Title: Charles Ehresmann

Author: Andrée C. Ehresmann

Characters: Charles Ehresmann

Dates: 1905-1979

This is the text of a talk given by Andrée C. Ehresmann at the Category meeting of Arnsberg, on October 27, 1979. It has been published in *Seminarberichte* 7 (FernUniversität Hagen), and reprinted in

"*Charles Ehresmann: Œuvres complètes et commentées*" Part I

(abbreviated in CE I). Slight corrections have been made, in particular references to Charles' works have been simplified, just indicating in which Part of the "Oeuvres" the papers are reprinted.

CHARLES EHRESMANN

by Andrée C. EHRESMANN

I have been deeply moved by your idea of dedicating this meeting to my husband. Please excuse me if this talk is somewhat informal, and if my english (which has never been very good) is particularly awful. But my husband died only one month ago, after a year long illness during which he needed incessant care. In any case, it would not have been possible for me to speak objectively about Charles, since we have been so closely related for 22 years.

Hence this talk will be mixing Mathematics and more personal recollections; but this is not too inadequate in a Category meeting, our life having been completely devoted to Mathematics, and more precisely to Category Theory and its Applications. Even when he was very ill, Charles often repeated that he wanted to do mathematics; he still said it during the two days of wakefulness which preceded his death.

§ l. His liking of Travels

The first thing that struck me when we met in 1957 was his marvelous blue eyes which, up to the end, kept their childish look. These eyes twinkled particularly when he spoke of his travels for mathematical purposes (we almost never took vacations, the last one being in 1966 when we toured the United States in Greyhound buses, talking about Mathematics most of the time).

For his first 'mathematical' journey, he spent some months at Göttingen in 1929-30, to study with Hermann Weyl. His thesis was written in Princeton, where he was a Proctor Visiting Fellow from 1932 to 1934.

In the fifties, he was proud of his about half year long lecturing in foreign universities ; most of the papers written at that time begin with a long list of towns in which he had given talks on the subject, and I gently teased him for that. He was very attracted by the oriental thinking he had just discovered in India and Iran; for him the most beautiful monument was the Tadj Mahal and (later on, in 1972) we converted ourselves to vegetarianism. He hoped to have an opportunity to visit China.

Up to 1966, we had some other long trips in America. Afterwards we preferred to stay in Paris or in Amiens partially because the "*Cahiers de Topologie et Géométrie Différentielle***" **took more of our time (they became a quarterly Journal in 1967).

But he liked to dream in front of maps, and one of the last times we went out we were looking for a new atlas. In fact, the only non-mathematical books he perused in his last years were atlases and the "*Annuaire de l'Association des anciens éléves de l'Ecole Normale Supérieure***" **(he remembered his years at the Ecole Normale Supérieure d'Ulm with a great fondness, and our last travel to Paris in May 1978 was for the yearly 'pot de la promotion').

§ 2. His conception of Mathematics

Though always serene (I almost never saw him out af temper), he had a communicative passion for Mathematics. His lectures (generally informal) were often followed by long discussions which inspired a great number of mathematicians.

We had many talks about the essence of Mathematics, especially in Kansas in 1966 where the paper "Trends toward unity in Mathematics" (CE III) was written. He said that I was a Platonist since I think (*) that the motivation for research work is the quest for the pre-existing idea (in Plato's sense) of a structure; while for him discovery was an entirely free creation, the value of which lies in its possible expansion.

We both agreed that Mathematics is an Art as well as a Science (perhaps more than a Science). And this thought was always his: he already exposed it to his pupils in the year he spent at Rabat as a secondary-level teacher in 1928-29; he was reminded of this by one of these pupils met fortuitly in a bus in Paris some 40 years later! (He had asked to go to Rabat to 'take a vacation' before undertaking research work, as he explained to me).

Charles was also interested in the History of Mathematics. We wrote a paper "Archiméde et la Science moderne" (CE III) for the commemoration of Archimedes' 2.000^{th} anniversary in Syracuse. One of the Greek mathematicians he most admired was Eudoxus, for his theory of ratios so similar to the Dedekind's definition of real numbers. From 1975 to 1978 he gave a course on the history and foundations of Mathematics in Amiens, in which he tried to show how the deep structure of the notions was gradually revealed. His last lecture was on December 4, 1978, the day before he fell on the street and could not deny his illness any more.

§ 3. His former works

On.our first rendez-vous in 1957, he introduced me to his works on Topology and Differential Geometry (he was then considered as one of the best geometers).

His thesis on the topology of homogeneous spaces (in 1934; CE I) was prepared under the direction of Elie Cartan, for whom he had a great admiration. Very modestly, he often said that many of his ideas were implicit in Cartan's works (but I could not find these ideas in the papers he showed me).

In Clermont-Ferrand during the war he introduced fibre bundles apart from Steenrod (the communications between France and the U.S.A. being then broken), and he used them in the early fifties to develop a beautiful theory of jet bundles, prolongations of manifolds and higher order connections (CE I). He also defined foliated manifolds and more general foliations (CE I).

While we were in Montréal in August 1961, a long paper "Structures feuilletées" (CE II) was written, full of important results (e. g. stability theorems for topological foliations, holonomy groupoids, 'unspreadings' of a foliation, a theorem on transverse foliations,...), but this paper is often ignored by the specialists, since it was not reviewed in the "*Mathematical Reviews*".

Charles was also eager to learn new things and I had no difficulty in convincing him of the beauty of the geometry of topological linear spaces and of infinite-dimensional polyhedrons on which I was preparing a thesis under the direction of G. Choquet. In memory of these first discussions, there is an icosahedron on our grave (besides the sketch of categories).

§ 4. How Charles came to categories

Long before knowing category theory, he had used groupoids, i. e., categories in which all the morphisms are invertible. (Oddly enough, groupoids, which were defined by Brandt in 1926 (*Math. Annalen*. 96), are often called Ehresmann's groupoids; and this somewhat irritated us since so many notions introduced by my husband are attributed to others or considered as 'universal knowledge' (as jets).) Indeed, groupoids intervene in fibre bundle theory in two different ways:

A. *Actions of a topological groupoid. *Denote by E a fibre bundle. The isomorphisms from fibre to fibre form a groupoid, which is equipped with a topology compatible with the maps domain, codomain, composition and inversion. This gives a topological groupoid (in the sense: internal groupoid in the category Top of topological spaces), which acts continuously on the topological space E. This topological groupoid G satisfies the axiom:

(LT) For each object, say *x*, of G (identified with a point of the base B of E ), there exists a local section *s*: U -> G of the codomain map on a neighborhood U of *x* such that *s(y): x* -> *y* for each *y* in U.

Conversely, to a topological groupoid satisfying (LT) (called a *locally trivial groupoid*) naturally corresponds a principal fibre bundle, and to its actions, the associated fibre bundles. This defines an equivalence from the category of fibre bundles to the category of actions of locally trivial groupoids. In this setting, connections, prolongations of manifolds,... are very easily defined.

More generally, the jets between all germs of manifolds form a (big) differentiable category (i.e., internal category in the category Diff of differentiable maps) and Charles described Differential Geometry as the study of this category and of the actions of its subcategories.

This 'categorical' point of view is indicated in a series of very concise papers from 1958 to 1969 (CE I). Charles always thought of writing a book on this subject, and he regretted to have spent so much time in Bourbaki's team in the forties instead of developing his own ideas.

B. *Local structures. *Fibre bundles may also be defined by atlases gluing together products, the transition functions between charts being compatible with the action of the structural group on the fibres. To unify the treatment of structures defined by such a 'gluing' process (as are topological or differentiable, analytic, foliated manifolds,...), Charles introduced the notion of *local structures*, which are those structures defined by an atlas compatible with a pseudogroup of transformations (i. e., a subgroupoid of the groupoid of homeomorphisms of a topological space whose set of morphisms, equipped with the 'restriction' order, satisfies conditions turning it into a 'local groupoid').

Improving his earlier results, in 1957 he wrote the paper "Gattungen von lokalen Strukturen" (CE III). During the correction of its proofs in 1958, I learnt the fundamentals of Category Theory... and also a little German. (Charles, whose first language was German, always discouraged me from studying Goethe's language saying that he could translate it for me and that it would be more useful to learn something he did not know.)

This paper was the starting point of many of the subsequent papers on discrete fibrations and extensions of functors, on ordered categories and completion theorems. Perhaps it will be clearer to see how all these questions occur on an example.

To define, say, differentiable manifolds, the problem is decomposed as follows : Let P: Diff_{-B} -> Top be the forgetful functor from the groupoid of diffeomorphisms between open subsets of a Banach space B ; it is a discrete fibration (defined by a 'species of structures') over a subgroupoid of the groupoid Top_{-g} of all homeomorphisms.

1° *Enlargement of the species of structures*: P is 'universally' extended into a discrete fibration P': D ->Top_{-g}, which is defined 'pointwise'. In modern terms P' is the discrete fibration associated to the Kan extension, along the insertion toward Top_{-g} of the Set-valued functor associated to the discrete fibration P.

2°* Local completion of a local functor*: On D there is an order deduced from the 'restriction order' on Diff_{-B}. With this order, D becomes a local groupoid (i. e., essentially, an internal groupoid in the category of distributive complete glb-lattices). However D is not (order-)complete over Top, in the sense that there exist P'-glb-compatible families of objects of D (i. e., P' preserves the glb for pairs of objects of the family) which have no lowest upper bound in D. The process of *local completion* consists in adding such lub. It leads from P' to the forgetful functor from the groupoid of diffeomorphisms between manifolds modeled on a Banach space. In fact, this step amounts to construct, over each topological space T, the sheaf associated to the presheaf determined by P' over T.

3° *Extension of a functor*: To get the category of differentiable manifolds, an analogous completion process is applied to the category of differentiable maps between the objects of Diff_{-B}. This construction is well described in several papers (CE II).

§ 5. Structured categories

As Charles came to categories from groupoids and to groupoids from groups, he 'felt' a category as a (small) set, equipped with a partially defined composition, rather than as a (big) class of sets Hom(E,E') (which is more usual when categories of structures are first considered). Hence it seems natural to equip the 'set' of morphisms of a category with some kind of structure, compatible with the maps domain, codomain and composition, as in the two preceding examples (a topology in the case A, a 'Iocal order' in the case B).

ln 1963 these reflections led to the definition of a P-*structured category,* where P : H -> Set is a forgetful functor. ln modern terms, it is an internal category in H ; but initially we thought of it as the data of a category C and of an object S of H sent by P to the set of morphisms of C in such a way as :

(i) The maps domain and codomain 'lift' into morphisms from S to a subobject of S; and (ii) The map composition 'lifts' into a morphism toward S from a P-subobject of the product SxS.

Hence the necessity of defining P-subobjects (CE III). (Oddly enough, it is the dual notion of quotient object which led us to re-discover the notion of a reflection of a category into a subcategory and, more generally, to the notion of a free object, via the construction of a comma category; this did not simplify the exposition of several papers where adjoint functors are used in this way!)

We marveled at the number of important examples which were unified :

1° *Topological categories*, already introduced in 1958, whose general theory is developed in a 1964 paper (CE II); in it classical results on the uniform structure of topological groups are adapted to 'microtransitive categories' (using quasi-uniform structures, which are a localization of uniform spaces), and prolongations of (quasi-)topological categories are constructed.

2° *Differentiable categories* (see § 4).

3° *Double categories* (internal categories in Cat), whose first example was the 2-category of natural transformations (already introduced in the paper "Catégories de foncteurs types" written while we were in Buenos-Aires for a four months stay in 1959), the second one being the category of commutative squares of a category A (or 'category of morphisms' of A ). Notice that, in our latest paper (CE IV), written during the summer of 1978, while Charles was already in poor health, we prove that all double categories 'are' subcategories of the double category of squares of a 2-category.

4° *Multiple categories*, defined in 1963 (CE III), whose theory is developed in our last series of papers in 1978-79 (CE IV) and for which theorems of existence of 'lax limits' (very conveniently defined in this frame) are proved by a short 'structural' method.

5° *Ordered categories*, and their specialization such as *local categories*; are considered in numerous papers (700 pages odd) that should be read after the "Guide sur les catégories ordonnées" (CE II). Among the main results, still to be exploited are those concerning local jets and atlases which are used to get (order)completion theorems for ordered categories (e, g., to construct the complete holonomy groupoid of a foliation) and for ordered functors. They culminate in a theorem of complete enlargement of a local functor, which generalizes the associated sheaf construction.

General theorems an structured categories are given in a series of papers from 1963 to 1969 (CE III et IV). For instance, to study the existence of colimits, we thought of constructing them as quotients of coproducts. But such quotients are scarce, even in Cat, hence the idea of defining *quasi-quotient objects* (CE III). Fine constructions of quasi-quotient structured categories are made in 1965 papers (CE III).

But in general only existence theorems may be obtained, and this led to develop theorems on existence of free objects (CE III et IV) in which the functor is 'extended to a higher universe' instead of 'restricted' like in the solution set condition of Freyd's Theorems.

These general theorems on P-structured categories require that P satisfies some good conditions, such as creation of some kinds of limits or colimits, existence of quasi-quotients, existence of a smallest subobject S' of an object S of H such that P(S') contains a given subset A of P(S) (sub-generating functors), or such that P(S') = A (called sub-spreading functors). Hence arose the questions :

1° Is it possible to define classes of functors by some properties, e. g., functors 'of a topological type' (= sub-spreading functors admitting limits = topological functors in Herrlich's sense), or 'of an algebraic type' (= sub-generating functors admitting quasi-quotients); the word 'algebraic' means here 'partially defined compositions' (and not everywhere defined compositions, as in monadic functors)? We thought that one of the future tasks of the mathematician would be to study such classes (as explained in "Trends toward unity in Mathematics"), and this prevision seems right enough if for instance we look at the recent works of the German school.

2° If a functor is not 'good enough', may it be extended universally in a good enough functor? The motivating example was the forgetful functor from Diff, which does not create kernels. This problem is tackled in 1967 and 1969 papers (CE IV), where universal completions of functors are constructed by transfinite induction. here again the German school seems to carry on.

§ 6. Sketches and sketched structures

The *idea of a category* consists in the graph formed by its maps

domain = a: C -> |C|, codomain = b, and composition = k: C_{*}C -> C ;

and such a graph in Set determines a category if it satisfies axioms expressing associativity, unitarity and the fact that C_{*}C is the pullback of (a,b). This remark led to 'isolate' the sketch of categories, whose underlying category is the full subcategory S_{Cat} of the simplicial category Simp with objects 0, 1, 2 and 3.

The realizations in a category H of this sketch are the internal categories in H (he initially called them generalized structured categories). Categories of internal categories are studied in different papers, specially in our long 1972 paper (CE III).

An internal category in H gives rise to a category object in Grothendieck's sense, but the inverse is valid only if H admits pullbacks. So arises the question: may we universally add pullbacks to a category H, so that both notions coincide? To answer it, theorems of completion of categories by some types of limits or colimits were devised in 1967. The completions were asked to be 'universal up to isomorphisms' for some choice of limits; the astonishing result is that they are also 'universal up to equivalences' for all limits of the given type (CE III). Later on (in our 1972 paper) we extended this result, replacing the choice of limits by a 'relational choice' (and then explicit constructions of the completion are necessary, for the existence theorems cannot be applied).

More generally, 'algebraic structures' may be sketched by the data of a neocategory (graph, equipped with some partial composition) and of cones on it; the corresponding structures (resp. internal structures in a category H) are the functors from the neocategory into Set (resp. into H) sending the cones onto limit-cones; this *theory of sketches *is developed in papers from 1966 on (CE IV). The interest of taking a neocategory with cones instead of a category with limit-cones (for instance, S_{Cat}, instead of Simp) is to get a 'finitely-presented' model. Completion theorems lead from the sketch to the prototype (category with limit-cones) and to the type, which is the 'complete' model (no more finitely presented).

Though categories of sketched structures are Iocally-presentable in the sense of Gabriel-Ulmer (whose work was published later on), the theory of sketches by itself is fruitful in many problems (**). General theorems on sketched structures may be found in (CE IV). Several theses prepared in our research team "Théorie et Application des Catégories" (Paris-Amiens) in the seventies are concerned with the theory of sketches.

§ 7. Discrete fibrations and enriched categories

Motivated by fibre bundle theory where both topological actions of a locally trivial groupoid and local discrete fibrations are considered (cf. § 4), Charles defined in 1957 the notion of a category C acting on set S, the points of which are called structures, hence the name *species of structures* over C ; and he proved the equivalence between the notions: species of structures, discrete fibrations (initially called 'foncteurs d'hypermorphismes'), functors toward Set.

Actions of categories may be defined by a sketch, so that internal actions in a category H are well-defined. They correspond to internal discrete fibrations in H (while the notion of a set-valued functor cannot be internalized). They are considered in several papers in the sixties (CE III), in particular topological or differentiable species of structures, and ordered species of structures; but a general theory is not written yet, though we knew several results on them. In 1963-64, I generalized topological fibrations into germs of fibrations for use in optimization problems.

To solve the problem of enlargement of a species of structures (mentioned in § 4 B), Charles did not extend the corresponding Set-valued functor (as in the more recent Kan extension theorems), but he took the situation 'upside-down' and extended the discrete fibration into another one. Then it is natural to replace the discrete fibration by any functor, and this led to the general theorems on extensions of functors (CE II and III), to which is devoted the fifth chapter of the book "*Catégories et structures*" (Dunod, 1965). These theorems also encompass the construction of categories of fractions (made for the first time, under the name of 'perfectionnement d'une catégorie' during our stay in São Paulo in 1960). These extension theorems have been later internalized (CE III).

Theorems on extensions of functors and on local completion of local functors were devised in such a way that I may apply them in Analysis to define distructures, which unify various concepts of 'generalized functions' (Schwartz distributions, Sata hyperfunctions, Mikusinski operators,...) and give for instance 'infinite-dimensional distributions' (A. Bastiani(-Ehresmann), "Systémes guidables et problèmes d'optimisation", Labo. Automatique Théorique, Univ. Caen, I à IV, 1963-64). The idea is to dissect the local definition of a distribution namely: the sheaf of distributions is the sheaf associated to the presheaf of 'formal' derivatives of continuous functions. This led to consider :

1° *Enriched species of structures*, called 'espéces de structures dominées' (CE III) which are simply functors taking their values into a concrete category, and more particularly :

2° *Species of morphisms* (or functors toward Cat), to which the notions of crossed product of a group acting on a group and of crossed homomorphisms were adapted in 1966, giving rise to the (split) fibration associated to a functor toward Cat, and to its first cohomology class, and then to a general theory of non-abelian cohomology (CE III);

3° Enriched species of structures in the category of discrete fibrations and the special case in which all the fibrations have the same base which gives a *pair of acting categories*; this notion is equivalent to that of a Benabou's distributor; when distributors are so considered as 'double fibrations', their composition is similar to the composition of atlases of a category (CE III);

4° *Partial actions of categories*, called systems of structures, enriched in a concrete category H, i. e., the fibres of the associated 'partial fibration' are equipped with objects of H (e.g., for distructures, take H = the category of Banach spaces). A theorem on extension of an enriched system of structures into an enriched species of structures is given in 1966 (it is used to define the analogue of 'formal derivatives').

An application of the notion of an enriched species of structures gives a special case of enriched categories: Let C be a category, C°xC acts on the set of morphisms of C (where C° is the opposite category), the corresponding Set-valued functor being Hom. To say that the action is enriched in a category H means that the sets Hom(E,E') are 'naturally' equipped with objects of H; this gives a way to add structures on a category, more adapted for 'big' categories. Examples also occur in Analysis (e. g. enriched categories in the category of Banach spaces).

However, we did not come across the notions of a closed category V, and of a V-category, on our own, because we only thought of enriched categories in a concrete category. When we discovered Eilenberg-Kelly's paper on closed categories (*Proc. Conf, on Categ. Algebra,* La Jolla, 1966), we extensively used monoidal closed categories.

For instance, we constructed monoidal closed structures on general categories of sketched structures, on categories of internal categories, on categories of topological ringoids, on the category of all multiple categories and on the category of n-fold categories (CE IV).

§ 8. Some critical comments

It may seem weird that the 2.000 pages odd written in the sixties on Category Theory are almost unknown, although some of them are still quite original. But this is understandable: during this period, we had few relations with categoricians, because Charles and I were essentially considered respectively as a Geometer and an Analyst (and anyway I was completely isolated from the world up to 1968). So we did not realize that most papers are difficult because of their abstraction, their heavy and unusual notations, their evolving terminology and their length (e. g., the important completion theorems of "Sur l'existence de structures libres et de foncteurs adjoints" (CE IV) carne after a first part devoted to very technical results). Moreover, we ignored notions that would have simplified the redaction, for instance adjoint functors are introduced as a complicated thing, general limits are used very late, and all the constructions are given 'pointwise' because we had difficulties in seeing global results such as the Yoneda Lemma.

The book "*Catégories et Structures*" also badly influenced people against Charles: it was received as an unsuccessful general treatment of Category Theory, while in our mind it had to show his personal 'feeling' of Category Theory. Hence the deliberate omission of fundamental notions such as general limits, adjoints in the usual way,... (given in the Appendices). Also it includes very elementary parts besides deeper results (such as the ends of Chapter 2 and 3, and the Chapter 5) ; it comes from the fact that some parts were taken from mimeographed courses, and others from research papers. The aim of the book "Álgèbre" (no more available) is much clearer.

From 1972 on, we gradually adopted a more usual terminology, and yet only our last series of papers is about standard on this point. Notice that we wrote less papers from 1970 to 1977; we had too many occupations then: teaching and administrative duties, direction of too big a research team (about 50 theses defended in 8 years!), entire publication of the "*Cahiers de Topologie et Géométrie Différentielle*", organization of several international meetings. So we could do personal research only during the summer vacation. In 1977, for different reasons (in particular the refusal of a '3ème cycle' in Amiens), we found time enough for a true research work again.

To be sure that my husband's work will not be forgotten, I am going to publish a complete commented edition of Charles' work (***). The comments will explain not only how the ideas evolved, but how the use of modern tools allows to go further on these subjects. Indeed, Charles thought that the future is more important than the past, and he often said to me that it comforted him to think his work would be pursued after his death, thanks to the 30 years of age between us.

Notes added in 2000:

(*) My philosophy has now changed, and become nearer from Charles' ideas.

(**) The category community was somewhat doubtful about the interest of sketch theory when I wrote this paper in 1979. The situation has much changed since then!

(***) This program has been fulfilled. The 4 parts (7 volumes) of "*Charles Ehresmann: Œuvres complètes et commentées*" have been published between 1980 and 1983 as Supplements to the "*Cahiers de Top. et Géom. Diff.*".