Seminarberichte Nr. 7

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Mathematik und

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Seminarberichte aus dem Fachbereich Mathematik der FernUniversität

07 – 1980

der Mathematik (Hrsg.)

Seminarbericht Nr. 7

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CHARLES EHRESMANN

19.4.1905 - 22.9.1979

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Preface

At the meeting of the "Nordwestdeutsches Kategorienseminar"

at Arnsberg from 0ctober 26 to 0ctober 28, 1979, the

participants were shocked by the sad news of the death of Charles Ehresmann. Despite the great loss she had just suffered, Hadame Ehresmann came to Arnsberg and held an

impressive lecture on the life and work of Charles Ehresmann, which is published in this volume. After this lecture,

Anders Kock expressed the unanimous wish of the participants

of this seminar to dedicate i t to the memory of Charles Ehresma~n.

List of participants Jiri Adamek

Fel CVUT

Suchbaterova 2, 16627 Praha 6/CSSR Harald Brandenburg

Institut für Mathematik, Freie Universität Berlin Hüttenweg 9, 1000 Berlin 33

Reinhard Börger

Fachbereich Mathematik, Fernuniversität Postfach 940, 5800 Hagen

Hans-Berndt Brinkmann

Fakultät für Mathematik, Universität Konstanz Postfach 5560, 7750 Konstanz

Andree Charles Ehresmann U.E.R. de Mathematiques

33 rue Saint-Leu, 80039 Amiens Cedex. France Detlev Franke

Parchimer Allee 45, 1000 Berlin 47 Alfred Fröhlicher

Institut de Mathematique

Universite Geneve, Geneve/Schweiz Georg Greve

Rene Guitart

Dep. of Math., University Paris 7 Toms 45/55, 5e etage

2, place Jussien, 75005 Paris Roswitha Harting

Math. Institut II, Universität Düsseldorf Universitätsstr. 1, 4000 Düsseldorf

N. Heldermann

Zentralblatt für Mathematik

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Horst Herrlich

Fachbereich Mathematik, Universität Bremen Kufsteinerstr., 2800 Bremen

Burkhard Hoffmann

Rudolf-Eberhard Hoffmann

Fachbereich Mathematik, Universität Bremen Kufsteinerstr., 2800 Bremen

E. Hotzel

Gesellschaft für Mathematik und Datenverarbeitung Postfach 1240, 5205 St. Augustin

Peter Johnstone

Dept. of Pure Ma.thematics, University of Cambridge 16 Mill Lane, Cambridge CB2 15B, England

Klaus-Heiner Kamps

Anders Kock

Mathematisk Institut Aarhus Universitet DK-8000 Aarhus Harald Lindner

Jean Marie McDill

California Polytechnic State University San Luis Obispo California 93407

Axel Möbus

Christopher J. Mulvey

Mathematics Division, University of Sussex Falmer, Brighton, England BNI 9QH

Hans-E. Porst

FB Mathematik, Universität Bremen Postfach 330 440, 2800 Bremen Dieter Pumplün

Günther Richter

Fakultät für Mathematik, Universität Bielefeld Postfach 8640, 4800 Bielefeld

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Michael M. Richter

Angewandte Math. und Informatik, RWTH Aachen Templergraben, 5100 Aachen

Jacques Riguet

Universite Rene Descartes

10 rue Jeanne d'Arc, 75013 Paris Horst Schubert

Math. Institut II, Universität Düsseldorf Universitätsstr. 1, 4000 Düsseldorf 1 Friedhelm Schwarz

Institut für Mathematik, Universität Hannover Welfengarten 1, 3000 Hannover 1

Dietmar Schweigert

FB Mathematik, Universität Kaiserslautern 6750 Kaiserslautern

Christian D. Spoerel

Fachbereich ET, Hochschule der Bundeswehr Hru~burg Postfach 700 822, 2000 Hamburg 70

Walter Sydow

Thomas Thode

Mathematisches Institut II, Universität Düsseldorf Universitätsstr. 1, 4000 Düsseldorf

Walter Tholen

Helmut Weberpals

Mathematisches Institut II, Universität Düsseldorf Universitätsstr. 1, 4000 Düsseldorf

Olaf Zurth

Institut für Mathematik I, Freie Universität Berlin Hüttenweg 9, 1000 Berlin 33

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Inhaltsverzeichnis Charles Ehresmann

von Andree Charles Ehresmann Liste des publications de Charles Ehresmann

von Andree Charles Ehresmann

Constructions de produits tensoriels et machines non-deterministes

von Rene Guitart

On exact orbit sequences von Philip R. Heath

und Klaus-Heiner Kamps Factorization theorems in topology and topos theory von Peter T. Johnstone

Formal manifolds and synthetic theory of jet bundles

von Anders Kock A non-commutative

Gelfand-Naimark theorem von Christopher J. Mulvey Preservation of coproducts by set-valued functors von Reinhard Börger

An extension theorem for monoidal closed topological categories

von Georg Greve

Lokal injektive Abelsche Gruppen und internes Coprodukt Abelscher Gruppen

1 7

25

31

37

55

85

91

107

von Roswitha Harting 121

An injective characterization

of compact metrizable spaces and Peano spaces

von Burkhard Hoffmann . . . 131

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Essentially complete T -spaces von Rudolf-E. Hoffmann°

Center and trace von Harald Lindner

Initial morphisms and monomorphisms von Dieter Pumplün

Varietal hulls of functors von Günther Richter

Syntaktische Kategorien und Wortprobleme

von Michael M. Richter

139

149

183

217

249

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by Andree CHARLES EHR:.,Sf1AHN

I have been deeply rnoved by your idea of dedicating this rneeting to rny husband. Please excuse rne if this talk is sornewhat informal, and if rny english

(which has never been very good) is particularly awful. Eut rny husband died only one rnonth ago, after a year long illness during which he needed incessant care.

In any case, it would not have been possible for rne to speak objectively about Charles, since we have been so closely related for so rnany years •••

Hence this talk will be melting Hathernatics and rnore personal recollections but this is not too inadequate in a Category rneeting, our life having been corn- pletely devoted to Mathernatics, and rnore precisely to Category Theory and its Applications. Even as he was very ill, Charles often repeated that he wanted to do Hathernatics; he still said it during the two days of wakefulness which pre- ceded his death.

1. His liking of travels.

The first thing that striked rne when we rnet in 1957 was his rnarvelous blue eyes which, up to the end, kept their childish looL These eyes twinkled particularly when he spoke of his travels for rnathernatical 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 rnonths in Goettingen, to study with Hermann Weyl. His thesis was written in Princeton, where he was Procter Visiting Fellow frot1 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 1) This is the text of the talk given by A.C. Ehresmann at the Category meeting of Arnsberg, on October 27, 1979.

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list of tor.,ms in which he lrnd given talks on the sutject, and I s;ently teased him for that. Ge was very attracted by the ori~ntal thinking he had just discovered in India and Iran; for lüm the most beautiful monument was the Tadj-:fahall 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 AIDerica. Afterwards we prefered to stay in :?aris or in Amiens, partially because the "Cahiers de Topologie et Geometrie Differentielle" took more of our time (they became a quarterly Journal in 1967, as explained in [ 139] 2

)). 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 eleves de l'Ecole Hormale Superieure"

(he remembered his years at the Ecole Normale Superieure 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 almest never saw him out of temper), he had a communicative passion for Mathematics. Eis 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'' [ 94] was written.

He said that I was a platonist since I think that the motivation for research work is the quest of the pre-existent 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 2) The numbers between brackets refer to the "Liste des publications de Charles Ehresmann", Cahiers Topo. et Geo. Diff. XX-3 (1979), 221-230.

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Science (perhaps more than a Science). And this thought was always his; he already exposed it to his pupils in the year te spent at Rabat as a secondary-level teacher in 1925::,-29; he was reminded of this by one of these pupils n,et for- tuitly in a bus in Paris some 40 years later ! (he had asked to go to RaLat to

"take a vacation¹¹ before undertaking research work, as he explained to me.)

Charles was also interested in the History of Hathematics. He wrote a paper

"Archimede et la Science moderne" [56] for the connnemoration of the Archimedes¹s

th . . S f

2000 anniversary in yracuse, One o the Greek mathematicians he most adrnired 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 profound 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 longer.

3, His forrner works.

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

Eis thesis [ 4,8]on the topology of homogeneous spaces (in 1934) was pre- pared 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 Elie 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 [ 14-18, 20]

apart from Steenrod (the counnunications 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 [ 22-24, 26- 38, 40-44, 46

1 48, 51}. He also defined foliated manifolds [ 19, 21, 30] and more general foliations [ 45] • Hhile we were in Montreal in August 1961, a long paper [ 541 was written, full of important results (e.g., stability theorems for

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topological foliations, holonomy groupoids, ¹¹unspreadings" of a foliation, a theorern on transverse foliations, ••• ), but this paper is often ignored oy the specialists, since it was not reviewed in the "Hatherrtatical Eeviews".

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 [ 49 1 on which I was preparing a thesis 3) under the direction of G. Choquet. In memory of these first discussions, there will be an icosahedron on our grave (as well as the sketch of categories).

4. How Criarles cam.e to categories,

Long before knowing category theory, he had to use grr,upoids (i.e. • cat- egories in which all the morphisms are invertible). Oddly enough, groupoids, which were defined by Brandt in 1926 4

) are often called Ehresmann's groupoids;

and this somewhat irritated us since so many important notions introduced by my husband are attributed to others or considered as "universal knowleoge"

(as j ets).

Indeed, groupoids intervene in fibre bundles theory in two different ways:

A) Actionsofa topological groupoid. Let E be a fibre bundle. The isomorphisms from fibre to fibre forma 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 total space E. This topological groupoid G satisfies the axiom:

(LT) For each object 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 neigh- borhood U of x such that s(y): x • y for each y in U.

3) "Polyedres convexes dans les espaces vectoriels topologiques'', Cahiers Topo, et Geo. Diff. 1 (1957-58),

4) "Ueber eine Verallgemeinerung des Gruppenbegriffes", Math. Ann. 96 (1926), 360,

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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 j ets 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 [ 116

J.

This "categorical" point of view is indicated in a series of very concise papers from 1958 to 1969 [ 50, 78, 101, 103, 105, 111 ] • 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 fourties, instead of developing his own ideas.

B) Local structures. Fibre bundles may also be defined by atlases "gluing together" products, the transition functions bet-v·een charts being compatible with the action of the structural group cn the fibres. To unify the treatment of structures defined by such a "gluing together" process (as are topological, 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 witf:.

the ¹¹restriction" order, satisfies conditions turning it into a "local groupoid").

Improving his earlier results [ 36, 39] he wrote the paper "Gattungen von lokalen Strukturen'' [ 47 ) • During the correction of its proofs in 1?58, I

learnt the fundamentals of Category Theory ••• and also a little Gerrnan. (Charles, whose first language was german, always discouraged me from studying thoroughly Goethe's language saying that he could translate it forme and that it would be

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more useful to learn somethin=:; he did not know. )

This paper was t:1e starting point of many of the suusequent papers on discrete fibrations and extensions of functors, on ordered categories and "local 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 PB: DiffB ^• Top be the forgetful functor from the groupoid of diffeomorphisms between cpen subsets of a Banach space; it is a discrete fibration (defined by a "species of structures") over a subgroupoid of the groupoid

all homeomorphisms.

Top g of

1° Enlargement of the species of structures: PB is "universally" extended into a discrete fibration P: D ^• Top , which is defined ''pointwise". In modern terms

g

Pis the discrete fibration associated to the Kan extension (along the insertion toward Topg) of the Set-valued functor associated to the discrete fibration PB.

2° Local completion of a local functor: On D there is an order deduced from the

"restriction order" on Diff • In this way, D becomes a local groupoid (i.e.,

B

essentially, an internal groupoid in the category of distributive complete A- lattices). But D is not (order-)complete over Top, in the sense that there exist A-compatible families (Ai)iEI of objects of D (i.e,,

P(A.AA.) = P(A.) nP(A.)

]_ J 1. J for each i, j € I )

which have no upper bound in D. The process of "local completion" consists in adding such upper bounds. lt leads from P to the forgetful functor from the groupoid of diffeomorphisms between manifolds modelled cn a Banach space. In fact, this step amounts to construct, over each topological space T, the sheaf associated to the presheaf determined by P

8 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 obj ects of Diff B • This construction is well described in [ llü ] •

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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') (wiüch is I!'ore 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 t;1e maps domain, codomain and composition, as in the two preceding examples (a topology in the case A, a ¹¹local order" in B).

In 1963 tI1ese reflections led to the definition of a P-structured category, where P: H • Set is a forgetful functor. In modern terms, it is an internal category in R but we thought of it as the data of a category C and of an object S of H sent by P on the set of morphisms of C in such a way as:

1° the maps domain and codomain "lift" into morphisms from S to a P-subobject, 2° the map composition "lifts" into a morphism toward S from a P-subobject of the product SxS •

Hence the necessity of defining P-subobjects [60, 66, 67, 69

J.

(Oddly enough, it is the dual notion of quotient object [61

J

which led us to the notion of a reflection of a category into a subcategory [ 66 J and, more generally~ to the notion of a free object, via the construction of a connna 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~ introduced in (50

J,

whose general theory is developed in [92

J;

in this paper, 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 [ 81] are constructed.

2° Differentiable categories (see 4).

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3° Double categories (internal categories in Cat ) [ 57, 58, 64, 117 ] , whose first example was the 2-category of natural transformations (already introduced in the paper ¹¹Categories de foncteurs type" [ 52] written while we were in Buenos-Aires for a four months stay in 1959), the second one being the category of commutative squares [55] of a category A (or Pcategory of morphisms of A ").

Notice that, in our latest paper [ 121 ] (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 (or "double category of quintets" in the terminology of [ 58, 64 ) ).

4° Yrt.1ltiple categories, defined in [57, 63], whose theory is developed in our last series of papers [ 119-121 ] and for which theorems of existence of "1 ax limits¹¹ (very conveniently defined in this frame) are proved by a short "structural" method,

5° Ordered categories and their specializations such as local categories, which are considered in numerous papers (700 pages odd) [53, 55, 62, 68, 71, 75, 76, 85

J

that should be read after the "Guide sur les categories ordonnees" [ 86 ] , Among the main results, still tobe exploited (1r11 show it elsewhere, e.g, in relation with topos theory) are those concerning local jets and atlases (as well as their generalizations: rockets and super-rockets) which are used to get (order-)completion theorems for ordered categories (e.g,, to construct the complete holonomy groupoid of a foliation in [ 75 ]) and for ordered functors. They culminate in the theorem of complete enlargement of a local functor [ 110 ] , which generalizes the associated sheaf construction,

Generaltheorems on structured categories are given in a series of papers frorn 1963 to 1969 [ 57, 63-66, 82.-84? 88, 100, 102, 109 ] , For instance, to study the existence of colimits, we thought of constructing them as quotients of coproducts. But such quotients are scarce [ 65, 66 ] , even in Cat [ 61, 80, 91]. Hence the idea of defining quasi-quotient objects [ 82, 100

J;

fine cons-

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tructions of quasi-quotient structured categories are made in f 65, S2-84, 100 and used to study a ¹'non-abelian" cohomology • But in general only existence theorems may be obtained, and this led to develop theorems on existence of free oojects [ 100, 102? 108 in which the functor is ¹¹extended to a higher universe"

instead of ¹¹restricted¹¹ like in the solution set condition of Freyd rs theorem.

These general theorems on P-structured categories require that P satisfies some ¹¹good¹¹ conditions, such as creation of some kinds of limits or colimits, existence of quasi-quotients, existence of a smallest subobject SA of an object S of H such that P(SA) contains a given subset A of P(S) (r-generating functors) or such that P(SA) = A (r--spreading functors. Hence arise the questions:

1° 1s it possible to define classes of functors by some properties, e.g., functors

11of a topological type" ( = , --spreading functors admitting limits - topological functors in Herrlich' s sense), or "of an algebraic type" (

= ,-

-generating functors admitting quasi-quotients); the word ¹¹algebraic" means here ''partially defined compositions"(and not everywhere defined compositions, as in monadic functors) ? We thought that one of the future task of the mathematician would be to srudy such classes (as explained in [ 94] ), 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 enough¹¹^, may it be extended universally in a good functor? The motivating example was the forgetful functor frcm

Diff

^which

does not create kernels. This problem is tackled in f 99, 107

J,

where universal completions of functors are constructed by transfinite induction. Here again the german school seems to carry on.

6, Sketched structures.

The idea of a category consists in the graph formed by its maps domain a ₁ codomain ß and composition

r ;

and such a graph in Set determines a category

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C o ~ C , C ß

if it satisfies axioms expressing associativity, unitarity and the fact that C~ is the pullback of (Cl'., ß) • This remark led to "isolate" the sketch of categories, whose underlying category is the full subcategory ZCat of the simplicial category L with objects

o,

1, 2, 3.

The realizations in a category h of this sketch are the internal categories in (called generalized structured categories in [ 93, 104, 105 ]).

Categories of internal categories are studied in ( 112, 113₁ 115

J •

An Lntern:J.l category in H gives rise tc a catcgory ubject in Grothendiec.k's sense [ 104 ] , but the inverse is valid only if H admits pulU,acks • So arised the question: may we universally add pullbacks to a category E so that botn notions coincide ? To ans wer i t, theorems of corr.pletion of categories by s01Ee types of limits or colimits were devised [ 102]. 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 equivalencesH for all limits of the given type. Later on [ 115] we extended this result₁ replacing the choice of liri1its by a "relational choice¹¹ (and then explicit constructions of the completion are necessary, for the existence theorems cannot be applied).

More generally, ¹¹algebraic 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 is developed in [ 93, 98? 106 ] • The interest of taking a neocategory with cones instead of a category with limit-cones (for instance, I: instead of t.,, ) is to get a ''finitely-presented" model. Completion theo_rems

Cat

led from the sketch to the prototype (category with limit-cones) and to the

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type [ 114, 115 ] , which is the "complete" model (no more finitely presented).

Though categories of sketched structures are locally-presentable cate·,;ories in the sense of Gahriel-Ulmer (whose work was published later on), the theory of sketcnes is fruitful in rr.any problems • General tr.eorems on sketched structures may be found in [ 106, 115

J ,

? • Discrete fibrations and em⁰iched categories 0

Motivated by fibre bundle theory wbere both topological actions of a locally trivial groupoid and local discrete fibrations are considered (cf. 4)?

Charles defined in [ 47 J the notion of a category C acting on a 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 fibration (called "foncteur d 'hypermorphismes" in [ 55 ] ) , 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 R (but the notion of a Set-valued functor cannot be internalized), They are considered in [59, 60, 64, 90

J,

but a general theory is not written yet, though we knew several results on them. The particular cases of topological or differentiable species of structures, and of ordered species of structures, are studied in [SO, 78, 101, 105;

75 J •

Topological fibrations were generalized into

"germs of fiorations" for use in optimization problems S) •

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 [ 47 ] • Then it is natural 5) A, Ehresmann, ''Systemes guidables et problemes d 'optimisation", Labo, Automa- tiqu.e theorique, Univ. Caen, I

a

IV (1963-64).

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to replace the discrete fibration by any functor, and this led to tl::e general theorems on extensions of functors [55, 72, 77, 70

J

to which is devoted the fifth chapter of "Categories et structures" [ 122 ] • These theorer:is also encom- pass the construction of categories of fractions (rnade for the first time, under the name of "perfectionnement d rune categorie" in the Appendix of [ 55 ] , written during our stay in Sao Paula in 1960). These extension t'1eorems are "internalized"

in [ 89, 90, 95, 96

J •

Theorems on extensions of functors and on local completion of local functors were devised in such a way that they may be applied in Analysis to define distructures 6

), which unify various concepts of ''generalized functions" (Schwartz distributions, Sato hyperfunctions, Mikusinski operators, ••• ) and give for instance "infinite-dimensional distributions" 7

). 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 us to consider:

1° Enriched species of structures, called "especes de structures dominees" in (64, 77, 122

J,

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 homomorphism are adapted in [ 70], giving rise to the (split) fibration associated to a functor toward Cat and to its first cohomology class; a general theory of non-abelian cohomology is developed in [ 7 3, 7 4, 91 ] (but i t may be much improved);

3° enriched species of structures in the category of discrete fibrations; the special case in which all the fibrations have the same base gives a pair of acting categories [ 77]; this notion is equivalent to that of a Benabours

6)

A.

Ehresmann, "Differentiabilite dans les espaces localement convexes; distruc-

tures'\ These Univ. Paris, 1962. . ^A ^•

7) A. Ehresmann, "Sur les distributions vectorielles", Labo. Automati-que Theori-que,;

Vniv. Caen, 1964.

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distributor; when distributors are so considered as "double fibrations" , their composition is similar to the composition of atlases of a category [ 75];

4^<> ^IIpar ia ^{t •} 1¹¹ ac ions o t • f categories, called syst~rns of structures [ • 122 ] , enriched in a concrete category H, i.e. the fibres of the associated "partial fibration¹¹ are equipped with objects of H (for distructures, TT is the category of Banach spaces). A theorem on extension of an enriched system of structures into an enriched species of structures is given in [ 87] (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: if C is a category,

c"I'

x C acts or. t:1e set of morphisms of C , the corresponding Set-valued functor being Homc. To say that the action is enriched in a category H means that the sets Homc(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 aotions of a closed category 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

S),

we exten- sively used monoidal closed categories. For instance, we constructed monoidal closed structures on general categories of sketched structures ( [ 115] Part II), on categories of internal categories [ 109, 112], on categories of topological ringoids [ 118

J,

on the category of all multiple categories [ 119

J

and on the category of n-fold categories [ 120, 121].

8, Some criticai corronentst

It may seem weird that the 2.000 pages odd written in the sixties on

Category Theory are almost un-known, although some of them are still quite original.

8) ¹¹Closed categories'', Proc. Conf. on categ, Aigebra La JoUa, Springer, 1966.

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But this is understandable: during this period, we had few relations with categ- orists, because Charles and I were essentially considered respectively as a Geo- meter 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 [ 102] come after a first part devcted 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 LeIIlIIla.

The book "Categories 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 how we felt 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 beside deeper results (such as the ends of Chapters 2 and 3, and the Chapter 5); this comes from the fact that some parts were retaken from mimeographed courses, and others from research papers. The aim of the book "Algebre" [ 123 ] (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 admi- nistrative duties, direction of too big a research team (about 50 theses defended in 8 years !), entire publication of the ¹¹Cahiers de Topologie et Geometrie Dif- ferentielle", organization of several meetings [ 142-144 ] • So we only could do personal research during the summ.er vacation. In 1977, for different reasons (in particular the refusal of a "3e cycle" in Amiens)~ we found time enough for a

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true research work again.

To oe sure that my husband's work will not be forgotten, I am going to publish (as "Supplements" to t~e "Cahiers") a complete commented edition of Charles' work. The comments will explain not only how the ideas evolved, Eut 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. But: it might be too heavy a weight forme and so I hope you will all help me to carry it on.

Appendix. Some biographicaZ notes.

Charles Ehresmann was born in Strasbourg on April 19, 1905, and died in Amiens on September 22, 1979.

As a student of the well-known "Ecole Normale Superieure" from 1924 to 1927, he passed the ¹¹Agregation" in 1927 and, after one year of service, he went to Rabat, as a secondary-level teacher (1928-29).

From 1929 to 1932, he had fellowships in Paris andin Goettingen. His thesis (Paris, 1934) was written during a two years stay at Princeton, as a Procter

Visiting Fellow. He spent the five following years in Paris, as a "Charge de

Recherches au C .N .R.S.". Then he was successively named Professor at the University of Strasbourg (1939-1955), at the Sorbonne, at the University Paris VII (up to his mandatary retirement in 1975) and, finally, at the University of Amiens,

Many foreign universities invited him for long stays, e.g., those of Rio de Janeiro (1952, as an expert of the UNESCO), Princeton (1953-54), York (1955), Bombay (Tata Institute, 1956-57), Mexico (1958, as an expert of the UNESCO), Buenos-Aires (1959), Sao Paulo (1960), Kansas at Lawrence (1966).

The Ur.iversity of Bologna named him Docteur honoris causa in 1967, and the

11Academie des Sciences de Paris¹¹ awarded him three prices, the last one ii: 1965.

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He organized international conferences in nerowicz; Grenoble (1963) with G. Reeb ; Paris

Strasbourg (1953) with A, Lich- Dijon (1967) -;-:ith G. Jcut,ert .Amiens (1973, 1975) and Chantilly (1975) with me, F:com 19.:,2 cn1 ".-1e anL:natE.d a regula:r ~~eminar and several srnall mathematical meetings in Strasbcurg, Paris and Amiens. 1Jnder his direction were prepar~d

26 ¹¹Doctorats drEtat¹¹: Reeb, Wu Wen Tsun, Calabi, Yen Chih Ta, P, LiE·ermann,

Haefliger, Srinivasacharyulu, F. ßenzecri, Nguyen Dinh Ngoc, Ve,~ Eecke, Eoudebine,

c.

de Barros, Joubert, Benabou, l(umpera, Ibiscü, S. Legrand, Stavroulakis, Ch2cron, Yuen, Machado, Foltz, Pradines, Lair, Coppey, the last one being Guitartts on

June 8, 1979 ;

about 50 ¹¹:Joctorats de 3e cycle".

He undertook the publication of the "Cahiers de Topologie et Geometrie Differentielle" in 1957 and of the series "Esquisses Mathematiques" in 1970. He was one of the sponsors of the "Zentralblatt für Mathematik" and one of t:1e editors of "Archiv für Me.thematik" and "Tensor".

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1. TRAVAUX DE RECHERCHE.

1. Les invariants integraux et la to pologie de l'espace proj ectif regle, C. R.

A. S. Paris 194 ( 1932 ), 2004-2006.

2. Sur la to pologie de certaines varietes algebriques, C. R. A. S. Paris 196 ( 1933 ), 152-154.

3. Un theoreme relatif aux espaces localement proj ectifs et sa generalisa- tion, C. R. A. S. Paris 196 ( 1933), 1354-1356.

4. Sur la topologie de certains espaces homogenes, Ann. of Math. 35 (1934 ), 3%-443. (These Paris 1934.)

5. Groupes d'homologie, Seminaire de Math. Paris, III-B ( 1935 ), 1- 25.

6. Sur les espaces localement homogenes, Enseignement Math. 35 ( 1936 ), 317-333.

7. Sur la notion d'espace complet en Geometrie differentielle, C. R. A. S.

Paris 202 (1936), 2033-2035.

8. Sur la topologie de certaines varietes algebriques reelles, ]. de Math.

XVI ( 1937), 69-100.

9. Les groupes de Lie

a

r parametres, Seminaire de Math. Paris, N E et F ( 1937), 1-61.

10. Sur les arcs analytiques d'un espace de Cartan, C. R. A. S. Paris 205 ( 1938 ), 1433-1436.

11. Sur les congruences paratactiques et les parallelismes dans les espaces proj ectifs,

C. R.

A. S. Paris 208 ( 1939), 153-155.

12. Sur la variete des generatrices planes d'une quadrique reelle et sur la topologie du groupe orthogonal

a

n variables, C. R. A. S. Paris 208 ( 1939 ),

321-323.

13. Sur la topologie des groupes simples clos, C. R. A. S. Paris 208 ( 193 9), 1263- 1265.

14. Sur les proprietes d'homotopie des espaces fibres,

C. R.

A. S. Paris 212 ( 1941 ), 945-948 ( avec

J.

FELDBAU).

15. Espaces fibre associes, C. R. A. S. Paris 213 ( 1941 ), 762- 764.

- 17 -

/

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16. Espaces fibres de structures corni=arables, C. R. A. S. Paris 214 ( 1942 ), 144-147.

17. Sur les espaces fibres associes

a

une variete differentiable, C. R. A. S.

Paris 216 ( 1943 ), 628-630.

18. Sur les applications continues d'un espace dans un espace fibrc ou dans un reveternent, Rul!. Soc. '1ath. France 72, Paris ( 1944 ), 27-54.

19. Sur les charnps d'elernents de contact de dirnension p cornpleternenc in- tegrables dans une variere continuernent differentia:-ile, C. R . .

!.

S. Paris 218 ( 1944 ), 955-956 (avec G. REES).

20. Sur la theorie des espaces fibres, Colt. Intern. Tupo. algc;bri11ue Paris, C. N. R . S. ( 1 94 7), 3 - 1 5 .

21. Sur les sections d'un charnp d'elernents de contact dans une variete differentiable, C. R . . 1. S. Paris 224 ( 1947), 444-445.

22. Sur les espaces fibres differentiables, C. R. A. S. Paris 224 0947), 1611- 1612.

23. Sur les varieres plongees dans une variete differentiahle, C. R.

1.

S Paris 226 ( 1948 ), 1879-1881.

24. Sur !es forrnes differentielles exterieures de degre 2, (. R. 1. S. Paris, 22 7 ( 1 94 8 ) , 4 2 0 - 4 2 l ( a v e c P . L 1B E R M A N N ) .

25. Sur !es extensions de groupes topologiqucs, C. R .. 1. S. Paris 228 (1949), 1551 -1553 (avec L. CALA.BI).

26. Sur le problerne d'equivalence des forrnes differentielles exterieures quadratiques, C. R. A. S. Paris 219 ( 194()), 697-699 ( avec P. LIBER- MANN ).

27. Sur Ja notion de connex10n infinitesirnak dans un espace f ibre et sur les espaces

a

^COnnexion de Canan, Cong res d'lnnsbruck, rvarh. Oest.

Math. (;ese!!schaft ( 1949), 22.

28. Les connexions infinitesimales dans un espace fibre differentiable, Coll.

de Topo. Rruxelles, C. B. R. M. ( 19':i0 ), 29- 55.

29. Sur les varieres presque cornplexes, Proc. Intern. Cong. of ¹tlaih. Har- vard ( 1950), Vol. 2,412-410 (et Seminaire Bourbaki 19'>0).

30 . .Sur ^j.]rheorie des varieres feuilletC'CS, N, nd. Mat. c :'l[lf!l. Ser. V, X-1-'.?

Rorne ( 19'51), 64--83.

31. Sur ks structures prcsguc hermitiennes isotrope.s, C, R. A. S. Paris 23:2 ( 1 9 5 1 ) , 12 81 - 1 2 8 3 ( a vc , P . LI n E R :\1 A. :\: l\ ) .

J2. Les prolongernents cl'urie variere differentiable, Aui !V Cong. Un. rnat.

ltaliana, Taorrnina Ott.(10',l ), 1-9.

33. Les prolongernents d'une variete differentiable, I: Calcul des jets, pro- longernent principal, C. R. A. S. Paris 233 (1951 ), 598-600.

·- ~8 -

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34. Les prolongements d'une variete differentiable, II: L'espace des jecs d'ordre r de Vn dans Vm, C.R.A.S. Paris 233 (1951), 777-779.

35. Les prolongements d'uoe variete differentiable, III: Transitivite des proloogernents, C. R. ,1. S. Paris 233 ( 1951 ), 1081-1083.

36. Structures locales et structures infinitesimales, C. R. A. S. Pari:s 234 ( 1952 ), 587 - 589.

~7. Lcs prolongernents d'uoe variete differenriable, IV: Elements rie contact et elements d'enveloppe, C. R. A. S. Paris 2~4 ( 1952 ), 1028-1030.

38. Lcs prolongements d'une variete differentiable, V: Covariants differen- riels, et prolongements d¹une structure infinitesimale,

C. R. A. S.

Paris

?3-i (1952), 1424-1425.

39. Structures locales, Ann. di Mat. ( 1954), 133- 142. ( Multigraphie Rome et Strasbourg, J 95:2.)

40. Introduction

a

la theorie des structures infinitesimales et des pseudogroupes de Lie, Colt. Intern. Geom. Diff. Strasbourg, C. N. R. S. ( 1953 ), 97-110.

41. Fxtension du calcul des jets aux jets non holonomes, C. R. A. S. Paris 239 ( 1954 ), 1762-1764.

42. Sur les structures infinitesimales regulieres & Sur les pseudogroupes de transformations de Lie, Proc. lnt. Cong. Amsterdam (1954), II, 478-479.

43. Applications de la notion de jet non holonome, C. R. A. S. Paris 240, ( 1955 ), 397-399.

44. Les prolongements d'un espace fibre differentiable, C. R. ,i S. Paris ( 1955 ), 1755-1757.

45. Sur les espaces feuilletes: theoreme de stabilite, C. R. A. S. Paris 243 ( 1956 ), 3i4 -346 ( avec SHIH WE ISHU ).

46 . .Sur les conncxions d'ordre superieur, Aui V Cong. Un. Mat. !taliarw Pavia-Torino (1956), 326-328.

-i 7. Gattungen von Lokalen Strukturen, ] ahres. d. Deutschen Math. 60-2 ( 1957), -19-77. (Traduit en fran~ais dans CTGD l) III, 1961.)

48. Sur les pseudogroupes de Lie de type fini, CR. A. S. Paris 246 ( 1958 ), 360- 362.

49. Sur les appuis d'une pyramide convexe et sur les polyedres convexes sans sommet, C. R. A. S. Paris 249 ( 1959), 2695 -2697 ( avec A. EHR I. S- MANN).

so.

Categories topologiques et categories differeotiables, Coll. Geom. Diff.

Globale Bruxelles, C. B. R . .M. ( 1959), 137-150.

1) CTGD se Iit «Cahiers de Topologie et Geometrie Differentielle».

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51. Gru_p:>ides diferenciables y pseudogrupos de Lie, Rev. Un. Mat. Argent.

19, Buenos-Aires ( 1 %0 ), 48.

52. Categorie des foncteurs types, Rev. Un. Mat. Argentina XX ( 1%0), 194- 209.

53. Categories inductives et pseudogroupes, Ann. Inst. Fourier X, Grenoble (1%0), 307-336.

54. Structures feuilletees, Proc. 5^th Can. Math. Cong., Montreal ( 1961 ), 109 -172.

55. Elargissements de categories, CTGD III (1961), 25-73.

56. Archimede et la Science moderne, Celeb. Archimedee, Syracuse ( 1%1 ), 25- 37.

57. Categories doubles et categories structurees, C. R.A.

S.

Paris 256 (1%~) 1198- 1201.

58. Categorie double des quintettes; applications eo variantes, C. R. A. S.

Paris 256 ( 1%3 ), 1891-1894.

59. Categories structurees d'operateurs, C. R. A. S. Paris 256 ( 1%3 ), 2080- 2083.

60. Sous-structures et applications K-covariantes, C. R. A.

S.

Paris 256 ( 1%3 ), 2280- 2283.

61. Structures quotient et categories quotient, C. R. A. S. Paris 256 (1963), 5031-5034.

62. Completion des categories ordonnees, C. R. A. S. Paris 257 ( 196~ ), 4110-4113.

63. Categories structurees, Ann. Ec. Norm. Sup. 80, Paris ( 1963 ), 349-426.

64. Quintettes et applications covariantes, CTGD V ( 1963 ), 1-21.

65. Categories structurees quotiem, CTGD V (1%3 ), 1- 5.

66. Structures quotient, Comm. Math. Helv. 38 ( 1963 ), 219 -283.

67. Teilstrukturen und Faktorstrukturen, Jahrestagung Deutschen Math. Ver.

Frankfurt ( 1963 ), 1.

68. Groupoides sous-inductifs, Ann. Inst. Fourier XIII-2, Grenoble ( 1%3 ), 1-60.

69. Sous-structures et categories ordonnees, Fund. Math. LIV ( 1964 ), 211- 228.

70. Produit croise de categories, C. R. A. S. Paris 258 ( 1964 ), 2461- 2464.

71. Completion des categories sous-p-elocales, C. R. A. S. Paris 259 (1%4), 701- 704.

72. Expansion d'homomorphismes en foncteurs, C. R. A. S. Paris 259 (1964), 1372- 1375.

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73. Cohomologie sur une categorie, C. R. A. S. Paris 259 ( 1 %4 ), 1683 - 1686.

74. Sur une notion generale de cohomologie, C. R. A. S. Paris 259 ( 1964 ), 2050- 2053.

75. Categories ordonnees, holonomie et cohomologie, Ann. Inst. Fourier XIV 1, Grenoble ( 1%4 ), 205- 268.

76. Completion des categories ordonnees, Ann. Inst. Fourier XIV- 2, Greno- ble ( 1%4 ), 89- 144.

77. Categories et Structures, Extraits, CTGD VI ( 1964 ), 1-31.

78. Prolongements des categories differentiables, CTGD VI ( 1%4 ), 1-8.

79. Expansion generale des foncteurs, C. R. A. S. Paris 260 ( 1965 ), 30-33.

80. Categorie quotient d'une categorie par une sous-categorie,

C.

R. A. S.

Paris 260 ( 1965 ), 2116- 2119.

81. Categories quasi-topologiques et leurs prolongements, Universite Paris ( 1965 ), 1-15.

82. Quasi-surj ections et structures quasi-quotient,

C.

R. A. S. Paris 261 ( 1965 ), 1577-1580.

83. Quasi-categories structurees, C. R. A. S. Paris 261 ( 1%5 ), 193?-1935.

84. Groupoides structures quasi-quotient et quasi-cohomologie, C. R. A. S.

261 ( 1965 ), 4583-4586.

85. Especes de structures sous-inductives, CTGD VII ( 1%5 ), 1-42.

86. Guide des categories ordonnees, CTGD VII ( 1%5 ), 43-49.

87. Expansion des systemes de structures domines, C. R. A. S. Paris 262 ( 1966 ), 8-11.

88. Adjonction de limites aux categories structurees, C. R. A. S. Paris 263 (1966 ), 655-658.

89. Quasi-elargissement d'un systeme de structures structure, C. R. A. S.

263 ( 1%6 ), 762- 765.

90. 1er theoreme d'expansion structuree, C. R. A. S. Paris 263 ( 1966), 863- 866.

91. Cohomologie

a

valeurs dans une categorie dominee, Call. Topo. algebri- que BruxeHes, C.B.R.M. (1%6), 21-80.

92. Categories topologiques I, H, III, lndig. Math. 28-1 ( 1966 ), 133-175.

93. On the definition of structured categories, Technical Report 10, Un. of Kansas, Lawrence ( 1966 ),

96

pages.

94. Trends toward unity in Mathematics, CTGD Vill ( 1%6 ), 1- 7.

95. 2e theoreme d'expansion structuree,

C.

R. A. S. Paris 264 ( 1%7 ), 5-8.

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96. Theoreme de quasi-expansion reguliere, C. R. A. S. Paris 264 ( 1967 ), 56- 59.

97. Problemes universels relatifs aux categories n-aues, C. R. A. S. Paris 264 ( 1967 ), 273 - 276.

98. Sur les structures algebriques, C. R. A. S. Paris 264 ( 1967 ), 840- 843.

99- Adjonction de limites

a

un foncteur fidele ou

a

une categorie, C. R. ,1.

S.

Paris 265 ( 1967 ), 29 6- 299 .

100. Structures quasi-quotients, Math. Ann. 171 ( 1967 ), 193- 363.

101. Proprietes infinitesimales des categories differentiables, CTGD rX-1 ( 1%7),

1-9.

102. Sur l'existence de structures libres et de foncteurs adj oints, CTGD IX- 1 - 2 ( 1 96 7 ) ,

3 3 -

1 8 0.

103. Sur les categories differentiables, Atti Conv. lnt. Geom. Diff. Bologna ( 1967 ), 31-40.

104. Categories structurees generalisees, C TGD X- 1 ( 1968 ), 139-168.

105. Categories structurees et categories differentiables, Revue Rouma1:ne Math. Pures et Appl. 7, XIII ( 1968 ), 967- 977.

106. Esquisses et types de structures algebriques, Bul. Inst. Polit. la~i XIV-1-2 (1968), 1-14.

107. Prolongements universels d'un foncteur par adjonction de limires, Dis- sertationes Math. LXIV, Varsovie ( 1969 ), 1-72.

108. Construction de structures libres, Lecture Notes in Math. 92, Springer ( 1969), 74-104.

109. Categories de foncteurs structures, CTGD XI-3 ( 1969), 329- 383 ( avec A. EHRESMANN ).

110. Elargissement complet d'un foncteur local, CTGD XI-4 (1969),405- 420.

lll. Espaces fibres et varietes differentiables, Univ. Paris VII (1969), 1- 44 ( avec A. EHRESMANN ).

112. Categories de foncteurs structures, Call. E. Cartan, Un. Paris VII (19")) 1 page.

113. Etudedes categories dans une categorie, Univ. Paris VII (1972), 1-45 ( avec :\. EHRESM1\NN ).

114. Sur le prolongement d'un prototype dans un type, Univ. Paris VII (1972), 1-12 ( avec A. EHRE SM ANN ).

115. Categories of sketched sr:ruc:tures, CTGD XIII-:? ( 1972 ), I 05 -214 (avec A. EHRESMANN ).

116. Categories in differential Geo metry, Resumes Coll. Amiens 1973, CTGD XIV - 2 (l 973 ), 175 - 177.

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117. Multiple functors, I: Limits relative to double categories, CTGD XV- 3 ( 1974 ), 215-292 (avec A. EHRESMANN ).

118. Tensor products of tofOlogical ringoids, CTGD XIX -1 ( 1978 ), 87-112 ( avec A. EHRESMANN ).

119- Multiple functo rs, II: The monoidal closed category of multiple categories, CTGD XIX-3 (1978), 295-334 (avec A. EHRESMANN).

120. Multiple functors, III: The cartesian closed category Catn, CTGn XIX- 3 (1978), 387-444 (avec A. EHRESMANN).

121. Multiple functors, IV : Monoidal closed structures on Catn, C TGD XX- I (1979), 59-104 (avec A. EHRESMANN).

2. LIVRES.

122. Categories et structures, Dunod, Paris, 1%5, 375 pages.

123. 4lgebre (Maitrise de Math. C3), C.D.U. Paris, 1%8, 168 pages.

3. CO URS MU L TIGRAPHIES.

124. Cinematique, lJniv. Strasbourg (a Clermont-Ferrand ), 1942.

125. Espaces fibres et structures infinitesimales, l er chapitre, Univ. Rio de Janeiro, 1952, 24 pages.

126. Categoiies diff erentiables et Geometrie differentielle, Chapitres I et II, Sem. Soc. Can. Math., Montreal, 1%1, 118 pages.

127. Cours de Topologie algebrique, Univ. Paris VU, 1970, 84 pages (avec

A. EHRESMANN ).

128. Topologie algebrique, Univ. Amiens, 1975, 150 pages (avec A. EHRES- MANN).

129. Histoire et Fondements des Mathematiques (3 chapitres), Univ.Amiens 1977-1978 ( avec A. EHRESMANN ).

4. DIVERS.

130. Analyse de l'ouvrage de Seifert & Threlfall: <•Lehrbuch der Topologie,>, Fnseignement Math. 34 ( 19 35 ), 404.

131. Analyse de l'ouvrage de Alexandroff & Hopf: <• Topo Iogie I •>, Enseign.

Math. 35(1936),403-

132. Analyse de l'ouvrage de Schouten & Struik: <<Einführung in die neu- eren Methoden der Differentialgeometrie,>, Bull .. Sc. Math. 60 ( 1936 ), 129-131.

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133. Revue critique des theses de

J.

CavaiHes: Methode axiomatique et for- malisme, Revue Philosophique 131 ( 1941 ), 81-86.

134. Analyse de Pouvrage d'E. Cartan: « Les systemes differentiels exte- rieurs et leurs applications geomerriques», Bull. Sc. Math. 70, Paris ( 1946 ), 117- 122.

135. Analyse de l'ouvrage d'E. Cartan: •Lec;ons sur la theorie des espaces de Riemaruu, Bull. Sc. Math. 70, Paris (1946 ), 149-151.

136. Travaux scien.tifiques de C. Ehresman.n, Univ. Strasbourg, 1955, 1-19.

137. Rapport so mmaire sur les ttavaux de M. A. Lichnerowicz, :1tti V Cong.

Un. Mat. ltalian.a Pavia-Torino ( 1956 ), 21-26.

138. Topologie algebrique, Fonnulaire de Math.

a

l 'usage des Physiciens et Ingenieurs, C. N. R. S. ( 1962 ), 202 -220 ( avec G. REE B ).

139. Vingt ans deja ... , CTGD XVID-4 ( 1977), 431-432 ( avec A. EHRES- MANN).

5. EDITION DE TRAVAUX.

140. Edition de Pouvrage posthume de

J.

CavaiUes: Sur la logique et Ia theorie de la Science, P. U. F ., 1947, 1e edition (avec G. CANG UILHEM ).

141. Edition du Colloque de Geometrie Differentielle de Strasbourg 1953, Coll. Intern. C. N. R. 5. (1953 ), 1-198 ( avec A. LICHNEROWICZ ).

142. Edition des Resumes du Colloque sur PAlgebre des categories Amiens 1973, CTGD XIV-2 ( 1973 ), 153-223 ( avec A. EHRESMANN ).

143. Edition des Resumes du 2e Colloque sur l'Algebre des Categories, Amiens 1975, CTGD XVI-3 ( 1975 ), 217-340 ( avec A. EHRESMANN).

144. Edition des Resumes des Journees T.A.C. de Chantilly, CTGD XVI- 4 ( 1975 ), 425-442 ( avec A. EHRESMANN ).

145. Publication des:

Receuils d'exposes du Colloque de Topologie de Strasbourg: 19

51,

19 52 et 1954 .

. CTGD, Volumes 1

a

20, depuis 1957 ( avec .A. EHRESMANN ) .

. Esquisses Mathematiques, Volumes 1

a

30, depuis 1970, Paris-Amiens ( avec .A. EHRESMANN ).

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- • ßD- i ~ - ~D Dill~_ 01b0

Rene GUITART, U. Paris

7

1 - Ehrig & c:ü., dans leur livre "Universal theory of auto;.1atas¹¹('l1eubner, 1974) indiq_uent comr:1ent, partant d¹une categorie monoidale fermee

(E,e,1,•··)

^a (~-lll)- decompositions, on associe

a

toute machine de lilealy A = (S~J0';J----!io>O) u.>1

morphisme

7:.:

S----l/l>o[J+,

o]

^{(avec J+} = 1 + J + J@J + ••• ) (la "run-map") puis son (E-J:vI)-ine,ge :Sehavior( A ) ~ [ J + ,

oJ

Ils enoncent alors le critere

est de Lt forme X = Behav=-or( A ) pour un certain A ssi

il existe und X@J~X tel q_ue

xe J---[J+, o] s

J

d

+

⁼ .k_,::_,ft shift X [ J+,

oJ

Probleme : Que devient ce cri tere et les thc§oremes d ^Iobservabili te et mini:nisation des ,,:achines deterministes ciuand on passe au:x machines P-fl,)Ues c ^Iest-a-dire

aux machines A = (PS ,et d J49S--s~~.PO) ou

J,o,s

sont toujours des objets de E mais ou P : E----,11,m,E est un foncteur,"donnant du flou",comme par exemple PX = 2X ou PX

= [O,l]

X (dans le cas E

=

Ens)?

Principe: Pour obtenir les resultats ¹¹non-deterministes¹¹ de Ehrig & al (2eme partie du livre) i l suffira d¹appliquer les resultats ¹¹deterministes¹¹ (lere partie du

livre)

a

la categorie monoidale

(t', ®)

^{que 1}¹on decrit plus loin

(JJJ.

Theoreme)o Exemple : Si E est un topos de Grothendieck, alors la minimisation des machines non-deterministss

(=

A-)_floues) dans E est possible (mais non unic:u::).

(1)

Conference

a

Arnsberg, en octobre

1979,

au Nordwestdeutsches Kategorienseminar;

.Une conference un peu differente sur le me'.Ile sujet a ete fai te en raai i Paris

7,

et une breve annonce des resul t.-,t::¹ a eu lieu cet ete 0, Oberwolfach. Le tout sora remanie et developpe dans l'article ¹¹TENSEURS EI' lVIACHINES"

a

^paraitre

dans Cahiers Top. Geo. Diff. XXI,l

(1980),

ou l'on trouvera en plus des deve- loppement sur l.es "bicategories graduees" et sur les "relations ternaires".

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