Max Kelly

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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[edit]

  • 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).

Preliminaries[edit]

  • Kelly, G. M.; Street, Ross (1974). "Review of the elements of 2-categories". Category Seminar (Proceedings Sydney Category Theory Seminar 1972/1973). SLNM. 420. doi:10.1007/BFb0063101.  "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] arising from adjunctions and 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[edit]

Categories with few structures, or many[edit]

Clubs[edit]

Coherence[edit]

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.

  • 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.

Lawvere theories, commutative theories, and the structure-semantics adjunction[edit]

  • 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 satisfying some smallness conditions, "applying Lawvere's “structure” functor to the hom-functor produces a Lawvere theory , called the canonical algebraic structure of ". --- 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[edit]

Monads[edit]

Monadicity[edit]

Operads[edit]

Presentations[edit]

  • 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 admits a presentation in terms of -objects Bc of ‘basic operations of arity c’ (where c runs through the finitely-presentable objects of ) and -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 , where is a natural number (as well as natural transformations between these and equations between derived operations)". Cf. Street, Ross (2012). "Kan extensions and cartesian monoidal categories": 1–8.  "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[edit]

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.

The property/structure distinction[edit]

  • 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[edit]

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.

  • 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[edit]

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[edit]

  • 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. MR 0396723.  See also: Kock, Anders (2013-12-05). "Fibrations as Eilenberg-Moore algebras": 1–24.  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 -bimodules and two-sided fibrations and cofibrations in -Cat: "The -modules turn out to amount to the bicodiscrete cofibrations in -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 . Their development of the theory of cofibrations is modeled on that in Street's "Fibrations in bicategories."
  • Streicher, Thomas (2014) [1999]. "Fibred Categories à la Jean Bénabou" (PDF): 1–95.  "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[edit]

Change of base and equipment[edit]

  • 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, spn, par, and pro for a suitable category , along with related constructs such as the -pro arising from a suitable monoidal category ."
  • 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 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."


  • Carboni, A.; Kelly, G. M.; Walters, R. F. C.; Wood, R. J. (2008). "Cartesian bicategories II". TAC. 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."

Factorization systems, reflective subcategories, localizations, and Galois theory[edit]

  • 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.; 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.
  • 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 ."

Actions and algebras[edit]

Also semidirect products.

  • 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[edit]

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

Adjunctions[edit]

  • 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"
  • 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[edit]

  • 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 of topological spaces might be cartesian closed; see section 1.5.
  • 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."

Homology[edit]

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".

Miscellaneous papers on other subjects[edit]

General references[edit]

External links[edit]