Max Kelly

Gregory Maxwell Kelly
Gregory Maxwell Kelly
Born	5 June 1930
Died	26 January 2007
Alma mater	University of Cambridge
Known for	Enriched category theory
Awards	Centenary Medal
	Scientific career
Fields	Mathematics
Institutions	University of Sydney
Thesis	Topics in Homology Theory (1957)
Doctoral advisor	Shaun Wylie
Doctoral students	Ross Street

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 PhD 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 ${\cal {V}}$ be a monoidal category, and denote by ${\cal {V}}$ -Cat the category of ${\cal {V}}$ -enriched categories. Among other things, Kelly showed that ${\cal {V}}$ -Cat has all weighted limits and colimits even when ${\cal {V}}$ 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 PhD 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: 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). A website archiving both CTGD and CTGDC is here.

Preliminaries

Kelly, G. M.; Street, Ross (1974). "Review of the elements of 2-categories". Category Seminar (Proceedings Sydney Category Theory Seminar 1972/1973). SLNM. 420. pp. 75–103. doi:10.1007/BFb0063101. ISBN 978-3-540-06966-9. "In §1 we rehearse the most elementary facts about [double categories and] 2-categories ... chiefly to introduce our notation and especially the operation of pasting that we use constantly. In §2 we use the pasting operation to give a treatment, which seems to us simpler and more complete than any we have seen, of the ['mates' bijection] $(bu,u'a)\cong (f'b,af)$ arising from adjunctions $f\dashv u$ and $f'\dashv u'$ in any 2-category, and of its naturality. In §3 we recall the basic properties of monads in a 2-category, and then mention some enrichments of these that become available in the 2-category of 2-categories (because it is really a 3-category).". Kan Extension Seminar discussion on 2014-03-09 by Dimitri Zaganidis

Some specific structures categories can bear

Kelly, G. M. (1965). "Tensor products in categories". J. 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. doi:10.1007/978-3-642-99902-4_22. ISBN 978-3-642-99902-4.

Kelly, G. M. (1986). "A survey of totality for enriched and ordinary categories". CTGDC. 27 (2): 109–132. MR 0850527.

Im, Geun Bin; Kelly, G. M. (1986). "A universal property of the convolution monoidal structure". JPAA. 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". JPAA. 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". Bull. Austral. Math. Soc. 31 (1): 41–59. doi:10.1017/S0004972700002264.

Clubs

Kelly, G. M. (1972). "An abstract approach to coherence". Coherence in Categories. SLNM. 281. pp. 106–147. doi:10.1007/BFb0059557. ISBN 978-3-540-05963-9. Mainly syntactic clubs, and how to present them. Closely related to the paper "Many-variable functorial calculus. I".

Kelly, G. M. (1974). "On clubs and doctrines". Category Seminar (Proceedings Sydney Category Theory Seminar 1972/1973). SLNM. 420. pp. 181–256. doi:10.1007/BFb0063104. ISBN 978-3-540-06966-9.

Kelly, G. M. (1992). "On clubs and data-type constructors". Applications of Categories in Computer Science (Proceedings of the London Mathematical Society Symposium, Durham 1991). Cambridge University Press. pp. 163–190. doi:10.1017/CBO9780511525902.010. ISBN 9780511525902. Kan Extension Seminar discussion on 2017-04-17 by Pierre Cagne

Garner, Richard (2006). "Double clubs". CTGDC. 47 (4): 261–317. arXiv:math.CT/0606733.

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. hdl:1911/62865.

Kelly, G. M. (1964). "On MacLane's conditions for coherence of natural associativities, commutativities, etc". J. Algebra. 1 (4): 397–402. doi:10.1016/0021-8693(64)90018-3.

Kelly, G. M.; Mac Lane, Saunders (1971). "Coherence in closed categories". JPAA. 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. ISBN 978-3-540-05963-9.

Kelly, G. M. (1972). "A cut-elimination theorem". Coherence in Categories. SLNM. 281. pp. 196–213. doi:10.1007/BFb0059559. ISBN 978-3-540-05963-9. 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". Category Seminar (Proceedings Sydney Category Theory Seminar 1972/1973). SLNM. 420. pp. 281–375. doi:10.1007/BFb0063106. ISBN 978-3-540-06966-9. 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". JPAA. 19: 193–213. doi:10.1016/0022-4049(80)90101-2.

Power, John (1989). "A general coherence result". JPAA. 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)". JPAA. 175 (1): 223–241. doi:10.1016/S0022-4049(02)00136-6. Kan Extension Seminar discussion on 2014-06-02 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". JPAA. 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 ${\rm {H}}={\rm {Hom}}_{A}:A^{\rm {op}}\times A\to {\rm {Set}}$ 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.

Local boundedness and presentability

Kelly, G. M.; Lack, Stephen (2001). " ${\cal {V}}$ -Cat is locally presentable or locally bounded if ${\cal {V}}$ is so". TAC. 8 (23): 555–575.

Monads

Street, Ross (1972). "The formal theory of monads". JPAA. 2 (2): 149–168. doi:10.1016/0022-4049(72)90019-9. MR 0299653. Kan Extension Seminar discussion on 2014-01-27 by Eduard Balzin

Blackwell, R.; Kelly, G. M.; Power, A. J. (1989). "Two-dimensional monad theory". JPAA. 59 (1): 1–41. doi:10.1016/0022-4049(89)90160-6. Kan Extension Seminar discussion on 2014-04-28 by Sam van Gool

Monadicity

Kelly, G. M. (1980). "Examples of non-monadic structures on categories". JPAA. 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". CTGDC. 38 (3): 179–191. MR 1474564.

Kelly, G. M.; Lack, Stephen (2000). "On the monadicity of categories with chosen colimits". TAC. 7 (7): 148–170.

Adamek, Jiri; Kelly, G. M. (2000). " ${\cal {M}}$ -Completeness is seldom monadic over graphs". TAC. 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. Kan Extension Seminar discussion on 2017-03-01 by Simon Cho

Presentations

Dubuc, Eduardo J.; Kelly, G. M. (1983). "A presentation of topoi as algebraic relative to categories or graphs". J. 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". JPAA. 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 ${\cal {A}}$ admits a presentation in terms of ${\cal {A}}$ -objects Bc of 'basic operations of arity c' (where c runs through the finitely-presentable objects of ${\cal {A}}$ ) and ${\cal {A}}$ -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". ACS. 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 ${\cal {A^{n}\to {\cal {A}}}}$ , where $n$ is a natural number (as well as natural transformations between these and equations between derived operations)". Cf. Street, Ross (2015). "Kan extensions and cartesian monoidal categories". Seminarberichte der Mathematik. 87: 89–96. arXiv:1409.6405. Bibcode:2014arXiv1409.6405S. "The existence of adjoints to algebraic functors between categories of models of Lawvere theories follows from finite-product-preservingness surviving left Kan extension. A result along these lines was proved in Appendix 2 of Brian Day's 1970 PhD thesis. His context was categories enriched in a cartesian closed base. A generalization is described here with essentially the same proof. We introduce the notion of cartesian monoidal category in the enriched context. With an advanced viewpoint, we give a result about left extension along a promonoidal module and further related results."

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". JPAA. 2 (3): 169–191. doi:10.1016/0022-4049(72)90001-1. MR 0322004. : There is a very significant Erratum ; Kan Extension Seminar discussion on 2014-02-15 by Fosco Loregian

Kelly, G. M. (1982). "On the essentially-algebraic theory generated by a sketch". Bull. Austral. Math. Soc. 26 (1): 45–56. doi:10.1017/S0004972700005591.

Kelly, G. M. (1982). "Structures defined by finite limits in the enriched context, I". CTGD. 23 (1): 3–42. MR 0648793. Kan Extension Seminar discussion of enriched, weighted limits on 2017-04-03 discussion of enriched, weighted limits 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". TAC. 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."

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. ISBN 978-3-540-04625-7.

Kelly, G. M. (1972). "Many-variable functorial calculus. I.". Coherence in Categories. SLNM. 281. pp. 66–105. doi:10.1007/BFb0059556. ISBN 978-3-540-05963-9. 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 ${\cal {V{\bf {-Mod}}}}$ whose objects are categories enriched in a cocomplete symmetric monoidal category ${\cal {V}}$ 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.

Fibrations, cofibrations, and bimodules

Gray, John W. (1966). "Fibred and Cofibred Categories". Proceedings of the Conference on Categorical Algebra (La Jolla 1965). pp. 21–83. doi:10.1007/978-3-642-99902-4_2. ISBN 978-3-642-99904-8.

Street, Ross (1974). "Fibrations and Yoneda's lemma in a 2-category". Category Seminar (Proceedings Sydney Category Theory Seminar 1972/1973). SLNM. 420. pp. 104–133. doi:10.1007/BFb0063102. ISBN 978-3-540-06966-9. MR 0396723. See also: Kock, Anders (5 December 2013). "Fibrations as Eilenberg-Moore algebras". pp. 1–24. arXiv:1312.1608 [math.CT]. Kock writes: "Street was probably the first to observe that opfibrations could be described as pseudo-algebras for a KZ monad [also known as lax-idempotent 2-monad]; in fact, in [F&YL], p. 118, he uses this description as his definition of the notion of opfibration, so therefore, no proof is given. Also, loc.cit. gives no proof of the fact that split opfibrations then are the strict algebras. So in this sense, Section 6 of the present article only supplements loc.cit. by providing elementary proofs of these facts."

Street, Ross (1980). "Fibrations in bicategories". CTGD. 21 (2): 111–160. MR 0574662., followed in 1987 by a four page correction and addendum. This paper discusses relations between ${\cal {V}}$ -bimodules and two-sided fibrations and cofibrations in ${\cal {V}}$ -Cat: "The ${\cal {V}}$ -modules turn out to amount to the bicodiscrete cofibrations in ${\cal {V}}$ -Cat." --- The paper by Kasangian, Kelly, and Rossi on cofibrations is closely related to these constructions.

Kasangian, S.; Kelly, G. M.; Rossi, F. (1983). "Cofibrations and the realization of non-deterministic automata". CTGD. 24 (1): 23–46. MR 0702718. Among other things, they develop the theory of bimodules over a biclosed, but not necessarily symmetric, monoidal category ${\cal {V}}$ . Their development of the theory of cofibrations is modeled on that in Street's "Fibrations in bicategories."

Streicher, Thomas (2018). "Fibred Categories à la Jean Bénabou". pp. 1–97. arXiv:1801.02927 [math.CT]. "The notion of fibred category was introduced by A. Grothendieck for purely geometric reasons. The "logical" aspect of fibred categories and, in particular, their relevance for category theory over an arbitrary base category with pullbacks has been investigated and worked out in detail by Jean Bénabou. The aim of these notes is to explain Bénabou’s approach to fibred categories which is mostly unpublished but intrinsic to most fields of category theory, in particular to topos theory and categorical logic."

Cosmoi

Street, Ross (1974). "Elementary cosmoi I". Category Seminar (Proceedings Sydney Category Theory Seminar 1972/1973). SLNM. 420. pp. 134–180. doi:10.1007/BFb0063103. ISBN 978-3-540-06966-9. MR 0354813.

Street, Ross (1980). "Cosmoi of internal categories". Trans. Amer. Math. Soc. 258 (2): 278–318. doi:10.1090/S0002-9947-1980-0558176-3. MR 0558176.

Change of base and equipment

Wood, R. J. (1982). "Abstract proarrows I". CTGD. 23 (3): 279–290. MR 0675339.

Wood, R. J. (1985). "Proarrows II". CTGDC. 26 (2): 135–168. MR 0794752.

Carboni, A.; Kelly, G. M.; Wood, R. J. (1991). "A 2-categorical approach to change of base and geometric morphisms I". CTGDC. 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". TAC. 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 rel ${\cal {K}}$ , spn ${\cal {K}}$ , par ${\cal {K}}$ , and pro ${\cal {K}}$ for a suitable category ${\cal {K}}$ , along with related constructs such as the ${\cal {V}}$ -pro arising from a suitable monoidal category ${\cal {V}}$ ."

Shulman, Michael (2008). "Framed bicategories and monoidal fibrations". TAC. 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.

Verity, Dominic (2011) [1992]. "Enriched categories, internal categories and change of base". Reprints in Theory and Applications of Categories. 20: 1–266. "[C]hapter 1 presents a general theory of change of base for category theories as codified into structures called equipments. These provide an abstract framework which combines the calculi of functors and profunctors of a given category theory into a single axiomatised structure, in a way which applies to enriched and internal theories alike."

Street, Ross; Walters, Robert (1978). "Yoneda structures on 2-categories". J. Algebra. 50 (2): 360–379. doi:10.1016/0021-8693(78)90160-6. MR 0463261., Kan Extension Seminar discussion on 2014-03-24 by Alexander Campbell

Carboni, A.; Walters, R. F. C. (1987). "Cartesian bicategories I". JPAA. 49 (1–2): 11–32. doi:10.1016/0022-4049(87)90121-6.

Carboni, A.; Kelly, G. M.; Walters, R. F. C.; Wood, R. J. (2008). "Cartesian bicategories II". TAC. 19 (6): 93–124. arXiv:0708.1921. Bibcode:2007arXiv0708.1921C. "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."

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 ${\cal {A}}$ of localizations of a category ${\cal {A}}$ , showing it to be a small complete lattice when ${\cal {A}}$ is complete with a (small) strong generator, and further to be the dual of a locale when ${\cal {A}}$ is a locally-presentable category in which finite limits commute with filtered colimits. We also consider the relations between Loc ${\cal {A}}$ and Loc ${\cal {A'}}$ arising from a geometric morphism ${\cal {A}}$ → ${\cal {A'}}$ ; 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 ${\cal {A}}$ , we consider the (often large) set Ref ${\cal {A}}$ 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. 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. ISBN 978-3-540-54706-8.

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 ${\cal {C}}$ , centrality being defined relatively to an "admissible" full subcategory ${\cal {X}}$ of ${\cal {C}}$ ."

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 ${\cal {V}}$ 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 ${\cal {V}}$ . We investigate the representability of the functor Act(-,X) in the case where ${\cal {V}}$ 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". 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 ${\cal {M}}$ of morphisms in a category ${\cal {A}}$ is closed under limits if, whenever $F,G:{\cal {K\to {\cal {A}}}}$ are functors that admit limits, and whenever $\eta :F\Rightarrow G:{\cal {K\to {\cal {A}}}}$ is a natural transformation each of whose components $\eta K:FK\to GK$ lies in ${\cal {M}}$ , then the induced morphism ${\displaystyle {\rm {lim}}~\eta$ also lies in ${\cal {M}}$ ."

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 $\phi$ -weighted colimits for all the weights $\phi$ in some class $\Phi$ ."

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. ISBN 978-3-540-04625-7.

Kelly, G. M. (1974). "Doctrinal adjunction". Category Seminar (Proceedings Sydney Category Theory Seminar 1972/1973). SLNM. 420. pp. 257–280. doi:10.1007/BFb0063105. ISBN 978-3-540-06966-9.

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 $P:{\cal {C\to {\cal {A}}}}$ and $T:{\cal {A\to {\cal {B}}}}$ 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". 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. 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 ${\bf {Top}}$ 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, 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 ${\cal {V\to {\cal {W}}}}$ of bicategories, more general than the original ones of Bénabou. When ${\cal {V={\bf {1}}}}$ , such a morphism is a category enriched in the bicategory ${\cal {W}}$ . 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 ${\bf {Caten}}$ of a simple kind whose objects are bicategories. It follows that a morphism from ${\cal {V}}$ to ${\cal {W}}$ in ${\bf {Caten}}$ induces a 2-functor ${\cal {V{\bf {-Cat}}}}$ to ${\cal {W{\bf {-Cat}}}}$ , while an adjunction between ${\cal {V}}$ and ${\cal {W}}$ in ${\bf {Caten}}$ induces one between the 2-categories ${\cal {V{\bf {-Cat}}}}$ and ${\cal {W{\bf {-Cat}}}}$ . Left adjoints in ${\bf {Caten}}$ are necessarily homomorphisms in the sense of Bénabou, while right adjoints are not. Convolution appears as the internal hom for a monoidal structure on ${\bf {Caten}}$ . The 2-cells of ${\bf {Caten}}$ 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 (11 April 2007). "Obituary : Polymath revelled in the mystery of numbers". Sydney Morning Herald. Retrieved 8 September 2017.
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. (eds.). 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. ISBN 978-1-4419-1523-8. 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. : 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

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