BIOGRAPHICAL MEMOIRS

Gregory Maxwell Kelly 1930–2007

This memoir was originally published in Historical Records of Australian Science, vol.21, no.2, 2010.

Letters in square brackets refer to the references at the end of the text.

Numbers in square brackets refer to the bibliography at the end of the text.

Introduction

Gregory Maxwell (‘Max’) Kelly (1930–2007) was educated at the University of Sydney (BSc 1951 with First Class Honours, University Medal for Mathematics, Barker Prize, and James King of Irrawang Travelling Scholarship) and the University of Cambridge (BA 1953 with First Class Honours and two Wright’s Prizes; Rayleigh Prize, 1955; PhD 1957). He returned to Australia as Lecturer in Pure Mathematics at the University of Sydney in 1957, became Senior Lecturer in 1961 and Reader in 1965. He was appointed Professor of Pure Mathematics first at the University of New South Wales in 1967 then at the University of Sydney in 1973, becoming Professor Emeritus in 1994. He introduced the mathematical discipline of category theory to Australia and continued active and influential research in the subject until the day of his death.

Professor Gregory Maxwell (‘Max’) Kelly was born in the inner Sydney suburb of Annandale on 5 June 1930. His father Owen Kelly was a radio operator on merchant ships plying the Pacific region before he married Rita McCauley who came from a farming family in Nelligen, New South Wales. After their marriage they together bought a business that collapsed during the Depression. Owen became a telegraphist with the Post Office and in his later years had a variety of jobs, the last of which was taxi-driving. Michael, born some seven years later, is Max’s only sibling.

Max received all his schooling at Bondi Beach where he was a student of the Marist Brothers throughout his primary and secondary education. He topped the New South Wales School Leaving Certificate Examination overall. He went on to win in 1951 the University Medal for Mathematics at the University of Sydney and to gain the James King of Irrawang Travelling Scholarship to study at Cambridge. There he obtained a BA with First Class Honours and two Wright’s Prizes in 1953, a Rayleigh Prize in 1955 and his PhD in 1957; the doctorate was in algebraic topology under the principal supervision of Shaun Wylie. Max also spent a term or more at Oxford where M.G. Barrett suggested a problem; Max’s solution became a chapter in his PhD thesis [1]. The thesis consisted of three separate parts published as [2], [3] and [4].

Figure 1. Max Kelly (on right) and his cousin
Vince McCauley at the latter’s graduation from
the University of Sydney, April 1966.
Figure 2. Max and Imogen Kelly at the
conferring of Imogen’s doctorate, June 2003.

Max returned to the University of Sydney in early 1957 as a Lecturer in Pure Mathematics; he was promoted to Senior Lecturer in 1961 and to Reader in 1965 (Fig. 1). For many years he served the New South Wales Department of Education as Assessor for the Leaving Certificate Examinations in Mathematics.

In November 1960 Max married Imogen Datson whom he met through friends of his brother. Imogen had come to Sydney from Broken Hill where she had taught for some years. She had planned to further her studies at the University of Sydney as an evening student while teaching at Newtown Demonstration School. After a year or so of juggling teaching, study and courtship, she decided to abandon her studies and devote herself to family life. Max and Imogen had four children—none of whom showed any great interest in mathematics—and the family now includes ten grandchildren. It was a source of great pride and delight to Max when, upon retirement from teaching, Imogen resumed her studies, gaining a PhD with a thesis on medieval and early modern English drama (Fig. 2). As Chancellor Santow remarked at the awarding of the degree, ‘You must have very interesting conversations in your household’.

Conversations were indeed lively in the Kelly household. With a seemingly natural flair for languages, Max spoke fluent French and Italian and frequently lectured in one or other of those languages when he travelled overseas. He had an abiding interest in etymology and a great love of literature. Language fascinated him. In his younger days, before debilitating back problems made extreme physical activity difficult, Max was a keen squash and table tennis player. He also enjoyed lunchtime games of Bridge in the Mathematics Department of the University of Sydney.

Max was solely responsible for introducing category theory into Australia at a time when the subject was in its infancy. The 1966 monograph ‘Closed Categories’ by Eilenberg and Kelly [14] set the stage for two more generations of Australian category theorists. This research stream reached maturity with Max’s 1982 book, Basic Concepts of Enriched Category Theory [41], and now finds application in many areas of mathematics, theoretical physics, computer architecture, software design and information management.

Eilenberg and Mac Lane completed the basic definitions of category theory in 1945. Although Max heard these definitions at Cambridge, the power of the subject was impressed upon him in 1962 when he heard some of Mac Lane’s deeper categorical ideas during Michael Atiyah’s lectures at Harvard. Soon Max had himself developed lasting ideas in the area. While visiting Tulane University in 1963–64, he met Eilenberg who insisted that Max remain in the USA for another year. Indeed Eilenberg, in one phone call, arranged a job at the University of Illinois for 1964–65. During the year at Tulane, Max also met Mac Lane, who recognized Max’s ability and arranged a visit to Chicago.

Max was a spontaneous lecturer, often referring to having found inspiration for the content that morning while in the shower or crossing the Sydney Harbour Bridge. Once Max had made a topic his own, he could provide a thorough account of it without notes at the drop of a hat. While teaching fourth-year honours in 1965, he asked the class whether they knew what was meant by the product of a family of sets. A lack of response prompted him to abandon the lecture he had planned. Upon the green board he wrote six forms of the Axiom of Choice (one form involving the non-emptiness of a product), and proceeded to prove the six equivalent. In that one lecture he completed five of the six steps, finishing the job next time. In that last step he used a technical lemma that he must have concocted on the day because I have not seen it anywhere else. I memorized this proof of the equivalence for the final examination; however, the examination question caused me some extra thinking by asking us to prove a different lemma for that last step.

In 1967, Max moved to become Professor at the University of New South Wales (UNSW). After visits to Columbia University from January to May 1968 with Eilenberg, and to the University of Chicago in 1970–71 with Saunders Mac Lane, Max returned to UNSW and arranged a sabbatical there for Peter Freyd from the University of Pennsylvania. During Freyd’s stay Max organized, with the strong support of Bernhard Neumann from the Australian National University, the first conference in Australia on category theory.

Max was elected a Fellow of the Australian Academy of Science in 1972 and moved back to the University of Sydney as Professor in 1973. He was a true academic: erudite in the classics, prolific researcher and publisher, editor for several journals, successful department head, traveller, linguist, raconteur and bon-vivant. He supervised five PhD students to completion: I was the first (1969), then Brian Day (1970) , Geoffery Lewis (1974), Robert Blackwell (1976) and Greg Bird (1984). Other supervisions included the MSc of Roger Eyland in 1962 and of Amnon Neeman in 1979.

Max was one of the first mathematicians to attract research funding from the Australian Government through the Australian Research Grants Committee and its successor the Australian Research Council, contributing to recognition of the legitimacy of funding for research fellows, visiting researchers and travel. Another way in which Max broke down the tyranny of distance for Australian category theory was to establish and maintain a Category Mailing List in those email-free days. Preprints and that List were typed using an IBM electric ‘golf-balls’typewriter. The List was photocopied on to address labels.

These initiatives led to an ongoing stream of researchers visiting category theorists in the Sydney area. Indeed, Max was invited to many overseas universities for research visits. His linguistic ability was helpful and much appreciated by his hosts. These visits and collaborations strongly influenced Max’s career and the direction his research followed. Periods spent at other universities were listed in his CV as follows: Massachusetts Institute of Technology (October–November 1962); Tulane University, New Orleans (1963–64); University of Illinois at Urbana (1964–65); Columbia University, New York (January–May 1968); University of Chicago (1970–71) ; McGill University, Montréal (September–December 1976; February–March 1986; June 1991); Université du Québec à Montréal (January–May 1977); Université Catholique de Louvain, Belgium (June 1977; May 1981; June 1983; August 1987; January–February 1993; July 1994; July 1995; May–June 1996; November 1996; May–June 1998); Università degli studi di Trieste, Italy (May 1980; June 1981; July 1983; April–May 1986; July 1990); Fernuniversität Hagen, Germany (June–July 1980; May 1981; February 1997); Athens, Thessalonika, Xanthi, Sofia, Brno, and Prague (May 1983); Aarhus, Denmark (June 1983); Chung-Ang University, Seoul, Korea (August–September 1984); Université de Paris Nord (June 1986); Polish Academy of Science, Warsaw (June 1986); Dalhousie University, Halifax, Canada (May–June 1987; July–August 1988; July 1989; August 1990; May–June 1993; July 2006); Università di Milano (July 1987; January–February 1989; July 1991; September–October 1992); University of Sussex (July 1988); University of Fribourg, Switzerland (June 1992); Georgian Academy of Science, Tbilisi (August 1992); Cambridge University, England (November–December 1992); University of Tours, France (July 1994); University of Santiago de Compostela, Spain (September 1995; January 1997); University of Vigo, Spain (September 1995); Universities of Coimbra and of Lisbon, Portugal (January 1998).

All of Max’s ninety or so scientific publications exhibit his obsession for completeness, beauty and accuracy. Michael Makkai (McGill) claims Max as a logician in his passionate insistence on precision and clarity in mathematics and his belief in, and search for, the grand order at the heart of the world. Much of Max’s work could be called higher-order universal algebra.

Max was very aware of how fortunate his life had been, and felt an obligation to give something back to the community. He gave freely of his time to aspiring young mathematicians and to those keen to learn. For example, frustrated by bureaucracies, he enlisted the power of the media and was able to borrow for a blind girl in the Catholic school system a mathematics textbook in Braille that had been gathering dust in a Department of Education office. This commitment to social justice was further evidenced by his involvement with Action for World Development and his efforts to help the Aboriginal community in Red-fern in Sydney. He befriended Father Ted Kennedy, Mum Shirl and others active in these movements. He also questioned the morality of the Vietnam War, making himself quite unpopular with some of the clergy of the day.

Many were moved by the words of encouragement Max offered young category theorists in his speech at the 2006 Category Theory Conference dinner in White Point, Nova Scotia. Max had an active and analytical mind to the very end. He attended the Category Seminar at Macquarie University two weeks before he died, excusing himself the next week because of an appointment. He started learning ancient Greek recently and in his last months was engaged in complex research on coherence theory, which he was typing despite failing eyesight. This research was completed and published by colleagues in Canada and Italy as [92]. The paper includes the remark: ‘G.M. (Max) Kelly died during the preparation of this paper. He was actively working on it on the day of his passing. The other authors express their gratitude for his work here and for so much more that he had shared with us as a friend and a colleague over many years. We regret too that he was unable to provide a proof reading of our final draft’.

A very successful conference was organized by George Janelidze at the University of Cape Town in January 2008 to commemorate the first anniversary of Max’s death. The proceedings will appear as [III]. The volumes of research papers [I] and [II] were also dedicated to Max.

Scientific Contribution

Max Kelly was the first researcher on category theory to be elected to the Australian Academy of Science. It therefore seems appropriate to provide some history of the subject itself.

With papers [a] and [b], top American mathematicians Eilenberg and Mac Lane founded category theory during the period 1942–45. Both became members of the US National Academy of Sciences; in 1973 Mac Lane became Vice-President of that body and President of the American Mathematical Society. More recently, Fields Medalist Vladimir Voevodsky declared [c] that categories were one of the most important ideas of twentieth-century mathematics.

As Max states on the first page of [35], in relation to Mac Lane’s work, ‘Major advances, once made, seem so inevitable that a younger generation, brought up familiar with these ideas, may not realize how great their impact was’. My intention here is to explain Max Kelly’s impact on his chosen field in the context of the times. I shall discuss how he was led to category theory and what his contribution was. In particular, his courage and ingenuity were shown by his leading role in the origins and development of enriched category theory.

Even by 1960 it was still the rare university mathematics course that mentioned the main categorical concepts: category, functor and natural transformation. Certainly, they were not mentioned during 1950–53 at the University of Sydney in the three courses Max studied on homology theory, Pontryagin duality and group representations. It is inconceivable nowadays that a course on any of these topics would not rely on categories.

Algebraic topology was popular at Cambridge so it was not surprising that Max, while commencing his postgraduate studies in 1953, came across the new book [d] by Eilenberg and Steenrod. Max had felt dissatisfied with earlier books but this one took the subject to a new and deeper level, as I shall try to explain.

Algebraic topology is concerned with the construction of invariants for topological spaces; topologically equivalent (homeomorphic) spaces should have the same invariant. The invariants started out as numbers such as Euler characteristic and Betti numbers. It was recognized, and particularly insisted upon by Emmy Noether, that algebraic structures (mainly groups or vector spaces), from which these numbers could be obtained, were the proper invariants to be studied. One such invariant of a space X is its homology: this is a sequence HX of abelian groups H0X, H1X, H2X, ..., also called a graded abelian group. Now I have said ‘one such invariant’, however, the truth is that there were many different constructions of what could justifiably be called homology, and the different constructions applied to different classes of space. Eilenberg and Steenrod wrote a list of properties that a homology construction should satisfy and proved that any two constructions satisfying the properties were essentially the same (isomorphic). So now there were axioms for homology.

The work of Eilenberg and Steenrod could barely have been expressed without some use of the language of category theory. Each class of space appropriate, for each homology construction, formed a category in which all the morphisms between the spaces (continuous functions in this case, not just homeomorphisms) also lived. Each construction was a functor H from a category of spaces to a category of graded abelian groups. The axioms involved natural transformations between functors. So it was from this book that Max learnt the very basic categorical notions.

Yet Max was not totally satisfied with the masterpiece [d]. It actually dealt not with spaces but with pairs of spaces, and axiomatized homology defined on these pairs. Max saw another step to be taken and wrote his first publication [2], which was also one of the three chapters of his PhD thesis [1]. Max established axioms that determined, uniquely up to isomorphism, the homology functor defined on single spaces. Clever topological constructions were involved in doing this and Max’s dexterity with universal techniques appeared at this early stage. The other two chapters of the thesis were published as [3] and [4].

Max had learnt some homological algebra from a short course at Cambridge taught by Davis Rees and from the book [f] by Car-tan and Eilenberg. This book uses only the basic elements of category theory yet it is clearly written in the spirit of the subject.

The review of [f] by Hochschild, which deserves quoting at length, articulates this spirit. It begins:

The title “Homological Algebra” is intended to designate a part of pure algebra which is the result of making algebraic homology theory independent of its original habitat in topology and building it up to a general theory of modules over associative rings. The particular formal aspect of this theory stemming from algebraic topology is that of a preoccupation with endomorphisms of square 0 in graded modules [that is, with chain complexes]. The conceptual flavor of homological algebra derives less specifically from topology than from the general ‘naturalistic’ trend of mathematics as a whole to supplement the study of the anatomy of any mathematical entity with an analysis of its behavior under the maps belonging to the larger mathematical system with which it is associated. In particular, homological algebra is concerned not so much with the intrinsic structure of modules but primarily with the pattern of compositions of homomorphisms between modules and their interplay with the various constructions by which new modules may be obtained from given ones.

The review concludes:

The appendix by D.A. Buchsbaum proposes an abstract framework of “exact categories” that is capable of accommodating the functor theory of this book as well as additional structural elements that one may wish to introduce. The proposed theory includes an abstract notion of duality which makes it unnecessary, at least in principle, to give separate treatments for covariance and contravariance and for projectivity and injectivity.

The appearance of this book must mean that the experimental phase of homological algebra is now surpassed. The diverse original homological constructions in various algebraic systems which were frequently of an ad hoc and artificial nature have been absorbed in a general theory whose significance goes far beyond its sources. The basic principles of homological algebra, and in particular the full functorial control over the manipulation of tensor products and modules of operator homomorphisms, will undoubtedly become standard algebraic technique already on the elementary level.

It is probably with such expectations that the authors have put so much missionary zeal into the systematization of their approach and the cataloguing of the basic results.

Note the central role ascribed to chain complexes since, as we shall see, these proved very important to Max. The idea of duality featured in the appendix and the importance of considering all ‘maps’ or morphisms amongst structures of the same kind are at the heart of category theory.

Max reminisced in [91] that it was late 1962, during lectures by Michael Atiyah at Harvard, that he first heard Mac Lane’s universal notion of product in a category. This idea was the beginning of a deeper theory, beyond merely a natural language.

Still interested in homology, contributing [5] and [10] to the subject, Max gave lectures at the University of Sydney in the early 1960s using, inter alia, the book [e] of Hilton and Wylie. He came to know the book well, and found a mistake in that first edition concerning the cohomology of a product of two spaces with coefficients in a general commutative ring. He adapted a result in another part of the book to provide a counter-example for the ring of integers. However this mistake in [e] was responsible for the rise of category theory in Australia (see the first paragraph of [91]).

The questions Max began to ask, arising from trying to understand the cohomology of a product, could not even be posed without categorical concepts. The chain complexes mentioned in the quote above from Hochschild were paramount. In particular, Max asked what it really meant for the homology functor to be a complete invariant for chain complexes. This led to publications [6], [7] and [8], which developed a considerable amount of category theory in its own right before turning to the new results in homological algebra. Ordinary mathematical structures are deemed ‘the same’ when they are isomorphic; categories are mathematical structures and isomorphism makes sense for them. However, there is a weaker notion, equivalence of categories, which is fundamental and which Max analysed thoroughly, introducing the idea much later to be called anafunctor by Makkai in [h]. He captured what it was for a functor to provide a complete system of invariants for objects of the domain category; he called these functors complete ([7], [8]). He implicitly recognized in [6] that additive categories were ‘rings with several objects’ by generalizing such notions as ideal and Jacobson radical to additive categories.

The germ of Max’s research on enriched category theory can be found in [8] with his concept of complex category. Most of category theory to that date had concentrated on additive categories. In a category, the morphisms from one object to another form a set, called a hom set. In an additive category, morphisms in each hom set can be added, forming an abelian group. A complex category does not merely ask for extra operations on the hom sets, rather, that each hom set should be replaced by a chain complex; that is, a complex category has its homs enriched in the category of chain complexes.

While on sabbatical at Tulane University, at the end of 1963, Max met Eilenberg who was lecturing at Las Cruces, New Mexico, on differential graded categories which he had invented with John C. Moore. Since these turned out to be the same as ‘complex categories’, Eilenberg immediately arranged a job for Max in Illinois for the next academic year. Soon after, Max met Mac Lane in Miami at an American Mathematical Society meeting where Max spoke on his paper [5]. Within a few weeks personal connections had been established that would direct Max’s research towards enriched category theory and ‘coherence’.

One reason that chain complexes form a suitable category on which to base hom enrichment is that there is an operation of tensor product of chain complexes which produces a new chain complex from two given ones. Mac Lane had studied general categories with tensor product. To match reality, such tensor products should not be associative nor commutative up to equality but only up to specific natural isomorphisms. It turns out it is desirable for these isomorphisms to satisfy infinitely many conditions. However, in [i], Mac Lane proved that this infinite class of conditions follows from finitely many conditions. Such results are called coherence theorems. On learning about this, Max was able in [9] to reduce Mac Lane’s finite list to two conditions for the associativity isomorphism and two more for the commutativity isomorphism.

Bénabou [j] also had been working on categories with tensor products and both Linton [k] and Bénabou [l] had ideas about enriching homs in suitable base categories. Max’s contribution [11] spurred Eilenberg into suggesting that they work together on the subject. Their first joint paper [13] was ground-breaking in many ways. They fundamentally extended category theory’s motivating concept of natural transformation and produced a calculus of substitution and composition for it. I have regretted that the paper did not include the diagrams that Max used to draw when speaking on the subject: these diagrams involved string-like linkages that are now understood as part of a bigger theory.

The first major conference on category theory was held at La Jolla, California, in June 1965. Eilenberg and Kelly reported on [14] and completed the 142-page document soon afterwards—involving many long letters between Sydney and New York. The paper developed the theory of two kinds of base categories for hom enrichment: closed categories and monoidal categories. The latter were categories with tensor product as previously mentioned. The former were categories with homs enriched in themselves; for this Linton [k] used the word ‘autonomous’. Most good examples were both closed and monoidal with the tensor and hom related by ‘adjunction’.

After [14], the structures for enriched category theory were established: enriched categories, enriched functors and enriched natural transformations.

Another more subtle contribution made by [14] was the use of Ehresmann’s 2-categories [m] to express their results: an example of what is now called ‘higher category theory’at work.Justas Max had found it necessary to use the language of category theory to express his ideas on homology, he now found the language of higher categories necessary to encapsulate his work on categorical structures. Ironically, the authors pointed out that these 2-categories were categories with homs enriched in the category of categories.

Now enriched category theory had its defining concepts. It needed Max’s courage and conviction to extend ordinary category theory to the enriched context. He did this beyond all expectations in papers such as [16], [17] and [32], culminating in his by-now classic book [41] (electronically available as [90]).

Some of the enrichment process was routine but a large part required insight that involved a re-thinking of ordinary category theory. In particular, the notion of conical limit, which sufficed for ordinary categories, needed supplementation in the enriched setting. Technically, Max realized that powers, also called cotensors, should be distinguished as limits and he introduced the concept of end. His book [41] culminates in a definitive theory for universally adjoining allocated limits to an enriched category.

Soon after writing his book he was able in [42] and [43] to enrich the whole subject of finite-limit theories which was developed by Ehresmann [n] and Gabriel and Ulmer [o] for ordinary categories. In my opinion, this was a great achievement. It is a case where a less courageous soul would be misled by the ordinary case. A cornerstone of the topic is the fact that a left Kan extension of a left exact functor should be left exact. It was known to be true when the target category was a topos. For enrichment one might expect to require a topos as the base monoidal category. However, surprisingly, Max proved it for a general locally finitely presentable base.

Figure 3. Diagram from ‘Closed categories’ by S. Eilenberg and G.M. Kelly (1966), p. 530.

Haunting Max was the dream to eliminate the large diagrams required in full proofs of theorems about monoidal and enriched categories. Such diagrams occurred in the La Jolla article [14] (see Fig. 3). The hope was to prove coherence theorems rendering the diagrams redundant. While Mac Lane was in Australia, he and Max were inspired by Lambek’s ideas in [p] to prove coherence theorems using Gentzen’s work in logic. This resulted in a major coherence theorem for symmetric closed monoidal categories published as [20]. Max, often jointly, proved other delicate specific coherence theorems: for example, [23], [27], [31] and [38].

Yet Max had more far-reaching ideas. He wanted more than individual coherence theorems: he wanted a whole theory of coherence. What are coherence theorems? In Chicago in 1970–71, he began a universal approach to the question, inventing the notion of club. At the same time and place, Max actively attended lectures by Peter May on what he came to call operads, published as [q]. Max saw a close connection between what he was trying to do with categories and what May was trying to do with topological spaces up to homotopy. Max produced a preprint. At the time, Mac Lane asked Max to expand more on the connection between clubs and operads; unusually for Max, he let that job slide and the work was not published. Under pressure of renewed interest in operads in category theory, the preprint was recently published as [86]. The paper was a barely noticed root for several major branches along which category theorists were to climb, such as Joyal’s theory of combinatorial species [r], classifying toposes [s] and Batanin’s higher operads [t].

The work on clubs appeared in [24], [25], [26], [29] and [30]. Max realized that a coherence theorem for a categorical structure was an assertion about free structures of that type. His clubs had the simplifying property that the free structure generated by a single object, appropriately augmented, determined all the free structures. This idea has proved paramount in work on higher category theory; see [u]. Such ideas surfaced later to do with the easier ordinary algebraic structures on sets. Max was invited to speak and write on the topic and produced [65] since it is of considerable relevance to computer science.

The importance of the cross-fertilization between homotopy theory and category theory is now well recognized. Yet it was a theme through Max’s work from the earliest times. Because of the analogy, he always insisted on using the homotopy symbol for categorical equivalence. Several techniques that he introduced into 2-dimensional universal algebra have their analogues in homotopy theory. No doubt the successful collaboration between Stephen Lack and Max [66], [67], [74], [77], [82], [85] gave Lack the grounding for his part in making these analogies mathematically precise [v].

Max collaborated extensively—with his students, with postdoctoral fellows, and with international colleagues. It is through these collaborations that some of Max’s basic and lasting contributions to category theory, ordinary and enriched, are permanently recorded. His papers [15], [22], [49] and [71] on factorization systems are a good example: he took a notion that had appeared in a rather exploratory way in work of Mac Lane and Isbell, completely pinned it down, and then used the concept creatively to solve problems. Those who collaborated with Max emerged with a deepened passion for precision, beauty and completeness in their research.

In short, category theory was the subject that matched Max’s approach to mathematics. Happily, it was there at the time he needed it. He collaborated with the founders and other key mathematicians, leaving an influential and stylish volume of work to motivate future mathematicians. The differential graded categories that he independently discovered are still a hot topic: for a sample of the developments since Max’s work the reader can look at Bon-dal and Kapranov [w], Drinfeld [x], Keller [y], [z] and [aa], Tabuada [ab] and [ac], Toen [ad], [ae] and [af] and Toen and Vaquie [ag]. There is also a survey in Keller’s invited address at the 2006 World Congress of Mathematicians; see [ah]. I am convinced that Max’s leading contributions to ordinary category theory, enriched category theory and higher universal algebra will stay at the heart of fundamental mathematics.

Acknowledgements

I am grateful to Imogen Kelly for her invaluable help with this and earlier biographical accounts. Steve Lack, George Janelidze, Margery Street and Richard Wood have provided thoughtful suggestions for improvements to earlier drafts, as have the two referees to the submitted version. I have only slightly adapted the words suggested.

References

  1. Samuel Eilenberg and Saunders Mac Lane, Natural isomorphisms in group theory, Proc. Nat. Acad. Sci. U. S. A. 28 (1942) 537–543.
  2. Samuel Eilenberg and Saunders Mac Lane, General theory of natural equivalences, Trans. Amer. Math. Soc. 58 (1945) 231–294.
  3. Vladimir Voevodsky, An Intuitive Introduction to Motivic Homotopy Theory, Clay Mathematics Videos (Mathematical Sciences Research Institute, Berkeley, 2002; available at http://claymath.msri.org/voevodsky2002.mov).
  4. Samuel Eilenberg and Norman E. Steenrod, Foundations of algebraic topology (Princeton University Press, Princeton, New Jersey, 1952) xv+328 pp.
  5. Peter Hilton and Shaun Wylie, Homology Theory (Cambridge University Press, 1960).
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  10. Jean Bénabou, Catégories avec multiplication, C. R. Acad. Sci. Paris 256 (1963) 1887–1890.
  11. Fred E. J. Linton, Autonomous categories and duality of functors, J. Algebra 2 (1965) 315–349.
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  13. Charles Ehresmann, Catégories structurées, Ann. Sci. École Norm. Sup. 89 (1963) 349–425.
  14. Charles Ehresmann, Esquisses et types desstructures algébriques, Bul. Inst. Politehn. Ia¸si (N. S. ) 14 (18) (1968) 1–14.
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  17. J. Peter May, The geometry of iterated loop spaces, Lectures Notes in Math. 271 (Springer-Verlag, Berlin-New York, 1972) viii+175 pp.
  18. André Joyal, Une théorie combinatoire desséries formelles. Adv. Math. 42 (1981) 1–82.
  19. Gavin C. Wraith, Lectures on elementary topoi. In: “Model theory and topoi (Conf., Bangor, 1973)”, Lecture Notes in Math. 445 (Springer, Berlin, 1975) 114–206.
  20. Michael A. Batanin, Monoidal globular categories as a natural environment for the theory of weak n-categories, Adv. Math. 136 (1998) 39–103.
  21. Mark Weber, Generic morphisms, parametric representations and weakly Cartesian monads, Theory Appl. Categ. 13 (2004) 191–234.
  22. Stephen Lack, Homotopy-theoretic aspects of2-monads, J. Homotopy Relat. Struct. 2 (2007) 229–260.
  23. Alexey Bondal and Mikhail Kapranov, Enhanced triangulated categories (Russian), Mat. Sb. 181 (1990) 669–683; English translation in Math USSR-Sb 70 (1991) 93–107.
  24. Vladimir Drinfeld, DG quotients of DG categories, J. Algebra 272 (2004) 643–691.
  25. BernhardKeller, DerivingDGcategories, Ann. Sci. Ecole Norm. Sup. 4 (1994) 63–102.
  26. Bernhard Keller, Invariance and localization for cyclic homology of DG algebras, J. Pure Appl. Algebra 123 (1998) 223–273.
  27. Bernhard Keller, On the cyclic homology of exact categories, J. Pure Appl. Algebra 136 (1999) 1–56.
  28. Goncalo Tabuada, Une structure de catégoriede modèles de Quillen sur la catégorie des dgcatégories, C. R. Math. Acad. Sci. Paris 340 (2005) 15–19.
  29. Goncalo Tabuada, Invariants additifs de DGcatégories, Int. Math. Res. Not. 53 (2005) 3309–3339.
  30. Bertrand Toen, Derived Hall algebras, Duke. Math. J. 135 (2006) 587–615.
  31. Bertrand Toen, The homotopy theory of DGcategoriesand derived Morita theory, Invent. Math. 167 (2007) 615–667.
  32. Bertrand Toen, Higher and derived stacks: a global overview, Proc. Sympos. Pure Math. 80 (2009) 435–487.
  33. Bertrand Toen and Michel Vaquie, Moduli of objects in DG-categories, Ann. Sci. Ecole Norm. Sup. 40 (2007) 387–444.
  34. Bernhard Keller, On differential graded categories, International Congress of Mathematicians2 (Eur. Math. Soc. , Zürich, 2006) 151–190.

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Publications Dedicated

  1. Aurelio Carboni, George Janelidze and Ross Street (Editors), Special Volume celebrating the 70th birthday of Professor Max Kelly, Journal of Pure and Applied Algebra 175 (1–3) (8 November 2002).
  2. John C. Baez and J. Peter May (Editors), Towards Higher Categories, Dedicated to Max Kelly (The IMA Volumes in Mathematics and its Applications; Springer, 2010).
  3. Martin Hyland, George Janelidze, Michael Johnson, Peter Johnstone, Stephen Lack, Ross Street, Walter Tholen and Richard Wood (Editors), Special Volume in Memory of Max Kelly, Applied Categorical Structures (to appear).

Ross Street. Department of Mathematics, Faculty of Science, Macquarie University, NSW 2109, Australia. Email: ross.street@mq.edu.au