M)-semi-sheaf on F(Γ) whose restriction to Cart is a K-presheaf. We define a (K, M)-generator D of distructures as follows. The (K, M)-semi-sheaf Du will be a restriction of B, namely the composite of B with the 'inclusion' Insu: Γuop → F(Γ)op.
Since B becomes a presheaf on Cart, for v: u' → u and e an object of Γu, B(v, ev) is a morphism from Be = Du(e) to Bev = Du'(ev) which we take for μv(e). To construct μv(y) for an y: ê → e in Γu, we take into account the two decompositions:
y(v, ê) = (v, y) = (v, ev)yv.
Since the restriction of B to Cart is a presheaf, the first decomposition implies (by transitivity) that there exists a p'(v,ê),y such that:
B(v, ê)B(y) = B(v, y)p'(v,ê),y and p'(v,ê),ymv,y = my.
From the second decomposition we deduce that Bv,y is the pullback of (myv,B(v, e)), the projections being mv,y and a qv,y such that B(yv)qv,y = B(v, y). Then we define μv(y) = qv,y p'(v,ê),y. The above diagram being commutative, this entirely defines a morphism (Γ(v), μv): Du → Du', whence a (K, M)-generator of distructures D, and we have φD = B, both K-presheaves having the same restrictions to the generating sub-categories Vert and Cart of F(Γ).
3. The bijection φ extends into an isomorphism from the category of (K, M)-generators of distructures with base Γ, onto the category of (K, M)-semi-sheaves on F(Γ) whose restriction to Cart are presheaves. Indeed, let γ: D → E be a natural transformation; for each object u of H, γ(u) is a morphism of (K, M)-semi-sheaves from Du to Eu and we take φγ(e) = γ(u)(e) for an object e in Γu.
To define φγ(v, x) for v: u' → u and x: e' → ev, we come back to the definition of φD(v, x) and φE(v,x) via the formation of the pullbacks PBx and PB'x. Since γ is a natural transformation, we have
μv(e)(γ(u')(ev)) = (γ(u)(e))μ'v(e) = φγ(e)μ'v(e)
where (Γ(v), μ'v): Eu → Eu'; whence a commutative square whose composite with PBx must factor through the pullback square PB'x to give a cube with other edges γ(u)(x) and the wanted φγ(v, x): φD(v, x) → φE(v, x). The commutativity of the diagram ensures that we have thus defined a morphism (Id, φγ): φD → φE. From the definition, it follows that the map sending γ to φγ defines the required isomorphism.
4. The isomorphism φ sending the reflective sub-category of presheaves of K-distructures with base Γ onto the reflective sub-category of K-presheaves on F(Γ), it preserves the reflections. Hence the presheaf of K-distructures D' associated to D has for image by φ the K-presheaf associated to φD.
This theorem shows how a (K, M)-generator of distructures D determines 'two structures' (whence the name distructure) of semi-sheaves with the same fibers:
• a 'vertical' (K, M)-semi-sheaf on the category Vert which is the 'union' of the semi-sheaves Du,
• a 'horizontal' K-presheaf on the category Cart, deduced from the morphisms between the Du.
These two structures are inter-connected since they both are restrictions of the 'global' (K, M)-semi-sheaf on F(Γ).
6. Locally convex distructures
The preceding double structure can be made more explicit in the case where K is the category Set, or more generally a concrete category..
a) Distructures valued in a concrete category
If K = Set and M is the class of inclusions of a subset in a set, then a (K, M)-generator of distructures D is simply called a generator of distructures and its associated presheaf of K-distructures D' a distructures presheaf.
The double structure can then be described as follows. Let Su be the (disjoint) union of the fibers (Du)e of the semi-sheaf Du and Σ the (disjoint) union of the sets Su for the various objects u of H. If Γ is the base of D and F(Γ) its associated fibration, the preceding theorem defines a semi-action of F(Γ) on Σ (corresponding to the presheaf φD). It has for restrictions:
• a 'vertical' semi-action of the sub-category Vert on Σ (gluing the semi-actions of the various Γu), with the fibers of D as 'horizontal' fibers; and
• a 'horizontal' action of the sub-category Cart with the same fibers.
Gluing the maps images by φD of the cartesian morphisms over a given v, we obtain a global 'horizontal' action of H on S; it corresponds to the presheaf S on H having for 'vertical' fibers the various Su and such that H(v) glues the various maps μv(e), where D(v) = (Γ(v), μv). The figure shows what we mean by 'vertical' and 'horizontal'.
For the distructures presheaf D' generated by D, the semi-action of Vert is also an action, so that the union Σ' of all its fibers is equipped with two actions: the vertical action of Vert with horizontal fibers (D'u)e, and the horizontal action of H with vertical fibers S'u uniting the horizontal fibers over u.
More generally, let K be a category with a faithful functor q to Set which preserves pullbacks, and M an ss-admissible class whose image by q consists of inclusions. If D is a (K, M)-generator of distructures, it admits an underlying generator of distructures, denoted by qD, whose fibers are the semi-sheaves qDu underlying the fibers of D (cf. Section 4). The vertical structure of qD (defined by the semi-action of Vert) underlies the vertical structure of D, defined by the (K, M)-semi-sheaf gluing the semi-sheaves Du. However the horizontal structure of qD is more precise than that of D, since the presheaf S on H is not underlying a K-presheaf, except if K admits coproducts which are preserved by q. In this special case, the K-presheaf on H has for fiber on u the coproduct of the fibers (Du)e for the various objects e of Γu.
For instance, if K = Top, the presheaf S lifts to a presheaf of topological spaces; or if K is the category Cat, it lifts to a presheaf of categories.
b) Locally convex distructures
Here we examine the case where K is the category Lcs of locally convex topological vector spaces, and M the class MLcs of inclusions m: E' → E of a vector sub-space E' of E with a locally convex topology finer than the topology induced by E.
Definition. A (Lcs, MLcs)-generator of distructures L is called a generator of locally convex distructures (abbreviated in generator of lc-distructures), and the presheaf of distructures L' it generates a locally convex distructures presheaf (abbreviated in lc-distructures presheaf).
Thus a generator of lc-distructures L on H is a presheaf of locally convex semi-sheaves. Its composite with the base functor gives a presheaf of categories Γ. The fiber Lu of L on an object u of H is a locally convex semi-sheaf on Γu and L(v) for v: u' → u is a morphism (Γ(v), μv): Lu → Lu'. From the preceding theorem, it follows that L 'extends' into the locally convex semi-sheaf φL on the fibration F(Γ) associated to Γ, thus interconnecting the two structures of L: the 'vertical' one corresponding to the locally convex semi-sheaf (on Vert) gluing the Lu's; and the 'horizontal' one consisting of an Lcs-presheaf on the cartesian morphisms of F(Γ).
The forgetful functor qLT: Lcs → Top preserves the limits and sends MLcs into the class MTop of continuous inclusions of a sub-space with a finer topology. It follows that L has an underlying (Top, MTop)-generator of distructures T with base Γ: the (Top, MTop)-semi-sheaf Tu is the composite qLTLu. Hence the fiber (Tu)e is the topology of the fiber (Lu)e of Lu, and, for y in Γu from ê to e, the continuous map Tu(y): (Tu)y → (Tu)ê maps s on Lu(x)(s).
By a construction similar to that of the presheaf S above we obtain the following presheaf of topologies Θ on H:
• Its fiber on u is the topological space Θu coproduct in Top of the topologies of the fibers of Lu;
• Θ sends v: u' → u to the continuous map Θ(v): Θu → Θu' gluing together the continuous maps µv(e) for the various objects e in Γu, so that Θ(v)(e, s) = (ev, µv(e)(s) for each s in the fiber (Lu)e.
We can apply the same construction to the lc-distructures presheaf L' generated by L; it leads to the Top-distructures presheaf T' whose fiber on u is the Top-presheaf qL'u; and to the Top-presheaf Θ' whose fiber Θ'u is the topology coproduct of the topologies on the fibers of L'u. As the functor qLT does not preserve colimits, T' is not the Top-presheaf associated to T, nor Θ' the Top-presheaf associated to Θ.
If a Grothendieck topology J is given on H, L also generates a Ps(Lcs)-sheaf L*, called the lc-distructures sheaf on H generated by L, which is the Ps(Lcs)-sheaf associated to L'. It is constructed hereafter. It has an underlying Top-sheaf Θ* and the following figure shows how the fibers increse from L to L' and to L*.
c) Construction of the lc-distructures presheaf and sheaf generated by L
Let L be a generator of lc-distructures with basis Γ. Using the description of the associated locally convex presheaf given in section 4, we can describe as follows the lc-distructures presheaf L' generated by L.
• For an object ê of the fiber Γu of Γ on u, the fiber (L'u)ê is the locally convex space constructed as follows. We take the locally convex direct sum Vê of the family of locally convex spaces (Lu)e indexed by the morphisms y: ê → e in Γu; its elements are linear combinations of pairs (y, s) where s is an element of (Lu)e. The relations
(y, σ) ~ (y’, σx) if y = y'x in Γu and σ is in (Lu)x
generate a congruence Rê on the vector space Vê. Then (L'u)ê is the locally convex space quotient of Vê by Rê. Its topology is the final locally convex topology for the sink (jy), where jy: (Lu)e → (L'u)ê maps s on the equivalence class [y, s] of (y, s). Let us note that it is linearly generated by the classes [y, s].
• For z: ê' → ê in Γu, the continuous linear map L'u(z): (L'u)ê → L'u(ê') is entirely defined by the fact that it sends [y, s] on [yz, s]. Thus we have well defined the Lcs-presheaf L'u.
• The morphism L'(v) image by L' of a v: u' → u in H is L'(v) = (Γ(v), μ'v) where μ'v(ê) is the unique continuous linear map from (L'u)ê to (L'u')ê such that
μ'v(ê)([y, s]) = [yv, μv(e)(s)].
• The natural transformation η from L to L' defining L' as a reflection of L in the sub-category DisH(Lcs) of lc-distructures pesheaves is defined as follows: η(u)(e): (Lu)e → (L'u)e is the continuous linear map sending s on [e, s].
Proposition. If H is equipped with a Grothendieck topology J and if Γ is a sheaf of categories, there is an lc-distructure sheaf generated by L, namely the PS(Lcs)-sheaf L* for J associated to L'; it hash Γ as its base and its underlying Top-sheaf Θ* is the Top-sheaf associated to Θ'.
Proof. The hypothesis that Γ is a sheaf of categories for J could be omitted, but then the base of L* would not be Γ but its asociated Cat-sheaf. To construct L* we first construct the Top-sheaf Θ* associated to Θ', by the method indicated in Section 4b, Corollary.
1. The elements of the fiber Θ*u are the complete compatible families on u; such a family is of the form (ev, σv)vεV where V is a covering sieve on u and (ev, σv) is an element in the fiber of Θ' on the domain dv of v, meaning that σv is an element of the fiber of L'dv on the object ev of Γdv. Moreover the σv must be compatible and the family 'maximal'. These conditions implies that the corresponding family (ev)v forms a complete compatible family of objects of Γ on u and, since Γ is a sheaf for J, it determines an object e of Γu admitting ev for image ev by Γ(v) for each v in V; thus the family (ev, σv)vεV can be written (e, σ), where σ = (σv)vεV, and the fiber (L*u)e has these families σ for elements. The topology is such that a basis of neighborhoods of 0 is formed by the sets Ω(Wv)vεV formed by the σ containing a σv in Wv, where V is a covering sieve on u and Wv is a convex neighborhood of 0 in (L'dv)ev for each v in V. For an x: e' → e in Γu the linear continuous map L*u(x) from (L*u)e to (L*u)e' sends σ on the family (σvxv)v, = σx, where σvxv = L'(xv)(σv). This defines the Lcs-presheaf L*u.
2. Let w: u° → u be a morphism in H. To define the morphism (Γ(w), μ*w): L*u → L*u°, for each v of the covering sieve V on u we consider the pullback of (v, w) with its projections v°, w°. The v° generate a covering sieve on u°. Then μ*w(e) is the continuous linear map from (L*u)e to (L*u°)ew sending the element σ = (σv)v of (L*u)e on the family (σv°)v°, where σv° = μ'w°(e)(σv).
The natural transformation η*: L' → L* is such that η*(u)(e) sends an element s of (L'u)e on the family (μ'w(e)(s))w of L*u(e) indexed by all the w with codomain u.
Proposition. If the locally convex spaces (Lu)e are complete, so will be the locally convex spaces (L'u)e (resp. (L*u)e if J is a Grothendicek topology on H) so that the lc-distructures presheaf L' (resp. sheaf L*) generated by L is a complete lc-distructures presheaf (resp. sheaf).
d) Locally convex distructures with base a presheaf of monoids
In many applications, in particular to define distributions as we will do in the next section,, we have a generator of lc-distructures L on H, whose base Γ is a sheaf of monoids, Thus each category Γu has a unique unit e, so that the Lcs-semi-sheaf Lu has a unique fiber (Lu)e; let us denote it by Lu. These locally convex spaces Lu are the fibers of a locally convex presheaf L on H: for v: u' → u in H, the linear continuous map L(u): Lu → Lu' is equal to μv(e), where L(v) = (Γ(v), μv). We call L the Lcs-presheaf underlying L.
Then, the Top-presheaf underlying L (obtained by 'forgetting' the vector structure on the fiber, keeping only their topology) is identical to the Top-presheaf Θ constructed in Section (b); indeed this presheaf had for fiber Θu the topology coproduct of the topologies of the fibers of Lu and now there is only one fiber.
In this case, a generator of lc-distructures on H can be defined as follows:
• a presheaf of monoids Γ and a presheaf of locally convex spaces L, both on H;
• for each object u of H a semi-action of Γu on the fiber Lu these semi-actions being preserved by the 'change of fiber' L(v).
Similarly, from the lc-distructures presheaf L' generated by L, we construct the underlying Lcs-presheaf L' on H whose fiber on u is the unique fiber of L'u, and the action of Γu on L'u, these actions being preserved by the 'change of fiber'. And, if H is equipped with a Grothendieck topology J, we also construct the Lcs-sheaf L* underlying the lc-distructures sheaf L* generated by L'; from the construction of L* we see that L* is the Lcs-sheaf associated to L'.
The situation which has motivated the introduction of distructures is the example of Schwartz distributions [19,20]. As already said, in my thesis, my aim was to define such distributions on infinite dimensional spaces by 'extending' the differential operators to continuous functions from a locally convex space E to a locally convex space E'. In fact Schwartz mentions in his book  that this was the initial idea, and he proves that, in the finite dimensional case he considers, each distribution glues together finite order distributions defined on open sub-spaces, a finite order distribution reducing to a higher derivative (in the distribution sense) of a continuous function.
To translate this in terms of distructures, the rough idea is to consider the monoid of 'derivations' and its semi-action on the continuous functions on an open set U of E. This leads to a generator of lc-distructures on the category H of open sets of E. Distributions correspond to the associated lc-distructures sheaf. In the case of finite-dimensional E, they give back the Schwartz vector-valued distributions .
a) The generator of distributions
In the usual definition of distributions defined through the dual of a space of infinitely differentiable functions, the choice of the concept of differentiability is important. Our definition is less dependent on this concept (this will be made more precise later on). What will be essential is the concept (more or less underlying the various differentiability concepts) of successive partial directional derivatives which are at the basis of the construction of any differential operator.
Let E and E' be two locally convex spaces, and f a continuous function from U to E', where U is an open set of E (with the induced topology). If αn is an element (vector) of E, we say that f has a derivative in the direction of αn on U if, for each a in U, there exists
∂f(a)/dαn = limk→0((f(a+
and if the function ∂f/dαn so defined is continuous from U to E'. By iteration, we define higher order directional derivatives.
To fix simple enough notations for such derivatives, we use the notion of a multiset α of vectors of E. It is an element of the free commutative monoid on the set of vectors of E, hence a collection of n possibly repetitive vectors α = <α1, α2, …αn>; the angular brackets indicate that the order is not taken into account and to be repetitive means that we may have αi = αj for two different indices; we call n the order of α.
We say that the function f from U to E' has an α-derivative, or more explicitly an n-th partial derivative with respect to α, denoted by f.α if, for each a in U, the restriction of f to the affine sub-space V = a+ΣiRαi is n-differentiable in a, with (f.α)(a) as its partial derivative with respect to the n-multiset <α1, α2, …αn>, and if f.α is continuous on U. The differentiability of f ensures that it has an n-linear differential Dnf on U which, because of the continuity of the partial derivatives, is symmetric.
Now we define the generator of distributions on E valued in E'.
• It is a generator of lc-distructures D on the category H (defining the order on the set) of open sets of E; we denote by (U', U) the morphism from the subset U' of U to U. This category is equipped with the usual Grothendick topology in which the covering sieves for U correspond to the open covers of U
• The base of D will be reduced to the constant functor on the free commutative monoid A on the set of vectors of E. Its elements are the multisets α, its unit is the multiset void, denoted o, and the composition merges the multisets. In fact, to have a smaller monoid, we may select an algebraic base of E and take only multisets on the vectors of this base; as the n-th differential of a function is supposed to be n-linear, we may only take n-th partial derivatives with respect to the vectors of this basis, the others being obtained by linear combinations; and the result will be independent of the choice of the basis.
• For an open set U of E, the locally convex semi-sheaf DU is defined as follows: its unique fiber (since A is a monoid) is the locally convex space C(U) of continuous functions from U to E', with the compact-open topology. For each multiset α in A, (DU)α is the vector sub-space of C(U) consisting of the functions f admitting an α-derivative on U, equipped with the compact-open topology for the functions and their successive derivatives up to the order n of α; this topology being finer than that induced by C(U), the inclusion (Du)α → C(U) belongs to the ss-admissible class MLcs. And DU(α): (DU)α → C(U) is the map sending a function f which has an α-derivative on this derivative f α. This defines a locally convex semi-sheaf DU since f has an αα'-derivative if and only if its α-derivative f. α has an α'-derivative.
• If U' is an open subset of U, the morphism D(U', U) from DU to DU' reduces to the map from C(U) to C(U') sending a continuous function g on U on its restriction g/U' to U'. This completes the definition of a generator of lc-distructures; the Lcs-presheaf underlying D (cf. Section 6) is the presheaf C of continuous functions from E to E'.
Definition. The generator of lc-distructures D defined above is called the generator of E'-valued distributions on E, and the (pre)sheaf of lc-distructures it generates is called the (pre)sheaf of (finite order) E'-valued distributions on E.
b) The (pre)sheaf of distributions
The general construction of the generated presheaf of distructures can be applied to D, where the fact that A has a unique unit simplifies the description. Thus in the presheaf of distributions D' generated by D, the presheaf D'U has a unique fiber, denoted by D'U and obtained as follows: we take the direct sum Σ of A copies of the vector space C(U); its elements are linear combinations of pairs (α, g) where α is in A and g a continuous function from U to E'. Then D'U is the quotient of Σ by the congruence R generated by the relations r:
(α'α, f) ~ (α', f.α) if f has an α-derivative f.α on U.
It means that an element of D'U is an equivalence class modulo R of such linear combinations. The topology is the finest locally convex topology for which the injections [α, -]: g |→ [α, g] become continuous from C(U) to D'U.
If U' is an open subset of U, D'(U', U) is determined by the linear continuous 'restriction map' D'(U', U) from D'U to D'U' which sends the equivalence class d = [α, g] on its restriction [α, g/U'] to U' (denoted d/U'), and the equivalence class of a linear combination on the linear combination of their restrictions. This describes the Lcs-presheaf D' underlying D'.
The monoid A acts on D'U as follows:
D'U(β)([α, g]) = [α, g]β = [αβ, g] for each β in A.
Remark. The equivalence relation depends on the selected notion of differentiability. Here we define (DU)α as the sub-space of functions f whose restriction to the affine sub-space generated by α is differentiable. We might take only those f which are n-differentiable in a stricter sense, for instance in the sense I have defined in [1,4]. Then the relation R becomes stricter. The advantage of the laxer definition used here is that it allows a more explicit description of R which the following theorem gives.
From now on, we suppose that E' is complete.
Theorem. If E' is complete, D'U identifies to the quotient of AxC(U) by the equivalence relation R' defined by:
(α, g) R' (α', g') if there exist β, β' in A and f, f' such that
g = f.β, g' = f'.β' , αβ = α'β' and (f – f').αβ = 0.
Denoting [α, g] the equivalence class of (α, g), the vector structure is determined by:
[α, g] + [α, h] = [α, g+h] and κ[α, g] = [α, κg] for a real κ.
Its topology is the finest locally convex topology which makes continuous the maps [α, -] from C(U) to D'U. The continuous linear map β-derivation D'U(β) from D'U to D'U maps [α, g] on [αβ, g]. For U' included in U, D'(U', U) is the continuous linear map sending [α, g] on the class [α, g/U']. If E is metrizable, then D'U is complete.
Proof. 1. First we prove that R' is a relation which contains r and is contained in the congruence R defined (before the theorem) on the direct sum Σ (it is possible since AxC(U) is a part of Σ).
At this end, we use the fact that, since E' is complete, each continuous function g to E' admits a β-anti-derivative, denoted by β-1g, for each multiset β = <β1, β2,..., βn> in A, meaning that it is of the form f.β for at least one function f having a β-derivative on U. Indeed, as E' is complete, each continuous g admits on U an E'-valued integral 
β-1g = ∫g dβ1dβ2...dβn
(defined up to a function b having 0 for β-derivative) which has g for its β-derivative. It follows that, for each β in A we have
(α', g) r (α'β, β-1g).
In particular, if (αα', f) r (α', f.α) and if we take g = f = g', β = o, β' = α, we get
αα'β = αα' = α'β', f = g.o, f.α = g'.α and (g' - g).αα' = 0,
thus R' contains the relation r.
On the other hand, if (α, g) R' (α', g'), using the β, β', f, f' which define R', we find:
(α, g) = (α, f.β) r (αβ, f), (α', g') = (α', f'.β') r (α'β', f'),
thus, in terms of the congruence R we have defined before the theorem on the sum Σ,
((α, g) - (α', g')) R ((αβ, f) – (αβ, f')) R (αβ, f – f') R 0,
hence (α, g) R (α', g').
2. R' is clearly reflexive and symmetric. To prove that it is an equivalence, it remains to prove that it is transitive. For this, let us suppose that
(α, g) R' (α', g') and (α', g') R' (α", g");
there exist β, β', f and f' such that g = f.β, g' = f'.β', αβ = α'β' and (f – f').αβ = 0, and also γ, γ', h and h' tels que g' = h.γ, g" = h'.γ', α'γ = α"γ' and (h - h').α'γ = 0. If we write ρ = γβ, ρ' = β'γ' we have
αρ = αγβ = α'β'γ = α"β'γ' = α"ρ'.
On the other hand, let us select
k = γ-1f, whence f = k.γ and g = k.βγ = k.ρ;
k' = β'-1h', whence h' = k'.β' and g" = k'.γ'β' = k'.ρ'.
Now h.γ = g' = f'.β', and we must prove that k –k' has a ρρ'-derivative 0. We may write
k – k' = γ-1f – γ-1f' + γ-1f' - β'-1h' = γ-1(f – f') + b + γ-1β'-1g' – β'-1h' + b'
where b.γ = 0 = b'.β'. However γ-1g' = h + b" with b".γ = 0. Hence
k – k ' = γ-1(f – f') + β'-1(h – h') + b + b' + β'-1b" + b"' with b"'.β' = 0.
f – f' has a derivative with respect to αβ = α'β' and h – h' for α'γ = α"γ', thus the two first terms have a derivative for αβγ = α'β'γ, their derivative being 0, and the other terms have also a derivative since they have 0 for derivative for γ or for β'. It follows that (k - k').αβγ = 0, as wanted.
3. Let us define the vector structure on the quotient of AxC(U) by R'. For each real κ, we define κ[α, g] = [α, κg]; it is independent on the choice of the representative (α, g). Let us note that AxC(U), looked at as a sub-space of the direct sum Σ of the family of A exemplars of C(U), is closed under the scalar multiplication, but only partially under the addition: it is not a vector sub-space of Σ though the addition of Σ induces a partial addition defined for elements with the same first term. However given two elements (α, g) and (β, h) of AxC(U) they are equivalent modulo R' (hence also modulo R as proved in the first part of the proof) to the elements (αβ, β-1g) and (αβ, α-1h) which can be added, the result being (αβ, β-1g + α-1h), in AxC(U). This allows to define directly in the quotient (by R' or by R):
[α, g] + [β, h] = [αβ, β-1g] + [αβ, α-1h] = [αβ, β-1g + α-1h].
It does not depend on the choice of the anti-derivatives: if we replace β-1g and α-1h by other anti-derivatives b and b' the added term [αβ, b + b'] is 0 since b.β = 0 = b'.α. As R' contains r and is included in the congruence R, we deduce that the result is also independent of the choice of representatives, and that AxC(U)/R' equipped with this addition becomes a vector space which can be identified to Σ/R, hence to D'U.
4. The topology of D'U is the final locally convex topology for the linear maps [α, -] from C(U) to D'U. If E' is complete and E metrizable, then C(U) is complete for the compact-open topology, so that D'U is also complete. A basis of neighborhoods of 0 is obtained as follows: for each α in A, let Kα be a compact included in U and Oα a neighborhood of 0 in E'; let Ω(Kα, Oα)α be the convex hull of the set of elements d of D'U admitting a representative of the form (α, g) for some g sending Kα into Oα. When (Kα, Oα)α vary, these sets form a basis of neighborhoods of 0 in D'U.
5. To achieve the description of the Lcs-presheaf D', let U' be an open subset of U. If we have (α, g) R' (α', g'), taking the restrictions of g and g' to U', we have also (α, g/U') R" (α', g'/U') where R" is the equivalence corresponding to U' (defined as R'). So we define, by passage to the quotients, a continuous linear map 'restriction' D'(U', U) from D'U to D'U' sending d = [α, g] on d/U' = [α, g/U].
Definition. An element of D'U is called a finite order E'-valued distribution on U. Its order is the smallest n such that it admits a representative [α, g] where the order of α is n. For each d = [α, g] in D'U the distribution D'U(β)(d) = [αβ, g] is called the β-derivative of d, denoted d.β.
A continuous function g on U can be identified to the distribution [o, g], but the topology induced by D'U on C(U) is finer than that of C(U). In particular a function f which has an α-derivative is thus identified to the distribution [o, f] of order 0, and its α-derivative f.α is identified to [o, f.α] = [α, f], which is also its α-derivative as a distribution, according to the above definition. Moreover, the topology induced by D'U on the locally convex space of α-derivable functions (DU)α is the compact-open topology for the functions and their derivatives, which is the topology we had initially considered on this space.
c) The sheaf of distributions
We denote by D* the lc-distructures sheaf generated by D. It is associated to D'. As the category H on which it is defined is the category of open sets of E, we can use the classical construction of the sheaf associated to a Lcs-presheaf on a topological space, through sections of the etale space of its germs (or local jets in Ehresmann's terminology). Let us develop this construction.
To the Lcs-presheaf D' on the topological space E, we first associate an etale space p: JD' → E whose stalk on an element a of E is defined as follows. Let H/a be the full sub-category of H having for objects the open sets U of E containing a, and D'/a the functor from H/a to Lcs restriction of D'. The colimit of D'/a is a complete locally convex space JD'a. An element of JD'a is an equivalence class <U, d> of pairs (U, d) where d is a finite order distribution on an U containing a, and
(U, d) ~ (U', d') if d/V = d'/V for some open subset V of U and U' containing a.
Definition. The equivalence class <U, d> in JD'a is called the local jet of d at a, denoted jλad.
(This terminology is that of Charles Ehresmann ; a local jet of d is often called a germ of d.)
Let JD' be the union of the spaces JD'a for the different elements a of E and p: JD' → E the map sending an element of JD'a on a. For each d in D'U, let jλd: U → JD' the map sending a to the locl jet of d at a. We equip JD' of the finest topology for which the maps jλd become continuous. It admits the images of these maps as a basis of its open sets. With this topology, JD' is an etale space on E through p. The monoid A acts fiberwise on JD' via the action jAα: (jλad) |→ jλa(d.α) for each α in A.
We associate to this etale space the sheaf Sec(p) of its sections, whose fiber on U is the set of continuous sections d of p on U.
Definition. A continuous section d of p on U is called a (general) distribution on U.
The sheaf D* of distributions is obtained by enriching the fibers of Sec(p) with a structure of locally convex spaces and with an action of A.
• For an open set U of E', we denote by D*U
the locally convex space defined as follows: Its elements are the distributions
on U; the sum d + d' is the section 'sum' which sends a to d(a) + d'(a). The topology is such that a
family of distributions di
converges to a distribution d if the local jets di(a) converge to d(a) in JD'a
for each a in U, and
this uniformly on each compact of E. An explicit description of its open sets
will be given later on (cf. also my
• The monoid A acts on D*U via the linear continuous maps D*U(α) sending d to jAα(d).
• For U' an open subset of U, D*(U', U) is the restriction map sending d to d/U'.
Proposition. D' is identified to a sub-lc-distructures presheaf of D*by identifying d in D'U to the section jλd: U → JD'. If E is metrizable and E' complete, D* is a sheaf of complete locally convex spaces.
preceding definition of a distribution makes explicit use of the 'points' of E,
via the local jets. We have given this definition because local jets of distributions
have an intrinsic interest; in particular they are used in my
To explicit it, let us say that two finite order distributions d in D'U and d' in D'U' are compatible if they have the same restriction to the intersection of U and U'. Then a complete compatible family of finite order distributions on U reduces to a maximal family Δ = (di)i of finite order distributions di on Ui which are compatible and such that U be the union of the Ui. The 'maximality' implies that it contains with a distribution its restriction and that a distribution on U' whose restrictions to a cover of U' belong to Δ also belongs to Δ.
Theorem (Local structure of a distribution). A distribution on U can be identified to a complete compatible family Δ = (di)i of finite order distributions di on U. If E is finite dimensional, the restriction of a distribution to a bounded open set of E is of finite order.
Proof. 1. Given such a family Δ = (di)i the compatibility ensures that all the di for which Ui contains a have the same local jet at a, and the map d sending a on this local jet is a section of the etale map p: JD' → E. Let us prove that this section is continuous for the etale topology; indeed, let jλd' (U') be a neighborhood of d(a); it implies that jλad' = d(a) = jλadi and, by definition of a local jet, d' and di have the same restriction to an open neighborhood V of a contained in U'. The image of V by d is then the same as the image of V by jλd', so that it is included in jλd'(U'); thus d is a continuous section of p on U, hence it is a distribution on U.
2. Conversely, let d' be a distribution on U and a an element of U. By definition of the topology of JD', d'(a) has an open neighborhood of the form jλda(U') for some finite order distribution da on U'. Since d' is a continuous section of p, there exists an open neighborhood Ua of a sent by d' into jλd'(U'), and (Ua, da) is a representative of d'(a).
These da for the different a in U are compatible; indeed, let (db, Ub) be a representative of d'(b) and c an element of the intersection Uab of Ua and Ub. Both da and db admit d'(c) as their local jet at c, so that there exists a neighborhood Vc of c contained in Uab on which da and db have the same restriction. It follows that da and db have the same restriction to the union Uab of the Vc so associated to the different elements c of Ua
The family (da)a generates a complete compatible family Δ on U, obtained by adding the finite order distributions on subsets of U compatible with all the da, in particular all the restrictions of an element of Δ and a distribution on U' whose restrictions to an open cover of U' belong to Δ. This maximality of Δ ensures that there is exactly one such family associated to d'. It follows that the map sending d' to Δ defines a 1-1 correspondence between the set D*U of distributions on U and the set of complete compatible families of D'U.
3. Now we suppose that E is finite dimensional. Let d be a distribution on U and U' a bounded open subset of U whose closure is contained in U. We associate to d the complete compatible family Δ = (di)i, di on Ui. The closure Ū' of U' admits an open covering by the traces of the Ui and, since it is compact (U' being bounded in the finite dimensional space E) we can extract from it a finite cover of Ū'; let U'1, ..., U'm the corresponding finite cover of U' and d'j the restriction of the distribution dj to U'j for j = 1,...,m. By the process already used, we can select representatives of those m distributions dj with the same first term, say dj = [α, hj]. Since the dj must have the same restriction to the intersection of two Uj it follows that the hj are restrictions of a same continuous map h on U', and the finite order distribution [α, h] on U' which glues together the dj must belong to the maximal family Δ. Thus the restriction of d to U' identifies to the finite order distribution [α, h].
Given the distribution d we'll write d ≈ (di)i, where (di)i is the corresponding complete compatible family.
The general construction of the associated Lcs-sheaf shows that the topology of the locally convex space D*U of distributions on U is obtained as follows: For each open subset Ui of U, let Ai be the set of distributions d ≈ Δ such that Δ contains an element di on Ui. We equip it with the topology inverse image of the topology of D'Ui (space of finite order distributions on Ui) by the map sending d to di. The locally convex topology of D*U is the finest topology making continuous the inclusion of Ai in D*U for each i. Given an open cover (Uj)j of U and a convex open neighborhood Wj of 0 in D'Uj for each Uj, we denote by Ω(Wj)j the set of distributions d ≈ Δ such that Δ contains a di belonging to Wi for each j. These sets, where the cover (Uj)j and Wj vary, form a basis of neighborhoods of 0 in D*U. We can also describe Ω(Wj)j in terms of local jets: it is the set of distributions d such that, for each a in Uj the local jet d(a) belongs to the set jλWj of jets of the elements of Wj.
d) Comparison with Schwartz distributions
Theorem. If E is finite dimensional and E' complete, there is an isomorphism S* from the Lcs-sheaf D* to the sheaf SD' of Schwartz E'-valued distributions, which has for restriction an isomorphism S' from D' to the presheaf SDf of Schwartz finite order distributions.
Proof. 1. Let DU be the space of the infinitely differentiable functions φ from E to R with compact support contained in U, with the compact-open topology for the functions and their derivatives. If d = [α, g] be an element of D'U of order n. We define
d(φ) = [α, g](φ) = (1/n)∫g(a)(φ.α)(a)da,
where the second member is a E'-valued integral which exists since E' is complete and φ has a compact support in U. By definition of the equivalence R' defining the finite order distributions, we deduce (using an integration by parts) that this is independent of the chosen representative of d. The map Sd: φ |→ d(φ) is linear; it is continuous from DU to E' because the α-derivation is continuous from DU to DU and the integral of g depends continuously (uniformly on each compact) of the continuous function φ.α (Bourbaki ). Thus Sd is a Schwartz E'-valued distribution .
2. This extends to any distribution d' on U. Indeed, since φ has a compact support K included in U, we can find a bounded open neighborhood U' of K included in U. From the first part of the proposition, the restriction d'/U' of d' to U' is a distribution of finite order, to which corresponds (d'/U')(φ) defined as above; this element of E' does not depend on the choice of U', so that we can denote it by d'(φ). The map Sd': φ |→ d'(φ) is the Schwartz E'-valued distribution associated to d'.
3. We are going to prove that, for each open set U of E' (not necessarily bounded) we have so defined an isomorphism S*U from D*U on the locally convex space SD'U of Schwartz E'-valued distributions sending d on Sd, whose restriction S'U to D'U maps this space on the space SDfU of finite order Schwartz distributions on U.
S'U is 1-1. Indeed, if d and d' are two finite order distributions which are different, we can find representatives [α, g] and [α, g'] of d and d' such that g be different from g'. Then it exists a function φ whose derivative φ.α has its support in an open subset of U on which g and g' differ, and d(φ) is different from d'(φ), whence Sd is dfferent from Sd'..
The topology of D'U being the final locally convex topology for the maps [α, -] from C(U) to D'U, to prove that S'U is continuous from D'U to SDfU, it is sufficient to prove that its composite with each of these maps is continuous; and that comes from the fact that S'U°[α, -] maps g on the above integral, and, for each φ, this integral depends continuously on g, It ensures that, if gn converge, the S([α, gn]) converge in SDfU.
Schwartz has proved that an E'-valued distribution, where E' is complete, is locally of finite order (, p. 90) and (, Proposition 24, p. 86) that a finite order distribution T on U is the finite sum of (distribution) derivatives βmgm of continuous functions gm. Since such a βmgm is the image by S'U of [βm, gm], it follows that their sum T is the image by S'U of the finite order distribution d sum of the [βm, gm]. Thus S'U defines an isomorphism from D'U on SDfU. Whence an isomorphism S' from D' to the presheaf SDf of Schwartz finite order E'-valued distributions.
Moreover Schwartz's result implies (though he did not use this terminology) that SD'U is the fiber on U of a sheaf SD'(E') on E which is the sheaf associated to SDf(E'). Therefore the isomorphism S' from the Lcs-presheaf D' to SDf extends into an isomorphism S* between their associated sheaves D* and SD'. In particular it gives an isomorphism S*E from D*E to the space of Scwhartz distributions SD' on E.
Remark. Distructures can be applied in many other cases, for instance a construction similar enough to that we have just done for distributions allows to define the analog of de Rham's currents on infinite dimensional manifolds. Other kinds of 'generalized functions' can also be defined through the construction of appropriate distructures (cf. examples in my thesis ).
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