Max Kelly

Gregory Maxwell "Max" Kelly (5 June 1930 – 26 January 2007), mathematician, founded the thriving Australian school of category theory.

A native of Australia, Kelly obtained his Ph.D. at Cambridge University in homological algebra in 1957, publishing his first paper in that area in 1959, Single-space axioms for homology theory. He taught in the Pure Mathematics department at Sydney University from 1957 to 1966, rising from lecturer to reader. During 1963–1965 he was a visiting fellow at Tulane University and the University of Illinois, where with Samuel Eilenberg he formalized and developed the notion of an enriched category based on intuitions then in the air about making the homsets of a category just as abstract as the objects themselves.

He subsequently developed the notion in considerably more detail in his 1982 monograph Basic Concepts of Enriched Category Theory (henceforth appreviated BCECT). Let be a monoidal category, and denote by -Cat the category of -enriched categories. Among other things, Kelly showed that -Cat has all weighted limits and colimits even when does not have all ordinary limits and colimits. He also developed the enriched counterparts of Kan extensions, density of the Yoneda embedding, and essentially algebraic theories. The explicitly foundational role of the category Set in his treatment is noteworthy in view of the folk intuition that enriched categories liberate category theory from the last vestiges of Set as the codomain of the ordinary external hom-functor.

In 1967 Kelly was appointed Professor of Pure Mathematics at the University of New South Wales. In 1972 he was elected a Fellow of the Australian Academy of Science. He returned to the University of Sydney in 1973, serving as Professor of Mathematics until his retirement in 1994. In 2001 he was awarded the Australian government's Centenary Medal. He continued to participate in the department as professorial fellow and professor emeritus until his death at age 76 on 26 January 2007.

Kelly worked on many other aspects of category theory besides enriched categories, both individually and in a number of fruitful collaborations. His Ph.D. student Ross Street is himself a noted category theorist and early contributor to the Australian category theory school.

The following annotated list of papers includes several papers not by Kelly which cover closely related work.

Structures borne by categories

• Kelly, G. M. (2005) [1982]. Basic Concepts of Enriched Category Theory. Reprints in Theory and Applications of Categories. 10. pp. 1–136. Originally published as London Mathematical Society Lecture Notes Series 64 by Cambridge University Press in 1982. This book provides both a fundamental development of enriched category theory and, in the last two chapters, a study of generalized essentially algebraic theories in the enriched context. Chapters: 1. The elementary notions; 2. Functor categories; 3. Indexed [i.e., Weighted] limits and colimits; 4. Kan extensions; 5. Density; 6. Essentially-algebraic theories defined by reguli and by sketches.

Many of Kelly's papers discuss the structures that categories can bear. Here are several of his papers on this subject. In the following "SLNM" stands for Springer Lecture Notes in Mathematics, while the titles of the four journals most frequently publishing research on categories are abbreviated as follows: JPAA = Journal of Pure and Applied Algebra, TAC = Theory and Applications of Categories, ACS = Applied Categorical Structures, CTGDC = Cahiers de Topologie et Géométrie Différentielle Catégoriques (Volume XXV (1984) and later), CTGD = Cahiers de Topologie et Géométrie Différentielle (Volume XXIV (1983) and earlier).

Functor categories and functorial calculi

Note that categories of sheaves and models are subcategories of functor categories, consisting of the functors which preserve certain structure. Here we consider the general case, functors only required to preserve the structure intrinsic to the source and target categories themselves.

• Eilenberg, Samuel; Kelly, G. M. (1966). "A generalization of the functorial calculus". J. Algebra. 3 (3): 366–375. doi:10.1016/0021-8693(66)90006-8. Compare to Street "Functorial Calculus in Monoidal Bicategories" below.

• Day, B. J.; Kelly, G. M. (1969). "Enriched functor categories". Reports of the Midwest Category Seminar III. SLNM. 106. pp. 178–191. doi:10.1007/BFb0059146.

• Kelly, G. M. (1972). "Many-variable functorial calculus. I.". Coherence in Categories. SLNM. 281. pp. 66–105. doi:10.1007/BFb0059556. Mainly semantic clubs. Closely related to the paper "An Abstract Approach to Coherence".

• Street, Ross (2003). "Functorial Calculus in Monoidal Bicategories". ACS. 11 (3): 219–227. doi:10.1023/A:1024247613677. "The definition and calculus of extraordinary natural transformations is extended to a context internal to any autonomous monoidal bicategory. The original calculus is recaptured from the geometry of the monoidal bicategory whose objects are categories enriched in a cocomplete symmetric monoidal category and whose morphisms are modules." Compare to Eilenberg-Kelly "A generalization of the functorial calculus" above.

Bimodules, distributeurs, profunctors, proarrows, fibrations, and equipment

In several of his papers Kelly touched on the structures described in the heading. For the reader's convenience, and to enable easy comparisons, several closely related papers by other authors are included in the following list.

Factorization systems, reflective subcategories, localizations, and Galois theory

• Kelly, G.M. (1969). "Monomorphisms, Epimorphisms, and Pull-Backs". J. Austral. Math. Soc. 9 (1–2): 124–142. doi:10.1017/S1446788700005693.

• Kelly, G.M. (1983). "A note on the generalized reflexion of Guitart and Lair". CTGD. 24 (2): 155–159. MR 0710038.

• Cassidy, C.; Hébert, M.; Kelly, G. M. (1985). "Reflective subcategories, localizations and factorization systems". J. Austral. Math. Soc. 38 (3): 287–329. doi:10.1017/S1446788700023624. , followed by Corrigenda. "This work is a detailed analysis of the relationship between reflective subcategories of a category and factorization systems supported by the category."

• Borceux, F.; Kelly, G.M. (1987). "On locales of localizations". JPAA. 46 (1): 1–34. doi:10.1016/0022-4049(87)90040-5. "Our aim is to study the ordered set Loc of localizations of a category , showing it to be a small complete lattice when is complete with a (small) strong generator, and further to be the dual of a locale when is a locally-presentable category in which finite limits commute with filtered colimits. We also consider the relations between Loc and Loc arising from a geometric morphism → ; and apply our results in particular to categories of modules."

• Kelly, G.M. (1987). "On the ordered set of reflective subcategories". Bull. Austral. Math. Soc. 36 (1): 137–152. doi:10.1017/S0004972700026381. "Given a category , we consider the (often large) set Ref of its reflective (full, replete) subcategories, ordered by inclusion."

• Kelly, G.M.; Lawvere, F.W. (1989). "On the Complete Lattice of Essential Localizations". Bulletin de la Société Mathématique de Belgique Series A. 41: 289–319. No copy of this could be found on the web as of 2017-09-29.

• Kelly, G. M. (1991). "A note on relations relative to a factorization system". Proceedings of the International Conference held in Como, Italy, July 22–28, 1990. SLNM. 1488. pp. 249–261. doi:10.1007/BFb0084224.

• Korostenski, Mareli; Tholen, Walter (1993). "Factorization systems as Eilenberg-Moore algebras". JPAA. 85 (1): 57–72. doi:10.1016/0022-4049(93)90171-O.

• Carboni, A.; Kelly, G. M.; Pedicchio, M. C. (1993). "Some remarks on Maltsev and Goursat categories". ACS. 1 (4): 385–421. doi:10.1007/BF00872942.  : Begins with basic treatment of regular and exact categories, and equivalence relations and congruences therein, then studies the Maltsev and Goursat conditions.

• Janelidze, G.; Kelly, G. M. (1994). "Galois theory and a general notion of central extension". JPAA. 97 (2): 135–161. doi:10.1016/0022-4049(94)90057-4. "We propose a theory of central extensions for universal algebras, and more generally for objects in an exact category , centrality being defined relatively to an “admissible” full subcategory of ."

• Carboni, A.; Janelidze, G.; Kelly, G. M.; Paré, R. (1997). "On Localization and Stabilization for Factorization Systems". ACS. 5 (1): 1–58. doi:10.1023/A:1008620404444.  : includes "self-contained modern accounts of factorization systems, descent theory, and Galois theory"

• Janelidze, G.; Kelly, G. M. (1997). "The reflectiveness of covering morphisms in algebra and geometry". TAC. 3 (6): 132–159. "Many questions in mathematics can be reduced to asking whether Cov(B) is reflective in C \downarrow B; and we give a number of disparate conditions, each sufficient for this to be so."

• Janelidze, G.; Kelly, G. M. (2000). "Central extensions in universal algebra: a unification of three notions". Algebra universalis. 44 (1–2): 123–128. doi:10.1007/s000120050174.

• Rosebrugh, Robert; Wood, R. J. (2001). "Distributive laws and factorization". JPAA. 175 (1–3): 327–353. doi:10.1016/S0022-4049(02)00140-8.

Actions and algebras

Also semidirect products.

• Kelly, G. M. (1980). "A unified treatment of transfinite constructions for free algebras, free monoids, colimits, associated sheaves, and so on". Bull. Austral. Math. Soc. 22 (1): 1–83. doi:10.1017/S0004972700006353. , followed by: "Two addenda to the author's ‘Transfinite constructions’ "

• Janelidze, G.; Kelly, G. M. (2001). "A Note on Actions of a Monoidal Category". TAC. 9 (4): 61–91.

• Borceux, F.W.; Janelidze, G.; Kelly, G.M. (2005). "On the representability of actions in a semi-abelian category". TAC. 14 (11): 244–286. "We consider a semi-abelian category and we write Act(G,X) for the set of actions of the object G on the object X, in the sense of the theory of semi-direct products in . We investigate the representability of the functor Act(-,X) in the case where is locally presentable, with finite limits commuting with filtered colimits."

• Borceux, Francis; Janelidze, George W.; Kelly, Gregory Maxwell (2005). "Internal object actions". Commentationes Mathematicae Universitatis Carolinae. 46 (2): 235–255. MR 2176890. "We describe the place, among other known categorical constructions, of the internal object actions involved in the categorical notion of semidirect product, and introduce a new notion of representable action providing a common categorical description for the automorphism group of a group, for the algebra of derivations of a Lie algebra, and for the actor of a crossed module." --- Contains a table showing various examples.

Limits and colimits

• Borceux, Francis; Kelly, G. M. (1975). "A notion of limit for enriched categories". Bull. Austral. Math. Soc. 12 (1): 49–72. doi:10.1017/S0004972700023637.

• Kelly, G. M.; Koubek, V. (1981). "The large limits that all good categories admit". JPAA. 22 (3): 253–263. doi:10.1016/0022-4049(81)90102-X.

• Im, Geun Bin; Kelly, G. M. (1986). "On classes of morphisms closed under limits" (PDF). J. Korean Math. Soc. 23 (1): 1–18. "We say that a class of morphisms in a category is closed under limits if, whenever are functors that admit limits, and whenever is a natural transformation each of whose components lies in , then the induced morphism also lies in ."

• Albert, M. H.; Kelly, G. M. (1988). "The closure of a class of colimits". JPAA. 51 (1–2): 1–17. doi:10.1016/0022-4049(88)90073-4.

• Kelly, G. M.; Paré, Robert (1988). "A note on the Albert–Kelly paper "the closure of a class of colimits"". JPAA. 51 (1–2): 19–25. doi:10.1016/0022-4049(88)90074-6.

• Kelly, G. M. (1989). "Elementary observations on 2-categorical limits". Bull. Austral. Math. Soc. 39 (2): 301–317. doi:10.1017/S0004972700002781. Kan Extension Seminar discussion on 2014-04-18 by Christina Vasilakopoulou

• Bird, G. J.; Kelly, G. M.; Power, A. J.; Street, R. H. (1989). "Flexible limits for 2-categories". JPAA. 61 (1): 1–27. doi:10.1016/0022-4049(89)90065-0.

• Kelly, G. M.; Lack, Stephen; Walters, R. F. C. (1993). "Coinverters and categories of fractions for categories with structure". ACS. 1 (1): 95–102. doi:10.1007/BF00872988. "A category of fractions is a special case of a coinverter in the 2-category Cat...."

• Kelly, G. M.; Schmitt, V. (2005). "Notes on enriched categories with colimits of some class". TAC. 14 (17): 399–423. arXiv:math.CT/0509102. "The paper is in essence a survey of categories having -weighted colimits for all the weights in some class ."

Adjunctions

• Kelly, G. M. (1969). "Adjunction for enriched categories". Reports of the Midwest Category Seminar III. SLNM. 106. pp. 166–177. doi:10.1007/BFb0059145.

• Kelly, G. M. (1974). "Doctrinal adjunction". Category Seminar (Proceedings Sydney Category Theory Seminar 1972/1973). SLNM. 420. pp. 257–280. doi:10.1007/BFb0063105.

• Im, Geun Bin; Kelly, G. M. (1986). "Some remarks on conservative functors with left adjoints" (PDF). J. Korean Math. Soc. 23 (1): 19–33. "We are interested here in those functors which, like the forgetful functors of algebra, are conservative and have left adjoints."

• Im, Geun Bin; Kelly, G. M. (1987). "Adjoint-triangle theorems for conservative functors". Bull. Austral. Math. Soc. 36 (1): 133–136. doi:10.1017/S000497270002637X. "An adjoint-triangle theorem contemplates functors and where and have left adjoints, and gives sufficient conditions for also to have a left adjoint. We are concerned with the case where is conservative - that is, isomorphism-reflecting"

• Kelly, G. M.; Power, A. J. (1993). "Adjunctions whose counits are coequalizers, and presentations of finitary enriched monads". JPAA. 89 (1–2): 163–179. doi:10.1016/0022-4049(93)90092-8. This is a duplicate of a reference in the section on structures borne by categories, which is the subject of the last two sections of the paper. However the first three sections are about "functors of descent type ", which are right adjoint functors enjoying the property stated in the title of the paper.

• Street, Ross (2012). "The core of adjoint functors". TAC. 27 (4): 47–64. "There is a lot of redundancy in the usual definition of adjoint functors. We define and prove the core of what is required. First we do this in the hom-enriched context. Then we do it in the cocompletion of a bicategory with respect to Kleisli objects, which we then apply to internal categories. Finally, we describe a doctrinal setting."

Miscellaneous papers on category theory

• Kelly, G. M. (1964). "On the radical of a category". J. Austral. Math. Soc. 4 (3): 299–307. doi:10.1017/S1446788700024071.

• Day, B. J.; Kelly, G. M. (1970). "On topological quotient maps preserved by pullbacks or products". Math. Proc. Camb. Phil. Soc. 67 (3): 553–553. Bibcode:1970PCPS...67..553D. doi:10.1017/S0305004100045850. This paper is in the intersection of category theory and topology: "We are concerned with the category of topological spaces and continuous maps." It is mentioned in BCECT, where it provides a counter-example to the conjecture that the cartesian monoidal category of topological spaces might be cartesian closed; see section 1.5.

• Kelly, G. M.; Street, Ross, eds. (1972). Abstracts of the Sydney Category Seminar 1972 (PDF). pp. 1–66. Some historical information on personnel matters, and early versions of ideas to be published formally later.

• Kelly, Max; Labella, Anna; Schmitt, Vincent; Street, Ross (2002). "Categories enriched on two sides (Dedicated to Saunders Mac Lane on his 90th birthday)". JPAA. 168 (1): 53–98. doi:10.1016/S0022-4049(01)00048-2. "We introduce morphisms of bicategories, more general than the original ones of Bénabou. When , such a morphism is a category enriched in the bicategory . Therefore, these morphisms can be regarded as categories enriched in bicategories “on two sides”. There is a composition of such enriched categories, leading to a tricategory of a simple kind whose objects are bicategories. It follows that a morphism from to in induces a 2-functor to , while an adjunction between and in induces one between the 2-categories and . Left adjoints in are necessarily homomorphisms in the sense of Bénabou, while right adjoints are not. Convolution appears as the internal hom for a monoidal structure on . The 2-cells of are functors; modules can also be defined, and we examine the structures associated with them."

• Kelly, G. M.; Lack, Stephen (2004). "Monoidal functors generated by adjunction, with applications to transport of structure". Fields Institute Communications. 43: 319–340. ISSN 1069-5265.
• Kelly, G. Maxwell (2007). "The beginnings of category theory in Australia.". Categories in Algebra, Geometry and Mathematical Physics. Contemporary Math. 431. Amer. Math. Soc. pp. 1–6. ISBN 978-0-8218-3970-6. A historical account.

Homology

The Biographical Memoir by Ross Street gives a detailed description of Kelly's early research on homological algebra, pointing out how it led him to create concepts which would eventually be given the names "differential graded categories" and "anafunctors".

• Kelly, G. M. (1959). "Single-space axioms for homology theory". Mathematical Proceedings of the Cambridge Philosophical Society. 55 (1): 10–22. Bibcode:1959PCPS...55...10K. doi:10.1017/S030500410003365X.

• Kelly, G. M. (1961). "The exactness of Čech homology over a vector space". Math. Proc. Camb. Phil. Soc. 57 (2): 428–429. Bibcode:1961PCPS...57..428K. doi:10.1017/S0305004100035398.

• Kelly, G. M. (1961). "On manifolds containing a submanifold whose complement is contractible". Math. Proc. Camb. Phil. Soc. 57 (3): 507–515. Bibcode:1961PCPS...57..507K. doi:10.1017/S0305004100035568.

• Kelly, G. M. (1963). "Observations on the Künneth theorem". Math. Proc. Camb. Phil. Soc. 59 (3): 575–587. Bibcode:1963PCPS...59..575K. doi:10.1017/S0305004100037257.

• Kelly, G. M. (1964). "Complete functors in homology I. Chain maps and endomorphisms". Math. Proc. Camb. Phil. Soc. 60 (4): 721–735. Bibcode:1964PCPS...60..721K. doi:10.1017/S0305004100038202.

• Kelly, G. M. (1964). "Complete functors in Homology: II. The exact homology sequence". Math. Proc. Camb. Phil. Soc. 60 (4): 737–749. Bibcode:1964PCPS...60..737K. doi:10.1017/S0305004100038214.

• Kelly, G. M. (1965). "A lemma in homological algebra". Math. Proc. Camb. Phil. Soc. 61 (1): 49–52. Bibcode:1965PCPS...61...49K. doi:10.1017/S0305004100038627.

• Kelly, G. M. (1965). "Chain maps inducing zero homology maps". Math. Proc. Camb. Phil. Soc. 61 (4): 847–854. Bibcode:1965PCPS...61..847K. doi:10.1017/S0305004100039207.

Miscellaneous papers on other subjects

• Dickson, S. E.; Kelly, G. M. (1970). "Interlacing methods and large indecomposables". Bull. Austral. Math. Soc. 3 (3): 337–348. doi:10.1017/S0004972700046037.

• Kelly, G. M.; Pultr, A. (1978). "On algebraic recognition of direct-product decompositions". JPAA. 12 (3): 207–224. doi:10.1016/0022-4049(87)90002-8.

General references

• Carboni, Aurelio; Janelidze, George; Street, Ross (8 November 2002). "Forward to Special Volume Celebrating the 70th Birthday of Professor Max Kelly". Journal of Pure and Applied Algebra. 175 (1–3): 1–5. doi:10.1016/S0022-4049(02)00125-1.  : contains list of 87 publications of Kelly from 1959 to early 2002

• Street, Ross (April 11, 2007). "Obituary : Polymath revelled in the mystery of numbers". Sydney Morning Herald. Retrieved 2017-09-08.
• Street, Ross (2008). "Editorial Notice: Max Kelly 5 June 1930 - 26 January 2007" (PDF). Theory and Applications of Categories. 20: 1–4.
• Street, Ross (2010). "Biographical Memoir : Gregory Maxwell Kelly 1930–2007". Australian Academy of Sciences.  : includes complete list of 92 publications from 1957 PhD thesis to posthumously published 2008 paper ; probably the most complete survey of Kelly’s career
• Baez, John C.; May, J. Peter, eds. (2010). Towards Higher Categories. The IMA Volumes in Mathematics and its Applications. 152. Springer-Verlag. doi:10.1007/978-1-4419-1524-5. ISBN 978-1-4419-1523-8.  : "This book is dedicated to Max Kelly, the founder of the Australian school of category theory"

• Street, Ross (2010). "An Australian Conspectus of Higher Categories". In Baez, J.; May, J. Towards Higher Categories. The IMA Volumes in Mathematics and its Applications. 152. Springer-Verlag. pp. 237–264. doi:10.1007/978-1-4419-1524-5_6. From the forward to the book: "[This paper], by Kelly’s student Ross Street, gives a fascinating mathematical and personal account of the development of higher category theory in Australia." The first quarter of the article contains information about the work of Kelly. It is available from the author here.

• Janelidze, George; Hyland, Martin; Johnson, Michael; et al., eds. (February 2011). "Forward to Special Issue Dedicated to the Memory of Professor Gregory Maxwell Kelly". Applied Categorical Structures. 19 (1): 1–7. doi:10.1007/s10485-010-9235-y.CS1 maint: Explicit use of et al. (link)  : contains list of publications of Kelly

External links

• O'Connor, John J.; Robertson, Edmund F., "Max Kelly", MacTutor History of Mathematics archive, University of St Andrews .
• Gregory Maxwell (Max) Kelly at the Mathematics Genealogy Project
• Max Kelly's Perpetual Web Page: a memorial page set up by Kelly's son Simon Kelly.
• "In Memory of Max Kelly": a post at The n-Category Café, containing praise from his fellow mathematicians
• G. M. Kelly at DBLP Bibliography Server