Max Kelly

Structures borne by categories

• Originally published as 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 limits and colimits; 4. Kan extensions; 5. Density; 6. Essentially-algebraic theories defined by reguli and by sketches.

Preliminaries

• "In §1 we rehearse the most elementary facts about 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 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.".

  Some specific structures categories can bear

• Categories with few structures, or many

• Clubs

• Mainly syntactic clubs, and how to present them. Closely related to the paper "Many-variable functorial calculus. I".

• Coherence

• :
• Mainly a technical result needed for proving coherence results about closed categories, and more generally, about right adjoints.
• 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.

• Lawvere theories, commutative theories, and the structure-semantics adjunction

• 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

• Monads

• Monadicity

• Operads

• Presentations

• "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' 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.
• 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 ". Cf. "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

• : There is a very significant ;
• , followed by

  The property/structure distinction

• "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 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 the classes of property-likes, fully property-likes, and lax-idempotents are each coreflective among all 2-monads."

  Functor categories and functorial calculi

• Compare to Street "Functorial Calculus in Monoidal Bicategories" below.
• Mainly semantic clubs. Closely related to the paper "An Abstract Approach to Coherence".
• "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

Fibrations, cofibrations, and bimodules

• See also: Kock writes: "Street was probably the first to observe that opfibrations could be described as pseudo-algebras for a KZ monad , 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."
• , followed in 1987 by a . 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.
• 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."
• "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

• Change of base and equipment

• "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."
• This paper generalizes the notion of equipment. The author writes: "The authors of 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 calls a starred pointed equipment.
• "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."
• ,
• "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

• , followed by . "This work is a detailed analysis of the relationship between reflective subcategories of a category and factorization systems supported by the category."
• "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 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."
• "Given a category, we consider the set Ref of its reflective subcategories, ordered by inclusion."
• No copy of this could be found on the web as of 2017-09-29.
• : Begins with basic treatment of regular and exact categories, and equivalence relations and congruences therein, then studies the Maltsev and Goursat conditions.
• "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."
• : includes "self-contained modern accounts of factorization systems, descent theory, and Galois theory"
• "Many questions in mathematics can be reduced to asking whether Cov is reflective in C \downarrow B; and we give a number of disparate conditions, each sufficient for this to be so."

• Actions and algebras

• , followed by:
• "We consider a semi-abelian category and we write Act 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 in the case where is locally presentable, with finite limits commuting with filtered colimits."
• "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

• "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."
• "A category of fractions is a special case of a coinverter in the 2-category Cat...."
• "The paper is in essence a survey of categories having -weighted colimits for all the weights in some class."

  Adjunctions

• "We are interested here in those functors which, like the forgetful functors of algebra, are conservative and have left adjoints."
• "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"
• 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.
• "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

• 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.
• Some historical information on personnel matters, and early versions of ideas to be published formally later.
• "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."
• A historical account.

  Homology

• Miscellaneous papers on other subjects

• General references

• : contains list of 87 publications of Kelly from 1959 to early 2002
• : includes complete list of 92 publications from 1957 PhD thesis to posthumously published 2008 paper ; probably the most complete survey of Kelly’s career
• : "This book is dedicated to Max Kelly, the founder of the Australian school of category theory"
• From the forward to the book: ", 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 .
• : contains list of publications of Kelly