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Max Kelly
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Name
Max Kelly
Max kelly cyclamen thirst 2011
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 V be a monoidal category, and denote by V-Cat the category of V-enriched categories. Among other things, Kelly showed that V-Cat has all weighted limits and colimits even when V 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.
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.
Preliminaries
The following paper introduces concepts which are used in many of the following papers.
Kelly, G. M.; Street, Ross (1974). "Review of the elements of 2-categories". Proceedings Sydney Category Theory Seminar 1972/1973. SLNM. 420. doi:10.1007/BFb0063101. 2014-03-09 discussion at Kan Extension Seminar by Dimitri Zaganidis
Some specific structures categories can bear
Kelly, G. M. (1965). "Tensor products in categories". Journal of Algebra. 2: 15–37. doi:10.1016/0021-8693(65)90022-0.
Eilenberg, Samuel; Kelly, G. Max (1966). "Closed categories". Proceedings of the Conference on Categorical Algebra, La Jolla 1965. Springer-Verlag. pp. 421–562. ISBN 978-3-642-99902-4. doi:10.1007/978-3-642-99902-4_22.
Kelly, G. M. (1986). "A survey of totality for enriched and ordinary categories". Cahiers de Topologie et Géométrie Différentielle Catégoriques. 27 (2): 109–132. MR 850527.
Im, Geun Bin; Kelly, G. M. (1986). "A universal property of the convolution monoidal structure". Journal of Pure and Applied Algebra. 43 (1): 75–88. doi:10.1016/0022-4049(86)90005-8.
Categories with few structures, or many
Foltz, F.; Lair, C.; Kelly, G. M. (1980). "Algebraic categories with few monoidal biclosed structures or none". Journal of Pure and Applied Algebra. 17 (2): 171–177. doi:10.1016/0022-4049(80)90082-1.
Kelly, G. M.; Rossi, F. (1985). "Topological categories with many symmetric monoidal closed structures". Bulletin of the Australian Mathematical Society. 31 (1): 41–59. doi:10.1017/S0004972700002264.
Clubs
Kelly, G. M. (1972). "Many-variable functorial calculus. I.". Coherence in Categories. SLNM. 281. pp. 66–105. doi:10.1007/BFb0059556. Mainly semantic clubs.
Kelly, G. M. (1972). "An abstract approach to coherence". Coherence in Categories. SLNM. 281. pp. 106–147. doi:10.1007/BFb0059557. Mainly syntactic clubs, and how to present them.
Kelly, G. M. (1974). "On clubs and doctrines". Proceedings Sydney Category Theory Seminar 1972/1973. SLNM. 420. pp. 181–256. doi:10.1007/BFb0063104.
Kelly, G. M. (1992). "On clubs and data-type constructors". Applications of Categories in Computer Science. Cambridge University Press. pp. 163–190. doi:10.1017/CBO9780511525902.010. 2017-04-17 discussion at Kan Extension Seminar by Pierre Cagne
Coherence
For an overview of Kelly's earlier and later views on coherence, see "An Abstract Approach to Coherence" (1972) and "On Clubs and Data-Type Constructors" (1992), listed in the section on clubs.
Mac Lane, Saunders (1963). "Natural Associativity and Commutativity". Rice University Studies. 49 (4): 28–46.
Kelly, G. M. (1964). "On MacLane's conditions for coherence of natural associativities, commutativities, etc.". Journal of Algebra. 1 (4): 397–402. doi:10.1016/0021-8693(64)90018-3.
Kelly, G. M.; Mac Lane, Saunders (1971). "Coherence in closed categories". Journal of Pure and Applied Algebra. 1 (2): 97–140. doi:10.1016/0022-4049(71)90013-2. : Erratum
Kelly, G. M.; Mac Lane, Saunders (1972). "Closed coherence for a natural transformation". Coherence in Categories. SLNM. 281. pp. 1–28. doi:10.1007/BFb0059554.
Kelly, G. M. (1972). "A cut-elimination theorem". Coherence in Categories. SLNM. 281. pp. 196–213. doi:10.1007/BFb0059559. Mainly a technical result needed for proving coherence results about closed categories, and more generally, about right adjoints.
Kelly, G. M. (1974). "Coherence theorems for lax algebras and for distributive laws". Proceedings Sydney Category Theory Seminar 1972/1973. SLNM. 420. pp. 281–375. doi:10.1007/BFb0063106. In this paper Kelly introduces the idea that coherence results may be viewed as equivalences, in a suitable 2-category, between pseudo and strict algebras.
Kelly, G. M.; LaPlaza, M. L. (1980). "Coherence for compact closed categories". Journal of Pure and Applied Algebra. 19: 193–213. doi:10.1016/0022-4049(80)90101-2.
Power, John (1989). "A general coherence result". Journal of Pure and Applied Algebra. 57 (2): 165–173. doi:10.1016/0022-4049(89)90113-8.
Lack, Stephen (1993). "Codescent objects and coherence (Dedicated to Max Kelly on the occasion of his 70th birthday)". Journal of Pure and Applied Algebra. 175 (1): 223–241. doi:10.1016/S0022-4049(02)00136-6. 2014-06-02 discussion at Kan Extension Seminar by Alex Corner
Lawvere theories, commutative theories, and the structure-semantics adjunction
Faro, Emilio; Kelly, G. M. (2000). "On the canonical algebraic structure of a category". Journal of Pure and Applied Algebra. 154 (1-3): 159–176. doi:10.1016/S0022-4049(99)00187-5. For categories A satisfying some smallness conditions, "applying Lawvere's “structure” functor to the hom-functor H=HomA:Aop×A→Set produces a Lawvere theory A∗, called the canonical algebraic structure of A". --- In the first section, the authors "briefly recall the basic facts about Lawvere theories and the structure-semantics adjunction" before proceeding to apply it to the situation described above. The "brief" review runs over three pages in the printed journal. It may be the most complete exposition in print of how Kelly formulates, analyzes, and uses the notion of Lawvere theory.
Monads
Street, Ross (1972). "The formal theory of monads". Journal of Pure and Applied Algebra. 2 (2): 149–168. doi:10.1016/0022-4049(72)90019-9. 2014-01-27 discussion at Kan Extension Seminar by Eduard Balzin
Blackwell, R.; Kelly, G. M.; Power, A. J. (1989). "Two-dimensional monad theory". Journal of Pure and Applied Algebra. 59 (1): 1–41. doi:10.1016/0022-4049(89)90160-6. 2014-04-28 discussion at Kan Extension Seminar by Sam van Gool
Monadicity
Kelly, G. M. (1980). "Examples of non-monadic structures on categories". Journal of Pure and Applied Algebra. 18 (1): 59–66. doi:10.1016/0022-4049(80)90116-4.
Kelly, G. M.; Le Creurer, I. J. (1997). "On the monadicity over graphs of categories with limits". Cahiers de Topologie et Géométrie Différentielle Catégoriques. 38 (3): 179–191. MR 1474564.
Kelly, G. M.; Lack, Stephen (2000). "On the monadicity of categories with chosen colimits". Theory and Applications of Categories. 7 (7): 148–170.
Adamek, Jiri; Kelly, G. M. (2000). "M-Completeness is seldom monadic over graphs". Theory and Applications of Categories. 7 (8): 171–205.
Operads
Kelly, G. M. (2005) [1972]. "On the operads of J.P. May". Reprints in Theory and Applications of Categories. 13: 1–13. 2017-03-01 discussion at Kan Extension Seminar by Simon Cho
Presentations
Dubuc, Eduardo J.; Kelly, G. M. (1983). "A presentation of topoi as algebraic relative to categories or graphs". Journal of Algebra. 81 (2): 420–433. doi:10.1016/0021-8693(83)90197-7.
Kelly, G. M.; Power, A. J. (1993). "Adjunctions whose counits are coequalizers, and presentations of finitary enriched monads". Journal of Pure and Applied Algebra. 89 (1-2): 163–179. doi:10.1016/0022-4049(93)90092-8. "Our primary goal is to show that - in the context of enriched category theory - every finitary monad on a locally finitely presentable category A admits a presentation in terms of A-objects Bc of ‘basic operations of arity c’ (where c runs through the finitely-presentable objects of A) and A-objects Ec of ‘equations of arity c’ between derived operations." -- Section 4 is titled "Finitary enriched monads as algebras for finitary monads"; section 5 "Presentations of finitary monads"; it makes a connection with Lawvere theories.
Kelly, G. M.; Lack, Stephen (1993). "Finite-product-preserving functors, Kan extensions, and strongly-finitary 2-monads". Applied Categorical Structures. 1 (1): 85–94. doi:10.1007/BF00872987. Using the results in the Kelly-Power paper "Adjunctions whose counits are coequalizers, and presentations of finitary enriched monads" "We study those 2-monads on the 2-category Cat of categories which, as endofunctors, are the left Kan extensions of their restrictions to the sub-2-category of finite discrete categories, describing their algebras syntactically. Showing that endofunctors of this kind are closed under composition involves a lemma on left Kan extensions along a coproduct-preserving functor in the context of cartesian closed categories, which is closely related to an earlier result of Borceux and Day." --- in other words, they study "the subclass of the finitary 2-monads on Cat consisting of those whose algebras may be described using only functors An→A, where n is a natural number (as well as natural transformations between these and equations between derived operations)".
Sketches, theories, and models
For a presentation, in the unenriched setting, of some of the main ideas in the last half of BCECT, see "On the Essentially-Algebraic Theory Generated by a Sketch". The first paragraph of the final section of that paper states an unenriched version of the final proclaimed theorem (6.23) of BCECT, right down to the notation; the main body of the paper is devoted to the proof of that theorem in the unenriched context.
Freyd, P. J.; Kelly, G. M. (1972). "Categories of continuous functors, I". Journal of Pure and Applied Algebra. 2 (3): 169–191. doi:10.1016/0022-4049(72)90001-1. : There is a very significant Erratum ; 2014-02-15 discussion at Kan Extension Seminar by Fosco Loregian
Kelly, G. M. (1982). "On the essentially-algebraic theory generated by a sketch". Bulletin of the Australian Mathematical Society. 26 (1): 45–56. doi:10.1017/S0004972700005591.
Kelly, G. M. (1982). "Structures defined by finite limits in the enriched context, I". Cahiers de Topologie et Géométrie Différentielle Catégoriques. 23 (1): 3–42. MR 648793. 2017-04-03 discussion of enriched, weighted limits at Kan Extension Seminar by David Jaz Myers, followed by 2017-04-03 discussion by the same commentator of other parts of the SFL article
The property/structure distinction
Kelly, G. M.; Lack, Stephen (1997). "On property-like structures". Theory and Applications of Categories. 3 (9): 213–250. "we consider on a 2-category those 2-monads for which algebra structure is essentially unique if it exists, giving a precise mathematical definition of `essentially unique' and investigating its consequences. We call such 2-monads property-like. We further consider the more restricted class of fully property-like 2-monads, consisting of those property-like 2-monads for which all 2-cells between (even lax) algebra morphisms are algebra 2-cells. The consideration of lax morphisms leads us to a new characterization of those monads, studied by Kock and Zoberlein, for which `structure is adjoint to unit', and which we now call lax-idempotent 2-monads: both these and their colax-idempotent duals are fully property-like. We end by showing that (at least for finitary 2-monads) the classes of property-likes, fully property-likes, and lax-idempotents are each coreflective among all 2-monads."
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.
Street, Ross (1974). "Fibrations and Yoneda's lemma in a 2-category". Proceedings Sydney Category Theory Seminar 1972/1973. SLNM. 420. pp. 104–133. MR 0396723. doi:10.1007/BFb0063102.
Street, Ross (1974). "Elementary cosmoi I". Proceedings Sydney Category Theory Seminar 1972/1973. SLNM. 420. pp. 134–180. MR 0354813. doi:10.1007/BFb0063103.
Street, Ross; Walters, Robert (1978). "Yoneda structures on 2-categories". Journal of Algebra. 50 (2): 360–379. doi:10.1016/0021-8693(78)90160-6., 2014-03-24 discussion at Kan Extension Seminar by Alexander Campbell
Street, Ross (1980). "Cosmoi of internal categories". Transactions of the American Mathematical Society. 258: 278–318. MR 558176. doi:10.1090/S0002-9947-1980-0558176-3.
Street, Ross (1980). "Fibrations in bicategories". Cahiers de Topologie et Géométrie Différentielle Catégoriques. 21 (2): 111–160. MR 574662., followed in 1987 by a four page correction and addendum. This paper discusses relations between V-bimodules and two-sided fibrations and cofibrations in V-Cat: "The V-modules turn out to amount to the bicodiscrete cofibrations in V-Cat." --- The paper by Kasangian, Kelly, and Rossi on cofibrations is closely related to these constructions.
Wood, R. J. (1982). "Abstract proarrows I". Cahiers de Topologie et Géométrie Différentielle Catégoriques. 23 (3): 279–290. MR 675339.
Kasangian, S.; Kelly, G. M.; Rossi, F. (1983). "Cofibrations and the realization of non-deterministic automata". Cahiers de Topologie et Géométrie Différentielle Catégoriques. 24 (1): 23–46. MR 702718. Among other things, they develop the theory of bimodules over a biclosed, but not necessarily symmetric, monoidal category V. Their development of the theory of cofibrations is modeled on that in Street's "Fibrations in bicategories."
Wood, R. J. (1985). "Proarrows II". Cahiers de Topologie et Géométrie Différentielle Catégoriques. 26 (2): 135–168. MR 794752.
Carboni, A.; Walters, R. F. C. (1987). "Cartesian bicategories I". Journal of Pure and Applied Algebra. 49 (1-2): 11–32. doi:10.1016/0022-4049(87)90121-6.
Carboni, A.; Kelly, G. M.; Wood, R. J. (1991). "A 2-categorical approach to change of base and geometric morphisms I". Cahiers de Topologie et Géométrie Différentielle Catégoriques. 32 (1): 47–95. MR 1130402.
Carboni, A.; Kelly, G. M.; Verity, D.; Wood, R. J. (1998). "A 2-Categorical Approach To Change Of Base And Geometric Morphisms II". Theory and Applications of Categories. 4 (5): 82–136. "We introduce a notion of equipment which generalizes the earlier notion of pro-arrow equipment and includes such familiar constructs as relK, spnK, parK, and proK for a suitable category K, along with related constructs such as the V-pro arising from a suitable monoidal category V."
Carboni, A.; Kelly, G. M.; Walters, R. F. C.; Wood, R. J. (2008). "Cartesian bicategories II". Theory and Applications of Categories. 19 (6): 93–124. "The notion of cartesian bicategory, introduced by Carboni and Walters for locally ordered bicategories, is extended to general bicategories. It is shown that a cartesian bicategory is a symmetric monoidal bicategory."
Shulman, Michael (2008). "Framed bicategories and monoidal fibrations". Theory and Applications of Categories. 20 (18): 650–738. This paper generalizes the notion of equipment. The author writes: "The authors of [CKW91, CKVW98] consider a related notion of 'equipment' where K is replaced by a 1-category but the horizontal composition is forgotten." In particular, one of his constructions yields what [CKVW98] calls a starred pointed equipment.
Factorization systems, reflective subcategories, localizations, and Galois theory
Kelly, G.M. (1969). "Monomorphisms, Epimorphisms, and Pull-Backs". Journal of the Australian Mathematical Society. 9 (1-2): 124–142. doi:10.1017/S1446788700005693.
Kelly, G.M. (1983). "A note on the generalized reflexion of Guitart and Lair". Cahiers de Topologie et Géométrie Différentielle Catégoriques. 24 (2): 155–159. MR 710038.
Cassidy, C.; Hébert, M.; Kelly, G. M. (1985). "Reflective subcategories, localizations and factorization systems". Journal of the Australian Mathematical Society. 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". Journal of Pure and Applied Algebra. 46 (1): 1–34. "Our aim is to study the ordered set Loc A of localizations of a category A, showing it to be a small complete lattice when A is complete with a (small) strong generator, and further to be the dual of a locale when A is a locally-presentable category in which finite limits commute with filtered colimits. We also consider the relations between Loc A and Loc A′ arising from a geometric morphism A → A′; and apply our results in particular to categories of modules."
Kelly, G.M. (1987). "On the ordered set of reflective subcategories". Bulletin of the Australian Mathematical Society. 36 (1): 137–152. doi:10.1017/S0004972700026381. "Given a category A, we consider the (often large) set Ref A 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.
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". Journal of Pure and Applied Algebra. 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". Applied Categorical Structures. 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". Journal of Pure and Applied Algebra. 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 C, centrality being defined relatively to an “admissible” full subcategory X of C."
Carboni, A.; Janelidze, G.; Kelly, G. M.; Paré, R. (1997). "On Localization and Stabilization for Factorization Systems". Applied Categorical Structures. 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". Theory and Applications of Categories. 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". Journal of Pure and Applied Algebra. 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". Bulletin of the Australian Mathematical Society. 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". Theory and Applications of Categories. 9 (4): 61–91.
Borceux, F.W.; Janelidze, G.; Kelly, G.M. (2005). "On the representability of actions in a semi-abelian category". Theory and Applications of Categories. 14 (11): 244–286. "We consider a semi-abelian category V 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 V. We investigate the representability of the functor Act(-,X) in the case where V 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". Bulletin of the Australian Mathematical Society. 12 (1): 49–72. doi:10.1017/S0004972700023637.
Kelly, G. M.; Koubek, V. (1981). "The large limits that all good categories admit". Journal of Pure and Applied Algebra. 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). Journal of the Korean Mathematical Society. 23 (1): 1–18. "We say that a class M of morphisms in a category A is closed under limits if, whenever F,G:K→A are functors that admit limits, and whenever η:F⇒G:K→A is a natural transformation each of whose components ηK:FK→GK lies in M, then the induced morphism limη:limF→limG also lies in M."
Albert, M. H.; Kelly, G. M. (1988). "The closure of a class of colimits". Journal of Pure and Applied Algebra. 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”". Journal of Pure and Applied Algebra. 51 (1-2): 19–25. doi:10.1016/0022-4049(88)90074-6.
Kelly, G. M. (1989). "Elementary observations on 2-categorical limits". Bulletin of the Australian Mathematical Society. 39 (2): 301–317. doi:10.1017/S0004972700002781. 2014-04-18 discussion at Kan Extension Seminar by Christina Vasilakopoulou
Bird, G. J.; Kelly, G. M.; Power, A. J.; Street, R. H. (1989). "Flexible limits for 2-categories". Journal of Pure and Applied Algebra. 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". Applied Categorical Structures. 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". Theory and Applications of Categories. 14 (17): 399–423. "The paper is in essence a survey of categories having ϕ-weighted colimits for all the weights ϕ in some class Φ."
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". 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). Journal of the Korean Mathematical Society. 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". Bulletin of the Australian Mathematical Society. 36 (1): 133–136. doi:10.1017/S000497270002637X. "An adjoint-triangle theorem contemplates functors P:C→A and T:A→B where T and TP have left adjoints, and gives sufficient conditions for P also to have a left adjoint. We are concerned with the case where T is conservative - that is, isomorphism-reflecting"
Kelly, G. M.; Power, A. J. (1993). "Adjunctions whose counits are coequalizers, and presentations of finitary enriched monads". Journal of Pure and Applied Algebra. 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". Theory and Applications of Categories. 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". Journal of the Australian Mathematical Society. 4 (3): 299–307. doi:10.1017/S1446788700024071.
Eilenberg, Samuel; Kelly, G. M. (1966). "A generalization of the functorial calculus". Journal of 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.
Day, B. J.; Kelly, G. M. (1970). "On topological quotient maps preserved by pullbacks or products". Math. Proc. Cambridge Phil. Soc. 67 (3): 553–553. 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 Top 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, G. M.; Lack, Stephen (2001). "V-Cat is locally presentable or locally bounded if V is so". Theory and Applications of Categories. 8 (23): 555–575.
Kelly, Max; Labella, Anna; Schmitt, Vincent; Street, Ross (2002). "Categories enriched on two sides (Dedicated to Saunders Mac Lane on his 90th birthday)". Journal of Pure and Applied Algebra. 168 (1): 53–98. doi:10.1016/S0022-4049(01)00048-2. "We introduce morphisms V→W of bicategories, more general than the original ones of Bénabou. When V=1, such a morphism is a category enriched in the bicategory W. 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 Caten of a simple kind whose objects are bicategories. It follows that a morphism from V to W in Caten induces a 2-functor V−Cat to W−Cat, while an adjunction between V and W in Caten induces one between the 2-categories V−Cat and W−Cat. Left adjoints in Caten are necessarily homomorphisms in the sense of Bénabou, while right adjoints are not. Convolution appears as the internal hom for a monoidal structure on Caten. The 2-cells of Caten are functors; modules can also be defined, and we examine the structures associated with them."
Street, Ross (2003). "Functorial Calculus in Monoidal Bicategories". Applied Categorical Structures. 11 (3): 219–227. doi:10.1023/A:1024247613677. Compare to Eilenberg-Kelly "A generalization of the functorial calculus" above.
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.
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. doi:10.1017/S030500410003365X.
Kelly, G. M. (1961). "The exactness of Čech homology over a vector space". Math. Proc. Cambridge Phil. Soc. 57 (2): 428–429. doi:10.1017/S0305004100035398.
Kelly, G. M. (1961). "On manifolds containing a submanifold whose complement is contractible". Math. Proc. Cambridge Phil. Soc. 57 (3): 507–515. doi:10.1017/S0305004100035568.
Kelly, G. M. (1963). "Observations on the Künneth theorem". Math. Proc. Cambridge Phil. Soc. 59 (3): 575–587. doi:10.1017/S0305004100037257.
Kelly, G. M. (1964). "Complete functors in homology I. Chain maps and endomorphisms". Math. Proc. Cambridge Phil. Soc. 60 (4): 721–735. doi:10.1017/S0305004100038202.
Kelly, G. M. (1964). "Complete functors in Homology: II. The exact homology sequence". Math. Proc. Cambridge Phil. Soc. 60 (4): 737–749. doi:10.1017/S0305004100038214.
Kelly, G. M. (1965). "A lemma in homological algebra". Math. Proc. Cambridge Phil. Soc. 61 (1): 49–52. doi:10.1017/S0305004100038627.
Kelly, G. M. (1965). "Chain maps inducing zero homology maps". Math. Proc. Cambridge Phil. Soc. 61 (4): 847–854. doi:10.1017/S0305004100039207.
Miscellaneous papers on other subjects
Dickson, S. E.; Kelly, G. M. (1970). "Interlacing methods and large indecomposables". Bulletin of the Australian Mathematical Society. 3 (3): 337–348. doi:10.1017/S0004972700046037.
Kelly, G. M.; Pultr, A. (1978). "On algebraic recognition of direct-product decompositions". Journal of Pure and Applied Algebra. 12 (3): 207–224. doi:10.1016/0022-4049(87)90002-8.
