RESEARCH WORKS  of Andrée C. EHRESMANN

         

Various domains of mathematics have required my attention and my research itinerary during more than 50 years can be divided in 3 periods: I worked successively on functional analysis, category theory and modeling of complex biological or cognitive systems. However I established strong connections between these domains as the red threads in the graph of my research work prove,

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My first research interest has been in the domain of topological real vector spaces and of algebraic analysis, meaning the study of algebraic structures underlying analysis problems.

 

In 1955, I was most stimulated by the undergraduate course of Gustave Choquet, in particular its parts on topology and vector analysis, and, the following year, I decided to work under his direction. At this time he was trying to prove the well-known theorem on extreme points of convex sets which bears his name, and he associated me with his research. I proposed a new proof in the (already known) case of finite dimensional spaces and tried to extend it to the infinite-dimensional case. Though I did not succeed, it had some influence on the intricate proof that Choquet gave (and he cited me in his first publication on the subject). This early research has been a very useful experience for it taught me the subtle techniques of topological vector spaces, in particular the difference between the finite and infinite cases.

After that, Choquet asked me to study a paper by Rosenbloom ([Rx], 1956). A remark in this paper, joined to my recent work on convexity, led me to the question: how to define a convex polyhedron in an infinite-dimensional space? This question is answered in my 3^rd cycle thesis [1] (the numbers refer to the list of publications): in an infinite-dimensional vector space E equipped with the finest locally convex topology, a polyhedron is a closed convex subset of E whose support cone in each x of C is closed (the support cone in x is the union of the half-lines from x containing at least one point of C other than x). A main theorem asserts that this property characterizes a convex polyhedron if E is finite dimensional.

To extend the theory of polyhedrons and pyramids to the infinite-dimensional case, I defined two species of pyramids: generalized simplicial pyramids, and strict pyramids  which are the intersection of their extreme supports (later interesting examples of them have been given by other authors). If (E,T) is a topological vector space, the T-closure of a pyramid is called a topological pyramid. Results obtained by Rosenbloom if E is the normed space of absolutely convergent series are generalized to topological pyramids, leading to various applications in optimization analysis.

These results are strengthened in the paper [5]: it is proved that a proper pyramid is the intersection of its extreme supports and that its polar cone is the weak closure of a convex pyramid. The Note [4] (with C. Ehresmann) gives an example of a pseudo-bounded convex polyhedron without any vertex.

 

In 1958, some problems on differential equations (in the frame of a consultant contract) led me to read Schwartz's theory of distributions. I was very stimulated by his two books [S1] and his paper [S2] on vector distributions. They naturally raised the following problem: How to extend this theory and define distributions generalizing differentiable maps from an infinite-dimensional topological vector space E to a topological vector space F?

As said above, my early work on locally convex spaces had attracted my attention on the different properties of finite and infinite-dimensional cases. It was evident that the initial definition of a distribution was to be modified since it uses continuous maps with compact support. Moreover there was no "good" notion of differentiable maps between topological vector spaces with the same stability properties as in the finite dimensional case. The first step was to define such a notion, and, in the first part of my thesis ([7], cf. its summary) I proposed the following definition (sometimes called "Bastiani-Michael differentiability"):

 

Let E and F be two topological vector spaces; a map f from an open U of E to F is differentiable if, for each x in U there exists a continuous linear map Df(x) from E to F such that:

However, if E is not metrizable, the theorem on the transitivity of prolongations is no more valid, because there is no topology on the space of continuous multi-linear maps such that: f is n-differentiable if and only if it is (n-1)-differentiable and if its (n-1)-differential is differentiable. To obtain a theorem of this kind, I had the idea to replace the topologies by quasi-topologies (or "Limesraüm" [Ko]); a quasi-topology is defined by the data, for each point x, of the filters which quasi-converge to x.

This idea is developed in the long (often cited) paper [11], where the set C(X,F) of continuous maps from a quasi-topological space X to a locally convex quasi-topological space F is equipped with the local convergence quasi-topology λ. This quasi-topology identifies with the compact-open topology if X is a topological locally compact space. In any case, it has the following properties, where E, F and G are locally convex quasi-topological spaces:

Having a convenient notion of differentiability, I had to take a characterization of Schwartz distributions which could be extended in the infinite-dimensional case. Schwartz has proved that a distribution is locally of finite order, i.e., is the generalized derivative of a continuous function, so that a distribution glues together finite order distributions. The idea at the basis of the second part of my thesis [7] is to define distributions by a double completion process: locally extension of differential operators to continuous functions, globally gluing of the finite order distributions so obtained.

To fulfill this program, I realized that it would be interesting to use the theory of categories. When I wrote my thesis In 1959-60, this theory was not well known except in algebraic topology or geometry, and it had not yet been applied in analysis. I had heard from it by Charles Ehresmann, and I was only familiar with the notions developed in his 1957 seminal paper "Gattungen of Strukturen" [CE]. Thus I had to adapt them to my problem. Among the notions I introduced in my thesis figure the categories of categories (equivalent to split fibrations, introduced by Grothendieck around the same time) and the categories of acting categories (in modern language, functors toward a category of presheaves), which later led Charles Ehresmann to study more generally what he called "dominated species of structures" [CE].

 

The essential notion is that of a topological vector inductive category (or civt) (civt), which, in modern terms, corresponds to a presheaf from an inductive category  (i.e., internal to the category of complete sub-lattices) C to the category of topological vector spaces. The civt is complete if the presheaf is a sheaf (for the order on C) and the fibers are complete topological vector spaces.

The main theorems of my thesis are completion theorems for a civt. In particular, given a civt C and a category of acting categories D on it(s fibers), I construct a solution to the universal problem of embedding C into a complete civt H, which still admits D as a category of acting categories. H is then called a species of distructures. The construction is in 2 steps:

This last construction (with E metrizable and F complete) is analyzed, developed and applied in the papers [10, 12-14], where various operations are defined on them: germs of distributions, associated canonical linear forms, value in a point. If E is ^{n-dimensional} and F is reflexive, the definition is equivalent to Schwartz's one [S1]. If moreover F = R, the filters can be replaced by series, and the construction is analog to that independently obtained by Mikusinski-Sikorski in 1961 [MS]. Let us note that these authors did not try to define infinite-dimensional distributions but to present the theory of distributions "in as simple a manner as would make it comprehensible to physicists and engineers".

In 2008 I have proposed a translation of the theory of distructures in a more modern language, based on the notion of a semi-sheaf [74, 75] (cf. Section 4 below).

The local study of distributions leads to a method of solving Cauchy problems with mixed type for partial differential equations. I have applied this method to give an explicit solution to the problem of Burmister in elasticity, in particular to compute the road deformation due to a car. This application (summarized in the Note [6]) is described in [8] (1962). However, the formulas (in terms of Fourier transforms of distributions) are too complex to be practically exploited.

 

In 1964 I accepted Pallu de la Barrière's proposal to be associated to the contract given by the DGRST ("Délégation Générale à la Recherche Scientifique et Technique") to his "Laboratoire d'Automatique théorique" at the "Université de Caen". The subject was the study control problems, a domain in full expansion at this period. The results of my study are given in 5 longs preprints [    ], summarized in a paper presented to the "Congrès d’Automatique Théorique" de Saclay (1965). [15]. As they were written while the work progressed, the general plan is not very clear and the texts are not homogeneous enough.

 

In this domain also I realized that the theory of categories could be an important tool. To model the structure underlying to the set of solutions of a partial differential system, in Part 1 [9] I introduced the notion of a noyau d'espèce de structures which, in modern terms (cf. the Calais presentation) corresponds to an internal to Top partial action of a category. A système guidable or control system, corresponds to an internal functor from the associated partial (discrete) fibration H to a category, its solutions are sections of this functor. A preferential control system is a control system with a functor from H to an ordered category; it models the algebrico-topological structures underlying to the optimization problems.

 

These notions allow giving a rigorous and general formulation of the heuristic "dynamic programming" method of Bellman [B]. Several examples are given (optimization problems for systems of differential equations, axiomatic of Roxin (Ro], of Bushaw [Bu],…).

Part II [12] studies differentiable control systems. The main theorem is an optimization theorem generalizing the Bellman method. In Part III [13] it is extended to more general control systems thanks to a simpler proof using germs of vector distributions; Pontrjagin's optimization theorem [P] for systems of differential equations can be deduced from it. Part IV [14] generalizes the notion of a well-posed Cauchy problem in the frame of control systems and proposes an algebraic method to study these problems.. In particular, I show how to apply the theorem of Part III to obtain the optimal solutions for systems of partial differential equations (on a Banach space) depending of parameters and with boundary conditions.

 

 

In my thesis and in the following works, problems of category theory had been studied in view of adapting its tools for analysis. As I have already mentioned they have led to the introduction of several notions, in particular:

Later, these notions have been generalized and developed in Charles  Ehresmann's papers written during long daily discussions in the sixties. They are at the basis of his theory of (concrete) internal categories and of dominated species of structures, and of his various completion and expansion theorems..

However, it is only in 1968 that, with Charles, we centered our efforts on category theory and were able to make our collaboration public. We created our joint research team "Théorie et Applications des Catégories" (Paris-Amiens), directed more than 40 theses and organized regular seminars and 2 international conferences (Amiens 1973, 1975). 

 

Our joint publications from 1968 to Charles' death in 1979 are more or less related to the completion of categories, and to the construction of monoidal closed structures on sketchable categories, in particular on categories of internal categories such as multiple categories.

In a long 1972 paper [19], we give an explicit construction of the universal completion (up to isomorphism) of a category by some kinds of limits or colimits, with preservation of some (co)limits, and we prove that it is also the solution of the same problem 'up to equivalence'. It strengthens and extends completion theorems of other authors, e.g. Isbell, Lambek, Tierney This construction is applied to describe the prototype and the type (or theory) of a (mixed) sketch.

The second part of the same paper is devoted to the study of the category of models of a sketch. One of the main results is the construction of a monoidal closed structure on a sketchable category; and in particular on the category of internal categories in a monoidal closed category. This last construction generalizes to the non-concrete case the construction (obtained in our paper [17]) of a cartesian closed structure on the category of internal categories in a concrete cartesian closed categories. Many theses prepared in our team are about sketch theory and sketchable categories, well before the interest these theories suscited in the eighties.

 

A series of 4 long papers [21, 23-25] develops the theory of multiple catégories. In [21] we define D-wise limits (where D is a multiple category) and we give existence theorems for these limits which generalize results of Gray, Bourn, Street on lax-limits of 2-functors; thanks to a more structural and simpler method.

We construct monoïdal closed structures on the category of multiple categories of any order [23] and on the category of n-fold categories. In this last case, the internal Hom is adjoint either to the "n-square functor" (for the Cartesian closed structure [24]; or the the "n-cube" functor (for a more complex monoidal closed structure [25]. As an application, we give a complete characterization of multiple categories in terms of 2-categories. In particular a 2-fold category is a double sub-category of the category of squares of a 2-category. It follows that the D-limits of a multiple functor are lax-limits with respect to a sub-category.

This series of papers has introduced a frame which has later been extensively developed, for instance by R, Brown and Higgins and their students (in homotopical algebra), or Grandis et Paré (multiple limits). To-day n-catégories are extensively applied in computer science and physics.

 

A large part of my time during this 1967-1983 period has been occupied with the direction of research students (cf. CV), and by the writing of several lecture notes [1', 3'-5'], in which I presented algebra and topology for undergraduates in a unified frame with the help of category theory.

I also wrote a book on set theory [18] which proposes an axiomatic based on the Zermelo-Fraenkel system, to which I add an axiom of universes (slightly weaker than Grothendieck's axiom). The well-ordered sets of natural numbers, of ordinal numbers and of cardinal numbers with respect to a universe are extensively studied, and explicit constructions are given for the classes of canonical ordinals and canonical cardinals, as well as for the 'classical' universes. I deduce from this construction that the given universe axiom implies Grothendieck's axiom.

 

After the death of my husband, I spent 3 years on the edition of Charles Ehresmann : Oeuvres complètes et commentées, published from 1980 to 1983, and divided into 4 Parts (7 volumes). Each volume (from 350 to 450 pages) contains a reprint of his papers relating to a particular domain, followed by long comments on these papers to replace them in their historical context and indicating later developments. It begins by a brief biography for the corresponding period, and ends with a Synopsis summing up the main results (translated in modern terms) to guide the reader.

 From 1932 to 1979, Charles Ehresmann has published about 150 papers on algebraic topology, differential geometry and category theory, which have had a strong influence on the development of these domains. In particular he has introduced the theory of (principal) locally trivial bundle spaces and (with his student Reeb) the theory of foliated manifolds. His notions of local structure and topology without points have been much studied later on, e.g. in the theory of locales and frames. He has laid a new foundation for differential geometry, based on his theory of jets and on his development of differentiable groupoids (his ideas are often cited in Synthetic differential geometry).

His work on differentiable groupoids and pseudogroups has paved the way to his interest in category theory. His papers in this domain cover the main part of Parts II to IV, and, having been associated by Charles to their development, I have written myself the comments and synopsis of Parts II [33, 35], III [28, 31] and IV [32, 37] (summing up to more than 450 pages). The comments are in English, under the form of notes, and their aim is:

The publication of the "Œuvres" has been well received, both in France and elsewhere. The reports by S. Mac Lane and J.W. Gray (in the Mathematical Reviews and the Zentralblatt für Mathematik) insist on the interest of the comments and synopsis which form a kind of history of the development of categories from 1960 to 1980. For instance Gray qualifies them of "remarkable addition to the literature of category theory" (Zentralblatt. f. Math. 452, p. 9).

 

The preparation of these comments has led to some new results completing some of the papers, such as:

 

When I had finished the commented edition of Charles' works, I felt the need of thinking on different problems where I could use my previous experience. It was prompted by discussions with Jean-Paul Vanbremeersch, a physician who was interested in explaining the complex responses of organisms to illness or senescence. When I organized the 1980 international conference on category theory in memory of Charles, I asked him for assistance in writing an explanation of category theory for non-mathematicians. It was during these initial interactions that he first suggested that categories might have applications for problems related to complexity. And this is how our study of memory evolutive systems began.

It necessitated a 3 years period to acquire some knowledge in biology and neuroscience. The examination of the literature revealed that there had not yet been any real work done on this subject. Although Robert Rosen [Ro] had promoted the use of category theory in biology, he considered only its basic notions and not its more powerful constructions.  Hence, we decided to try using category for modeling natural complex systems such as living systems and social or cognitive systems. It was the beginning of a 25 years collaboration, of which the main results are exposed in our book "Memory Evolutive Systems: Emergence, Hierarchy, Cognition" (Elsevier  2007). The journal Axiomathes (Springer) has devoted his volume 19-3 (Sep. 2009) to reviews on this book, with some comments of us on them [81].

Here I'll just give a rough idea of the model, referring to our site on MES for a summary of this work and to our various articles on MES reproduced in Articles.

 

A MES proposes a model for a living system characterized by the following properties:

The configuration of the system at a given time t of its timescale is represented by a category K(t) whose objects represent the states of the components ot the system existing at t, the morphisms being their interactions around t. The change from t to t' is modeled by a transition functor from a sub-category of K(t) to K(t'); these transitions satisfy the transitivity axiom for semi-sheaves. A complex object is modeled by the colimit of a pattern (or diagram) which represents its internal organization. The categories are hierarchical, meaning that the objects are divided into several levels 'of complexity', an object of a given level being the colimit of at least one pattern of objects of strictly lower levels. The transitions result from a complexification process [41], adding or suppressing some (co)limits (adapted from the completion theorems of [19]).

An important result is a characterization of emergence: successive complexification processes of a category K lead to the emergence of components of strictly increasing complexity order if and only if K satisfies the Multiplicity Principle [57]: there are objects C which are the colimit of two patterns of strictly lower levels such that the identity of C is not the gluing of an cluster between them.

 

Since 1999, we develop the model MENS for cognitive systems [72, 77, 78]. It is a MES based on the evolutive system Neur modeling the neuronal system of an animal. Its components, called category-neurons, model classes of synchronous hyper-assemblies of neurons. We show how more and more complex cat-neurons can emerge, representing mental objects or higher cognitive processes. For higher animals, we describe the formation of a semantic memory, leading to the extension of the archetypal core, a particularly intricate sub-system of the memory (whose anatomical basis has been recently found by neuroscientists [HC]). The archetypal core represents a personal memory at the basis of the self and it is is essential for the development of conscious processes which we characin extension of the temporal landscape through a retrospection and a prospection process [60, 64, 77, 87, 96]. Quantifications are added to MENS in [82, 97] by showing how to extend the propagation delays and strengths of synapses to the morphisms between cat-neurons of any level.

In [79, 80, 83, 94] .we give an application of MES to the study of the complexity in Art, looked at under 3 aspects: 1. Intrinsic complexity of an artwork in itself; 2. Complexity of the creation of the work, studied in the MENS of the artist, accounting for the interactions between his sensibility, experience and the artistic and social context; 3. Complexity of the diffusion of the artwork, studied in the MES of the society, possibly leading to the emergence of a new artistic current.

In [85, 96, 102, 103, 104] MES are applied to anticipation and creativity problems.

 

By composition with R, D generates a K-presheaf of distructures D*. This presheaf is explicitly constructed in the case where K is the category Lcs of locally convex vector spaces and M consists of in inclusions of a vector sub-space equipped with a finer locally convex topology.

 

This construction is applied to define the E'-valued distributions on E, where E and E' are two locally convex spaces. For that, D is defined as follows; H is the category of opens U of E, G(U) is the free commutative monoid generated by E, and D(U) is the (Lcs, M)-semi-sheaf associated to the partial action of G on the locally convex space of continuous maps from U to E':: (u, g) -> partial derivative of g with respect to u, if it exists. Then D generates the Lcs-presheaf D* of finite order E'-distributions on E, and the sheaf D" associated to D* is the sheaf of E'-distributions on E. If E is metrizable and E' complete, a main theorem constructs a finite order E'-distribution as an equivalence class of pairs (?, g), and an E'-distribution as a compatible class of finite order E'-distributions. If E is finite dimensional, these distributions are in 1-1-correspondence with the Schwartz E'-valued distributions.

[B] Bellman, R. (1962), Adaptive control processes, Prineton Univ. Press.

[BK] Binz, E & Keller, H.H. (1966), Funktionenraume in der Kategorie der Limesräume, Ann. Acad. Sci. Penn. A, 383, 1–21.

[Bu] Bushaw, D. (1963), Dynamical polysystems and optimization, RIAS Report 63-10.

[CE] Ehresmann, C. (1980-83), Charles Ehresmann: Œuvres complètes et commentées, Amiens.

[FB] Frölicher, A. and Bucher, W. (1966), Calculus in vector spaces without norm, Lecture Notes in Math. 30, Springer.

[GL] Guitart, R. & Lair, C. (1980), Calcul syntaxique des modèles et calcul des formules internes, iagrammes, 4.

[HC] Hagmann, P., Cammoun, L., Gigandet, X., Meuli, R., Honey, C.J., Van J. Wedeen and Sporns, O., 2008, Mapping the Structural Core of Human Cerebral Cortex, PLoS Biology 6, Issue 7, 1479-1493.

[Ko] Kowalsky, H.J. (1954), Limesräume und Komplettierung, Math. Nachr. 12, 301-340.

[MS] Mikusinski, J. & Sikorski, R. (1961), The elementary definition of distributions, Rozprawy Mat. Varsovie, 1-46.

[P] Pontrjagin, Boltyanskii, Gamkrelidze & Mischenko (1962), The mathematical theory of optimal processes, Int. Sci. Pub. New York.

[R] Rosenbloom, P. (1951), Quelques classes de problèmes extrémaux, Bull. SMF 79, 1-58.

[Ro] Rosen, R., 1958, The representation of biological systems from the standpoint of the theory of categories, Bull. Math. Biophys. 20, 245-260.

[Rx] Roxin, O. (1962), Axiomatic theory of control systems, RIAS Report 62-16.

[S1] Schwartz, L. (1950), Théorie des distributions, tomes 1 and 2, Hermann, Paris.

[S2] Schwartz, L. (1957), Théorie des distributions à valeurs vectorielles 1, Ann. Inst. Fourier VII, 1-141.