**-SEMI-SHEAVES, DISTRUCTURES AND**

**SCHWARTZ DISTRIBUTIONS**

by Andrée
C. EHRESMANN

Université de Picardie Jules Verne, Amiens

ehres@u-picardie.fr

2008

**0. Introduction**

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 ([7], 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 » [10] that he did make the link with the categories
Eilenberg and

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 S* _{e}* (hence, in modern terms, to
a presheaf on C

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 » [11]
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) [1] 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 [3] 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 *[13], and in a
Note of 1966 [14] 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 C^{op}; 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 [18]. (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:

·
For
each object *e* of C, F(*e*) is the
identity of a set F* _{e}*,
called the

·
For
*x: e' **→** e*, F(*x*)
is a map from a subset F* _{x}*
of F

·
(*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 F* _{e}* and

**Remark.**** **This can be compared to the notion
of a *system of structures* (C.
Ehresmann [13]) on the opposite category C^{op}. 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 F* _{x}*, for an

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 *F_{x}* by *F(*x*) *is equal to the intersection
*I* _{xx'} of *F

**Proof**. If *s* is in I* _{xx'}*
the composites

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, *p*_{0},
*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 F* _{x'}*.

The transitivity ensures that this law
defines a category whose objects are the pairs (*e, s*) such that *s* is in F* _{e}*. The map sending (

(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 [13], 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 F* _{e}* is the inverse image
of

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 F* _{x}* to F*

µ(*x*)(*s*)λ(*x*) = µ(*e'*)(*sx*)
for *x: e' → e *and*
s* in F* _{x}*.

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 = λ*p*.

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
[21] which decomposes any functor *p*:
H → C as the
composite *p' _{° }j* of a
fibration

A(*s*) = Hom_{C}(*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

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 F* _{x}* in the fiber F

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 [3], 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 [14] 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:

·
F(*e*) for an object *e* of C is the identity of an object F* _{e}* of K, called the

·
For
each *x: e' **→** *e in C, the morphism F(*x*): F* _{x}*
→ F

·
(Transitivity)
If *xx'* is defined in C, there exists
a pullback I* _{xx'}* of (F(x),

F(*x’*)*p _{xx'}* = F(

where *p _{xx'}*
and

A (K, M)-semi-sheaf F such that F* _{x}* = F

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:

- A functor λ from C to C*.

·
A map µ from C to K which sends *x: e' → e* to a morphism µ(*x*): F* _{x}*
→ F*

*m**_{λ}_{(x) }µ(*x*) = µ(*e’*) *m _{x}* and
F*(λ(

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

G* _{ê}*(

**•**** **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
γ* _{z}*: Sub(

• We
define a morphism (Id_{C}, ρ): F → F' by taking for ρ(*ê*) the
injection *j _{ê}* of F(

F'(z)ρ(*z*) = F'(z)ρ(*ê*)*m _{z} = *F'(

• It
remains to prove that (Id_{C}, 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 (Id_{C}, ρ) gives (λ, μ). For λ' we
take λ itself. For an object *ê*
of C, we have a natural transformation *n*
from G* _{ê}* to F*

*n(y)
= *F*(λ(*y*))μ(*e*)
and *n(t)*
= F*(λ(*y*))μ(*e*)*m _{x}*

It is a
natural transformation since

*n(y)m _{x} = n(t) = * F*(λ(

This
natural transformation glues into the morphism μ'(*ê*) from the colimit F'* _{ê}*
of G

μ(*ê*) = *n(ê)* = μ'(*ê*)*j*_{ê} =
μ'(*ê*)ρ(*ê*).

It is a
natural transformation since

*n(y)m _{x} = n(t) = * F*(λ(

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*)*j _{y}*
= μ'(

= F*(λ(*yz*))μ(*e*)
(by definition of μ'(*ê'*)
)

= F*(λ(*z*))F*(λ(*y*))μ(*e*) = F*(λ(*z*))μ'(*ê*)*j _{y}* (by definition of μ'(

whence μ'(*ê'*)F'(*z*) = F*(λ(*z*))μ'(*ê*) since the *j _{y}* are the injections toward the colimit F'

μ'(*ê*)*j _{y}*
= μ'(

for each *y, *and thus μ'(*ê*) = μ"(*ê*).
This completes the proof that (Id_{C}, 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 *Σ* _{ê} of
the family *(F

**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 G

** **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 [14]. 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 I* _{xx'}*
of (F(x),

The
difference with a semi-sheaf is that here we do not impose that I* _{xx'}* be the intersection of F

**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" [7] (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) → (f*_{1}*, m*_{1}*) *of KM is defined by an *m* *in
M such that *m = m*_{1}*m* *and *f = f*_{1}*m**.

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 3^{rd} 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 *C^{op}*
to *KM, *hence also as a 2-functor l*F
*from an appropriate 2-category *C_{2}
*to *KM*.* *And the associated
K-presheaf is a* KM**-wise colimit of
the 2-functor l*F*.*

The
category C_{2} has been constructed by several authors (Gray [17],
Vaugelade [22]). 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 *l*F admits a KM*-wise colimit, so that F
has an associated K-presheaf.

b) *Completely
regular ordered categories*

In fact
in Charles' Note [14] as well as in my comment on it [7], 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 [12] 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 [7]) 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 (s* _{i}*)

(v) If *s =* V_{i}s_{i}_{ } for *f:
s' → s*, then *f = *V*s _{i}f. *

**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 (ss _{i})_{i}*

The axioms of a
pretopology are easily verified, using the fact that *s _{i}f *is a projection of the pullback of (

**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 *q*F whose fibers are the images by
*q* of those of F and *q*F(*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 *q*F. However the* *underlying presheaf *q*F' to F' is generally not the presheaf (*q*F)' associated to *q*F, except if *q* also preserves colimits (for example
if K is the category Top); there is only a non-invertible morphism from (*q*F)' to *q*F'.

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 *[1] 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 M_{Lcs}
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, M_{Lcs})-semi-sheaf.

Thus a locally convex semi-sheaf F
on C has for fiber F* _{e}* a
locally convex space, and F(

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 G

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 [1] 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 [10]). 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 (*c _{i})_{i} *of

For each pair (*i, j*) of indices, let *e _{ij}*
be the pullback of (

**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 [16] 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

• its index
set, denoted Iσ, is a covering sieve of *e* for J; for each morphism *x:*
*e _{x} →*

• if *y* has *e _{x}* for codomain, the 'restriction' σ

• two elements
σ* _{x}* and σ

• σ* _{x}* belongs to σ if there exists a covering sieve W
of

2. For each x with
codomain *e*, let A* _{x}* be the subset of E formed by the σ for which Iσ contains

We equip
A* _{x}* with the locally convex
structure inverse image by

More
explicitly: its vector structure is such that *k*σ = (*k*σ* _{x}*)

3. For *z: ê → e*,
the continuous linear
map F*(*z*): F** _{e}* → F*

The natural
transformation η': F' → F* defining the reflection into Sh_{(C,J)}(Lcs) is defined as follows: η'(*e*): F'* _{e}* → F*

**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 *F_{e}*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, M_{Cat})-semi-sheaf
F on C, where M_{Cat} is the class of inclusions of a sub-category in a
category. Thus F associates to each object *e*
of C a category F* _{e}* and to
an

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 (F

(*y**, bx) ~ (yx, b) *for* * each *y: ê **→** e *and *b* in F* _{e}*.

(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

The *Evolutive Systems* which we have introduced with Jean-Paul
Vanbremeersch [8] 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 F* _{t}* models the configuration of
the system around

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, M_{Cat})-semi-sheaf
on a category Time. Otherwise, it is an evolutive system whose fibers are
hierarchical categories, the transition functors respecting the levels [8].

The theory of Memory Evolutive
Systems developed with Jean-Paul Vanbremeersch (cf. our book [8], summarized on
our internet site [9]) 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 D* _{u}* of D is a (K, M)-semi-sheaf on the category Γ

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 GDis_{H}(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 Dis_{H}(K) be
its sub-category of K-distructures presheaves. Since PS(K)
is reflective in SS(K, M), we have:

**Proposition.**** ***The category *Dis_{H}(K) *is a reflective sub-category of *GDis_{H}(K, M). *The fiber** *D'* _{u }of the
reflection *D'

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 Γ* _{u}*; a
morphism from

*v**: u' → u* is in H and *x: e' → ev* *= *Γ(*v*)(*e*) is in Γ* _{u'}*;

and its composite with
(*v', x'*): *e" → e'*, where *v':
u" → u', *is *(vv', xv'.x'):
e" → e* (the dot denotes the composition in Γ* _{u"}*)

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 Γ* _{u}*; for

• the sub-category Cart of cartesian morphisms which are of
the form (*u, e*) where* e *is an object of Γ* _{u}*.

These
sub-categories generate F(Γ) since we have: (*v, x) = (v, ev)x* for *x *in Γ* _{u'}*.

**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 φD* _{e}*
on the object

φD(*v, x*) = D* _{u'}*(

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'*) = D* _{u"}*(

The
transitivity of φD means that the pullback I' of (*m _{v',x'},* φD(

Similarly,
considering the composite pullback square PB_{m}_{°}PB_{3}
and the adjacent square PB* _{xv'.x'}*
connected by

For a cartesian morphism (*v, e*),
the above construction implies φD(*v, e)* = μ* ^{v}*(

If D takes its values in the
category of K-presheaves, so that it is already a presheaf of K-distructures,
then *m _{x}* for

*m _{v,}*

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 [7]). It relies on the
fact that the fibration F(Γ) is a lax colimit of Γ looked at as a 2-functor from H^{op}
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 Ins* _{u}*:
Γ

**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 D* _{u}* will be a
restriction of B, namely the composite of B with the 'inclusion' Ins

Since B becomes a presheaf on Cart, for
*v: u' **→** u *and *e*
an object of Γ* _{u}*, B(

*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

From the second decomposition we deduce
that B* _{v,y}*
is the pullback of (

**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 D* _{u}*
to E

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 PB* _{x}* and PB'

μ* ^{v}*(

where (Γ(*v*), μ'* ^{v}*):
E

**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 D* _{u}*,

• a 'horizontal' K-presheaf on the
category Cart, deduced from the morphisms between the D_{u}__.__

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 S* _{u}*
be the (disjoint) union of the fibers (D

• a 'vertical' semi-action of the sub-category Vert
on Σ (gluing
the semi-actions of the various Γ* _{u}*),
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 S* _{u}* and such
that H(

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}*)

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 *q*D, whose
fibers are the semi-sheaves *q*D_{u}* *underlying the fibers of D (cf. Section
4). The vertical structure of *q*D (defined
by the semi-action of Vert) underlies the vertical structure of D, defined by
the (K, M)-semi-sheaf gluing the semi-sheaves D* _{u}*. However the horizontal structure of

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 M_{Lcs} 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, M_{Lcs})-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 L* _{u}* of L on an object

The
forgetful functor *q _{LT}*: Lcs → Top preserves the limits and sends M

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 Θ_{u} coproduct in Top of the topologies of
the fibers of L* _{u}*;

• Θ sends *v:
u' **→** u* to the
continuous map Θ(*v*):
Θ* _{u}* → Θ

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 *q*L'* _{u}*;
and to the Top-presheaf Θ' whose fiber 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 Γ* _{u}*
of Γ on

(*y**, **σ*) ~ (*y’, **σ**x*) if * y =
y'x *in Γ* _{u
}*and

generate a congruence R* _{ê}* on the vector space V

• For *z:
ê' **→** ê *in Γ* _{u}*, the
continuous linear map L'

• The morphism L'(*v*) image by L' of a *v: u' **→** u* in H is L'(*v*) = (Γ(*v*), μ'* ^{v}*) where μ'

μ'* ^{v}*(

• The
natural transformation η from L to L' defining L' as a reflection of L in
the sub-category Dis_{H}(Lcs) of lc-distructures pesheaves is defined
as follows: η(*u*)(*e*): (L* _{u}*)

**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

2. Let *w: u° → u *be a
morphism in H. To define the morphism (Γ(*w*), μ*^{w}): L** _{u}* → L*

The natural
transformation η*: L'
→ L* is such that η*(*u*)(*e*) sends an element *s* of (L'_{u}__)__* _{e}* on the family (μ'

**Proposition.**** ***If the locally convex spaces *(L* _{u}*)

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 Γ* _{u}* has
a unique unit

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 Θ* _{u}* the topology coproduct of the topologies of
the fibers of L

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 Γ* _{u}* on the fiber

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 Γ

**7. Distributions**

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 [19] 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 [20].

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 α_{n}* *is an element (vector) of E, we say
that *f has a derivative in the direction
of *α* _{n}* on U if,
for each

∂*f(**a)/d*α_{n}* = *lim_{k}_{→}_{0}((*f(a+**k*α_{n}* – f(a))/**k*)

and if the function ∂*f/d*α* _{n}* 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 = <α_{1}, α_{2}, …α* _{n}*>; the angular brackets indicate that the
order is not taken into account and to be repetitive means that we may have 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*+Σ_{i}**R**α_{i} is *n*-differentiable* *in *a*,
with (*f.*α)(*a*) as its partial derivative with respect to the *n-*multiset
<α_{1}, α_{2}, …α* _{n}*>, and if

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 D_{U} 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, (D_{U})_{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 (D* _{u}*)

• If U' is an
open subset of U, the morphism D(U', U) from D_{U} to D_{U'} 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 (D__ _{U})_{α}__ as the sub-space of
functions

** **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,
*κ**g*] 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^{-1}*g*, for each multiset β =
<β_{1}, β_{2},..., β* _{n}*> in A, meaning that it is of the form

β^{-1}*g =* ∫*g* dβ_{1}dβ_{2}...dβ_{n}

(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^{-1}*g*).

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^{-1}*f*, whence *f*
= *k.*γ and *g* = *k.*βγ = *k.*ρ;

and

*k**'* = β'^{-1}*h'*,
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^{-1}*f* – γ^{-1}*f'* + γ^{-1}*f'* - β'^{-1}*h'*
= γ^{-1}(*f – f'*) + *b* + γ^{-1}β'^{-1}*g'* – β'^{-1}*h'*
+ *b'*

where *b.*γ = 0 = *b'.*β'. However γ^{-1}*g'* = *h + b"* with *b".*γ = 0. Hence

*k – k '* = γ^{-1}(*f – f'*) + β'^{-1}(*h
– h'*) + *b + b' +* β'^{-1}*b" + 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, κ*g*];
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^{-1}*g*) and (αβ, α^{-1}*h*) which can be added, the result being (αβ, β^{-1}*g* + α^{-1}*h*),* *in*
*AxC(U). This
allows to define directly in the quotient (by R' or by
R):

[α, *g*] + [β, *h*] = [αβ, β^{-1}*g*] + [αβ, α^{-1}*h*] = [αβ, β^{-1}*g* + α^{-1}*h*].

It does not depend on the choice of the
anti-derivatives: if we replace β^{-1}*g* and α^{-1}*h* 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
K_{α} be a compact included in U and O_{α} a
neighborhood of 0 in E'; let Ω(K_{α},
O_{α})_{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 K_{α}
into O_{α}. When (K_{α}, O_{α})_{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 (D_{U})_{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'

(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

(This terminology is that of Charles
Ehresmann [7]; 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

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

**Definition.**** **A
continuous section ** d **of

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
** d**
on U; the sum

• The monoid A acts on __D__*_{U}
via the linear continuous maps D*_{U}(α) sending ** d** to

• For U' an open subset of U, __D__*(U', U) is the
restriction map sending ** d** to

**Proposition.**** **D'* is
identified to a sub-lc-distructures presheaf of *__D__**by identifying d in *__D__'_{U}
*to the section j ^{λ}d*: U
→ JD'

* *

The
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 = (*d _{i}*)

**Theorem (Local structure of a distribution).**** ***A
distribution on* U *can be identified
to a complete compatible family* Δ = (*d _{i}*)

**Proof.**** 1. **Given such
a family Δ = (*d _{i}*)

2. Conversely, let ** d' **be a distribution on U
and

These *d _{a }*for the different

The family (*d _{a}*)

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

Given the
distribution ** d** we'll write

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 U* _{i}*
of U, let A

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 SD ^{f} of Schwartz finite order distributions. *

**Proof.**** 1. **Let *D*_{U} 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 S*d*: φ |→ *d*(φ) is linear; it is continuous from *D*_{U} to E' because the α-derivation is continuous from *D*_{U} to *D*_{U} and the integral of *g *depends continuously (uniformly on each compact) of the
continuous function φ.α (Bourbaki [6]). Thus S*d* is a Schwartz E'-valued distribution [20].

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

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 S

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 S*d* 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 *SD ^{f}*

Schwartz
has proved that an E'-valued distribution, where E' is complete, is locally of
finite order ([20], p. 90) and ([20], Proposition 24, p. 86) that a finite
order distribution T on U is the finite sum of (distribution) derivatives β* _{m}g_{m}* of continuous
functions

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 *SD ^{f}*(E').
Therefore the isomorphism S' from the Lcs-presheaf

**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 [1]).

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