G. M. Kelly

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G. Maxwell Kelly (1930–2007), often cited as G. M. Kelly, was an Australian mathematician recognized for his foundational work in category theory, especially enriched category theory. He spent much of his career at the University of Sydney, where he nurtured graduate students and researchers in the field. Kelly’s book “Basic Concepts of Enriched Category Theory” (1982) remains a significant reference in the area, introducing many key ideas and methods that continue to influence modern developments in category theory.

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Basic concepts of enriched category theory

1982-01-01

G. M. Kelly

Received by the editors 2004-10-30. Transmitted by Steve Lack, Ross Street and RJ Wood. Reprint published on 2005-04-23. Several typographical errors corrected 2012-05-13. 2000 Mathematics Subject Classification: 18-02, 18D10, 18D20. Received by the editors 2004-10-30. Transmitted by Steve Lack, Ross Street and RJ Wood. Reprint published on 2005-04-23. Several typographical errors corrected 2012-05-13. 2000 Mathematics Subject Classification: 18-02, 18D10, 18D20.

Coherence for compact closed categories

1980-12-01

G. M. Kelly , Miguel L. Laplaza

Categories of continuous functors, I

1972-09-01

Peter Freyd , G. M. Kelly

Two-dimensional monad theory

1989-07-01

Robert Blackwell , G. M. Kelly , John Power

On MacLane's conditions for coherence of natural associativities, commutativities, etc.

1964-12-01

G. M. Kelly

A unified treatment of transfinite constructions for free algebras, free monoids, colimits, associated sheaves, and so on

1980-08-01

G. M. Kelly

Many problems lead to the consideration of “algebras”, given by an object A of a category A together with “actions” T k A → A on A of one or … Many problems lead to the consideration of “algebras”, given by an object A of a category A together with “actions” T k A → A on A of one or more endofunctors of A, subjected to equational axioms. Such problems include those of free monads and free monoids, of cocompleteness in categories of monads and of monoids, of orthogonal subcategories (= generalized sheaf-categories), of categories of continuous functors, and so on; apart from problems involving the algebras for their own sake. Desirable properties of the category of algebras - existence of free ones, cocompleteness, existence of adjoints to algebraic functors - all follow if this category can be proved reflective in some well-behaved category: for which we choose a certain comma-category T/A We show that the reflexion exists and is given as the colimit of a simple transfinite sequence, if A is cocomplete and the T k preserve either colimits or unions of suitably-long chains of subobjects. The article draws heavily on the work of earlier authors, unifies and simplifies this, and extends it to new problems. Moreover the reflectivity in T/A is stronger than any earlier result, and will be applied in forthcoming articles, in an enriched version, to the study of categories with structure.

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Elementary observations on 2-categorical limits

1989-04-01

G. M. Kelly

With a view to further applications, we give a self-contained account of indexed limits for 2-categories, including necessary and sufficient conditions for 2-categorical completeness. Many important 2-categories fail to be … With a view to further applications, we give a self-contained account of indexed limits for 2-categories, including necessary and sufficient conditions for 2-categorical completeness. Many important 2-categories fail to be complete but do admit a wide class of limits. Accordingly, we introduce a variety of particular 2-categorical limits of practical importance, and show that certain of these suffice for the existence of indexed lax- and pseudo-limits. Other important 2-categories fail to admit even pseudo-limits, but do admit the weaker bilimits; we end by discussing these.

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Adjunctions whose counits are coequalizers, and presentations of finitary enriched monads

1993-10-01

G. M. Kelly , John Power

Some remarks on Maltsev and Goursat categories

1993-01-01

A. Carboni , G. M. Kelly , Maria Cristina Pedicchio

Galois theory and a general notion of central extension

1994-11-01

George Janelidze , G. M. Kelly

Monomorphisms, Epimorphisms, and Pull-Backs

1969-02-01

G. M. Kelly

As the applications of category theory increase, we find ourselves wanting to imitate in general categories much that was at first done only in abelian categories. In particular it becomes … As the applications of category theory increase, we find ourselves wanting to imitate in general categories much that was at first done only in abelian categories. In particular it becomes necessary to deal with epimorphisms and monomorphisms, with various canonical factorizations of arbitrary morphisms, and with the relations of these things to such limit operations as equalizers and pull-backs.

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On topological quotient maps preserved by pullbacks or products

1970-05-01

Brian J. Day , G. M. Kelly

We are concerned with the category of topological spaces and continuous maps. A surjection f : X → Y in this category is called a quotient map if G is … We are concerned with the category of topological spaces and continuous maps. A surjection f : X → Y in this category is called a quotient map if G is open in Y whenever f −1 G is open in X . Our purpose is to answer the following three questions: Question 1. For which continuous surjections f : X → Y is every pullback of f a quotient map? Question 2. For which continuous surjections f : X → Y is f × l z : X × Z → Y × Z a quotient map for every topological space Z ? (These include all those f answering to Question 1, since f × l z is the pullback of f by the projection map Y × Z → Y .) Question 3. For which topological spaces Z is f × 1 Z : X × Z → Y × Z a qiptoent map for every quotient map f ?

Reflective subcategories, localizations and factorizationa systems

1985-06-01

Charles Cassidy , M. Hébert , G. M. Kelly

Abstract This work is a detailed analysis of the relationship between reflective subcategories of a category and factorization systems supported by the category. Abstract This work is a detailed analysis of the relationship between reflective subcategories of a category and factorization systems supported by the category.

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A universal property of the convolution monoidal structure

1986-11-01

Geun Bin IM , G. M. Kelly

A generalization of the functorial calculus

1966-05-01

Samuel Eilenberg , G. M. Kelly

Flexible limits for 2-categories

1989-11-01

Gregory J. Bird , G. M. Kelly , John Power +1 more

Adjunction for enriched categories

1969-01-01

G. M. Kelly

Enriched functor categories

1969-01-01

Brian J. Day , G. M. Kelly

Chain maps inducing zero homology maps

1965-10-01

G. M. Kelly

The purpose of this paper is to prove Theorems 1 and 2 below by exhibiting them as special cases of a more general result, Theorem 3, which admits of a … The purpose of this paper is to prove Theorems 1 and 2 below by exhibiting them as special cases of a more general result, Theorem 3, which admits of a simple proof.

On the radical of a category

1964-08-01

G. M. Kelly

In [1] the concept of completeness of a functor was introduced and, in the cse of additive * categories and and an additive functor T : → , a criterion … In [1] the concept of completeness of a functor was introduced and, in the cse of additive * categories and and an additive functor T : → , a criterion for T (supposed surjective) to be complete was given in terms of the kernel of T : this was that for each object A of the ideal A should be containded in the (Jacobson) radical of A . (The meaning of this notation and nomemclature is recalled in § 2 below). The question arises whether in any additive category there is a greatest ideal with this property, so that the canonical functor T : → / is in some sense the coarsest that faithfully represents the objects (but not the maps) of .

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ON PROPERTY-LIKE STRUCTURES

1997-01-01

G. M. Kelly , Stephen Lack

A may bear many monoidal structures, but (to within a isomorphism) only one of category with finite products. To capture such distinctions, we consider on a 2-category those 2-monads for … A may bear many monoidal structures, but (to within a isomorphism) only one of category with finite products. To capture such distinctions, we consider on a 2-category those 2-monads for which algebra is 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.

On clubs and doctrines

1974-01-01

G. M. Kelly

V -CAT IS LOCALLY PRESENTABLE OR LOCALLY BOUNDED IF V IS SO

2001-01-01

G. M. Kelly , Stephen Lack

We show, for a monoidal closed category V =( V0, ⊗ ,I ), that the category V -Cat of small V -categories is locally λ-presentable if V0 is so, and … We show, for a monoidal closed category V =( V0, ⊗ ,I ), that the category V -Cat of small V -categories is locally λ-presentable if V0 is so, and that it is locally λ-bounded if the closed category V is so, meaning that V0 is locally λ-bounded and that a side condition involving the monoidal structure is satisfied. Many important properties of a monoidal category V are inherited by the category V -Cat of small V -categories. For instance, if V is symmetric monoidal, V -Cat has a canonical symmetric monoidal structure, as was observed already in (4). Much later (7, Remark 5.2), it was realized that if V is only braided monoidal then V -Cat still has a canonical monoidal structure, although it need not have a braiding unless the braiding on V is in fact a symmetry. Similarly, it is straightforward to show that V -Cat is monoidal closed when V is closed and complete, and that V -Cat is complete when V is so. All of these results are essentially routine; the less trivial fact that V -Cat is cocomplete when V is so was first proved in (11).

A notion of limit for enriched categories

1975-02-01

Francis Borceux , G. M. Kelly

For a V -category B , where V is a symmetric monoidal closed category, various limit-like notions have been recognized: ordinary limits (in the underlying category B 0 ) preserved … For a V -category B , where V is a symmetric monoidal closed category, various limit-like notions have been recognized: ordinary limits (in the underlying category B 0 ) preserved by the V -valued representable functors; cotensor products; ends; pointwise Kan extensions. It has further been recognized that, to be called complete , B should admit all of these; for which it suffices to demand the first two. Hitherto, however, there has been no single limit-notion of which all these are special cases, and particular instances of which may exist even when B is not complete or even cotensored. In consequence it has not been possible even to state , say, the representability criterion for a V -functor T : B → V , or even to define , say, pointwise Kan extensions into B , except for cotensored B . (It is somewhat as if, for ordinary categories, we had the notions of product and equalizer, but lacked that of general limit, and could not discuss pullbacks in the absence of products.) In this paper we provide such a general limit-notion for V -categories.

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Cartesian Bicategories II

2007-01-01

A. Carboni , G. M. Kelly , R. F. C. Walters +1 more

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 … 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.

Tensor products in categories

1965-03-01

G. M. Kelly

On clubs and data-type constructors

1992-06-26

G. M. Kelly

The diagrams for symmetric monoidal closed categories proved commutative by Mac Lane and the author in [19] were diagrams of (generalized) natural transformations. In order to understand the connexion between … The diagrams for symmetric monoidal closed categories proved commutative by Mac Lane and the author in [19] were diagrams of (generalized) natural transformations. In order to understand the connexion between these results and free models for the structure, the author introduced in [13] and [14] the notion of club, which was further developed in [15] and applied later to other coherence problems in [16] and elsewhere.The club idea seemed to apply to several diverse kinds of structure on a category, but still to only a restricted number of kinds. In an attempt to understand its natural limits, the author worked out a general notion of “club”, as a monad with certain properties, not necessarily on Cat now, but on any category with finite limits. A brief account of this was included in the 1978 Seminar Report [17], but was never published; the author doubted that there were enough examples to make it of general interest.During 1990 and 1991, however, we were fortunate to have with our research team at Sydney Robin Cockett, who was engaged in applying category theory to computer science. In lectures to our seminar he called attention to certain kinds of monads involved with data types, which have special properties : he was calling them shape monads, but in fact they are precisely examples of clubs in the abstract sense above.

Notes on enriched categories with colimits of some class

2005-01-01

G. M. Kelly , Vincent Schmitt

Given a class Phi of weights, we study the following classes: Phi^+ of Phi-flat weights which are the psi for which psi-colimits commute in the base V with limits with … Given a class Phi of weights, we study the following classes: Phi^+ of Phi-flat weights which are the psi for which psi-colimits commute in the base V with limits with weights in Phi; and Phi^-, dually defined, of weights psi for which psi-limits commute in the base V with colimits with weights in Phi. We show that both these classes are saturated (i.e. closed under the terminology of Albert-Kelly or Betti's coverings). We prove that for the class P of all weights P^+ = P^-. For any small B, we defined an enriched adjunction a` la Isbell [B,V]^op -> [B^op,V] and show how it restricts to an equivalence (P^-(B^op))^op ~ P^-(B) between subcategories of small projectives.

Finite-product-preserving functors, Kan extensions, and strongly-finitary 2-monads

1993-01-01

G. M. Kelly , Stephen Lack

A note on relations relative to a factorization system

1991-01-01

G. M. Kelly

A 2-CATEGORICAL APPROACH TO CHANGE OF BASE AND GEOMETRIC MORPHISMS II

1998-01-01

A. Carboni , G. M. Kelly , Dominic Verity +1 more

We introduce a notion of equipment which generalizes the earlier notion of pro-arrow equipment and includes such familiar constructs as relK, spnK, parK ,a nd proK for a suitable category … We introduce a notion of equipment which generalizes the earlier notion of pro-arrow equipment and includes such familiar constructs as relK, spnK, parK ,a nd proK for a suitable category K, along with related constructs such as the V-pro arising from a suitable monoidal category V. We further exhibit the equipments as the objects of a 2-category, in such a way that arbitrary functors F : L ✲ K induce equipment arrows relF : relL ✲ relK, spnF : spnL ✲ spnK, and so on, and similarly for arbitrary monoidal functors V ✲ W. The article I with the title above dealt with those equipments M having each M(A, B) only an ordered set, and contained a detailed analysis of the case M = relK; in the present article we allow the M(A, B) to be general categories, and illustrate our results by a detailed study of the case M = spnK. We show in particular that spn is a locally-fully-faithful 2-functor to the 2-category of equipments, and determine its image on arrows. After analyzing the nature of adjunctions in the 2-category of equipments, we are able to give a simple characterization of those spnG which arise from a geometric morphism G.

Algebraic categories with few monoidal biclosed structures or none

1980-05-01

F. Foltz , C. Lair , G. M. Kelly

On locales of localizations

1987-01-01

Francis Borceux , G. M. Kelly

A survey of totality for enriched and ordinary categories

1986-01-01

G. M. Kelly

Coinverters and categories of fractions for categories with structure

1993-01-01

G. M. Kelly , Stephen Lack , R. F. C. Walters

Complete functors in homology I. Chain maps and endomorphisms

1964-10-01

G. M. Kelly

There is a sense in which the homology group HA of a free Abelian chain complex A may be said to be a ‘complete system of invariants’ of A , … There is a sense in which the homology group HA of a free Abelian chain complex A may be said to be a ‘complete system of invariants’ of A , to within chain equivalence; certainly any graded Abelian group G is isomorphic to HA for a suitable A , and if HA and HB are isomorphic then A and B are chain equivalent. Such a result is useful in showing that it is fruitless to seek other homotopy invariants of A ; whatever depends only on the homotopy class of A depends only on HA , so that we can, for instance, predict the existence of a formula giving H ( A ⊗ G ), to within isomorphism, in terms of HA and G . The theorem on the existence and uniqueness to within chain equivalence of projective resolutions of modules is a variant of the above theorem, more general in one direction and more special in another.

On the ordered set of reflective subcategories

1987-08-01

G. M. Kelly

Given a category A , we consider the (often large) set Ref A of its reflective (full, replete) subcategories, ordered by inclusion. It is known that, even when A is … Given a category A , we consider the (often large) set Ref A of its reflective (full, replete) subcategories, ordered by inclusion. It is known that, even when A is complete and cocomplete, wellpowered and cowellpowered, the intersection of two reflective subcategories need not be reflective. Supposing that A admits (i) small limits and (ii) arbitrary (even large) intersections of strong subobjects, we prove that an infimum ∧ i C i in Ref A must necessarily be the intersection ∩ i C i . Accordingly Ref A is not in general, even for good A , a complete lattice. We show, however, under the same conditions on A , that Ref A does admit small suprema ∨ i C i , given by the closure in A of the union ∪ i C i under the limits of type (i) and (ii) above.

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Single-space axioms for homology theory

1959-01-01

G. M. Kelly

Eilenberg and Steenrod( 1 ) give a set of axioms for the homology theory of pairs of spaces and their maps, and prove that these axioms are categorical on triangulable … Eilenberg and Steenrod( 1 ) give a set of axioms for the homology theory of pairs of spaces and their maps, and prove that these axioms are categorical on triangulable pairs. Here we give a set of axioms for the homology theory of single spaces and their maps, that is, for absolute rather than relative homology. This axiomatization is shown to be essentially equivalent to that of Eilenberg and Steenrod, the relative homology groups being suitably denned in terms of the absolute groups.

Complete functors in homology II. The exact homology sequence

1964-10-01

G. M. Kelly

This is the second paper of a series, and supposes familiarity with the results and the notations of the first(l), which we refer to as I. This is the second paper of a series, and supposes familiarity with the results and the notations of the first(l), which we refer to as I.

Monoidal functors generated by adjunction, with applications to transport of structure

2004-01-01

G. M. Kelly , Stephen Lack

Interlacing methods and large indecomposables

1970-12-01

Spencer E. Dickson , G. M. Kelly

The method of interlacing of modules, like amalgamation of groups, is a way of getting new objects from old. Briefly, the interlacing module we consider is a certain factor module … The method of interlacing of modules, like amalgamation of groups, is a way of getting new objects from old. Briefly, the interlacing module we consider is a certain factor module of a direct sum of copies (finite or infinite) of an original module M . The conditions given in a previous paper by the first author in order that the interlacing module (using finitely many copies) be indecomposable are here greatly weakened, and we further allow the number of copies of the original to be infinite. R. Colby has shown that if R is a left artinian ring, the existence of a bound on the number of generators required for any indecomposable finitely-generated left R -module implies that R has a distributive lattice of two-sided ideals. This result is extended to rings whose identity is a sum of orthogonal local idempotents. For these rings the same distributivity is proved in case every indecomposable interlacing module of the above type which begins with an indecomposable projective M is finitely-generated. A consequence is that any finite-dimensional algebra over a field having infinitely many two-sided ideals has infinite-dimensional indecomposables.

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On the essentially-algebraic theory generated by a sketch

1982-08-01

G. M. Kelly

By a sketch we here mean a small category S together with a small set φ of projective cones in S , each cone φ ∈ φ being indexed by … By a sketch we here mean a small category S together with a small set φ of projective cones in S , each cone φ ∈ φ being indexed by a small category L φ . A model of S in any category B is a functor G : S → B such that each G φ is a limit-cone. Let F be any small set of small categories containing all the L φ . A small category T admitting all F -limits (that is, an F -complete small T ) is called an F -theory; it is considered as a sketch in which the distinguished cones are all the F -limit-cones. It is an important result of modern universal algebra, due originally to Ehresmann, that each sketch S = ( S , φ) with every L φ ∈ F determines an F -theory T , with a generic model M : S → T of S , such that composition with M induces an equivalence M* between the category of T -models in B and that of S -models in B , whenever B is F -complete. We give a simple proof of this result – one which generalizes directly to the case of enriched categories and indexed limits; and we make the new observation that the inverse to M* is given by (pointwise) right Kan extension along M .

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The exactness of Čech homology over a vector space

1961-04-01

G. M. Kelly

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Observations on the Künneth theorem

1963-07-01

G. M. Kelly

Let A be a right Λ-complex and C be a left Λ-complex where Λ is a ring which is both left and right hereditary; a principal ideal domain is a … Let A be a right Λ-complex and C be a left Λ-complex where Λ is a ring which is both left and right hereditary; a principal ideal domain is a special case of such a ring. This paper is concerned with the Künneth theorem expressing the homology H(A⊗C) of the product complex A⊗C (= A⊗ Λ C) in terms of H(A) and H(C) . Any notations we do not explain are those of (1).

On the canonical algebraic structure of a category

2000-12-01

E. Faro , G. M. Kelly

Reflective subcategories, localizations and factorization systems: Corrigenda

1986-10-01

Charles Cassidy , M. Hébert , G. M. Kelly

Reflective Subcategories, Localizations and Factorization Systems: Corrigenda C. Cassidy, M. Hébert and G. M. Kelly, 1980 Mathematics subject classification (Amer. Math. Soc.): 18 A 20. Reflective Subcategories, Localizations and Factorization Systems: Corrigenda C. Cassidy, M. Hébert and G. M. Kelly, 1980 Mathematics subject classification (Amer. Math. Soc.): 18 A 20.

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A note on the Albert–Kelly paper “the closure of a class of colimits”

1988-03-01

G. M. Kelly , Robert Paré

Topological categories with many symmetric monoidal closed structures

1985-02-01

G. M. Kelly , Fabio Rossi

It would seem from results of Foltz, Lair, and Kelly that symmetric monoidal closed structures, and even monoidal biclosed ones, are quite rare on one-sorted algebraic or essentially-algebraic categories. They … It would seem from results of Foltz, Lair, and Kelly that symmetric monoidal closed structures, and even monoidal biclosed ones, are quite rare on one-sorted algebraic or essentially-algebraic categories. They showed many such categories to admit no such structures at all, and others to admit only one or two; no such category is known to admit an infinite set of such structures. Among concrete categories, topological ones are in some sense at the other extreme from essentially-algebraic ones; and one is led to ask whether a topological category may admit many such structures. On the category of topological spaces itself, only one such structure - in fact symmetric - is known; although Greve has shown it to admit a proper class of monoidal closed structures. One of our main results is a proof that none of these structures described by Greve, except the classical one, is biclosed. Our other main result is that, nevertheless, there exist topological categories (of quasi-topological spaces) which admit a proper class of symmetric monoidal closed structures. Even if we insist (like most authors) that topological categories must be wellpowered, we can still exhibit ones with more such structures than any small cardinal.

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On the monadicity over graphs of categories with limits

1997-01-01

G. M. Kelly , I. J. Le Creurer

Donnee une classe petite M de categories petites, notons Cat M la categorie dont les objets sont les petites categories qui admettent, pour tout M ∈ M, des M-limites (choisies), … Donnee une classe petite M de categories petites, notons Cat M la categorie dont les objets sont les petites categories qui admettent, pour tout M ∈ M, des M-limites (choisies), et dont les morphismes sont les foncteurs qui preservent (strictement) ces limites; notons Gph la categorie des graphes (petits); et notons U: Cat M → Gph le foncteur d'oubli qui envoie une categorie a M-limites sur son graphe sous-jacent. Pour certaines classes M il est connu que ce foncteur U est monadique; mais les demonstrations emploient pour chacune de ces M une astuce differente. Nous demontrons que U est au moins de descente si chaque M ∈ M est une categorie librement engendree par un graphe, et que U est alors monadique quand ce graphe est acyclique.

Examples of non-monadic structures on categories

1980-07-01

G. M. Kelly

Cartesian Bicategories II

2007-01-01

A. Carboni , G. M. Kelly , R. F. C. Walters +1 more

Notes on enriched categories with colimits of some class (completed version)

2005-01-01

G. M. Kelly , Vincent Schmitt

The paper is in essence a survey of categories having $ϕ$-weighted colimits for all the weights $ϕ$ in some class $Φ$. We introduce the class $Φ^+$ of {\em $Φ$-flat} weights … The paper is in essence a survey of categories having $ϕ$-weighted colimits for all the weights $ϕ$ in some class $Φ$. We introduce the class $Φ^+$ of {\em $Φ$-flat} weights which are those $ψ$ for which $ψ$-colimits commute in the base $\V$ with limits having weights in $Φ$; and the class $Φ^-$ of {\em $Φ$-atomic} weights, which are those $ψ$ for which $ψ$-limits commute in the base $\V$ with colimits having weights in $Φ$. We show that both these classes are {\em saturated} (that is, what was called {\em closed} in the terminology of \cite{AK88}). We prove that for the class $\p$ of {\em all} weights, the classes $\p^+$ and $\p^-$ both coincide with the class $\Q$ of {\em absolute} weights. For any class $Φ$ and any category $\A$, we have the free $Φ$-cocompletion $Φ(\A)$ of $\A$; and we recognize $\Q(\A)$ as the Cauchy-completion of $\A$. We study the equivalence between ${(\Q(\A^{op}))}^{op}$ and $\Q(\A)$, which we exhibit as the restriction of the Isbell adjunction between ${[\A,\V]}^{op}$ and $[\A^{op},\V]$ when $\A$ is small; and we give a new Morita theorem for any class $Φ$ containing $\Q$. We end with the study of $Φ$-continuous weights and their relation to the $Φ$-flat weights.

Notes on enriched categories with colimits of some class

2005-01-01

G. M. Kelly , Vincent Schmitt

Monoidal functors generated by adjunction, with applications to transport of structure

2004-01-01

G. M. Kelly , Stephen Lack

V -CAT IS LOCALLY PRESENTABLE OR LOCALLY BOUNDED IF V IS SO

2001-01-01

G. M. Kelly , Stephen Lack

On the canonical algebraic structure of a category

2000-12-01

E. Faro , G. M. Kelly

A 2-CATEGORICAL APPROACH TO CHANGE OF BASE AND GEOMETRIC MORPHISMS II

1998-01-01

A. Carboni , G. M. Kelly , Dominic Verity +1 more

ON PROPERTY-LIKE STRUCTURES

1997-01-01

G. M. Kelly , Stephen Lack

On the monadicity over graphs of categories with limits

1997-01-01

G. M. Kelly , I. J. Le Creurer

Galois theory and a general notion of central extension

1994-11-01

George Janelidze , G. M. Kelly

Adjunctions whose counits are coequalizers, and presentations of finitary enriched monads

1993-10-01

G. M. Kelly , John Power

Finite-product-preserving functors, Kan extensions, and strongly-finitary 2-monads

1993-01-01

G. M. Kelly , Stephen Lack

Coinverters and categories of fractions for categories with structure

1993-01-01

G. M. Kelly , Stephen Lack , R. F. C. Walters

Some remarks on Maltsev and Goursat categories

1993-01-01

A. Carboni , G. M. Kelly , Maria Cristina Pedicchio

On clubs and data-type constructors

1992-06-26

G. M. Kelly

A note on relations relative to a factorization system

1991-01-01

G. M. Kelly

Flexible limits for 2-categories

1989-11-01

Gregory J. Bird , G. M. Kelly , John Power +1 more

Two-dimensional monad theory

1989-07-01

Robert Blackwell , G. M. Kelly , John Power

Elementary observations on 2-categorical limits

1989-04-01

G. M. Kelly

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A note on the Albert–Kelly paper “the closure of a class of colimits”

1988-03-01

G. M. Kelly , Robert Paré

Adjoint-triangle theorems for conservative functors

1987-08-01

Geun Bin IM , G. M. Kelly

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 … 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; then P has a left adjoint under various combinations of completeness or cocompleteness conditions on C and A , with no explicit condition on P itself. We list systematically the strongest results we know of in this direction, augmenting those in the literature by some new ones.

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On the ordered set of reflective subcategories

1987-08-01

G. M. Kelly

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On locales of localizations

1987-01-01

Francis Borceux , G. M. Kelly

A universal property of the convolution monoidal structure

1986-11-01

Geun Bin IM , G. M. Kelly

Reflective subcategories, localizations and factorization systems: Corrigenda

1986-10-01

Charles Cassidy , M. Hébert , G. M. Kelly

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A survey of totality for enriched and ordinary categories

1986-01-01

G. M. Kelly

Reflective subcategories, localizations and factorizationa systems

1985-06-01

Charles Cassidy , M. Hébert , G. M. Kelly

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Topological categories with many symmetric monoidal closed structures

1985-02-01

G. M. Kelly , Fabio Rossi

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Two addenda to the author's ‘Transfinite constructions’

1982-10-01

G. M. Kelly

Since the author's article “A unified treatment of transfinite constructions …”, in Volume 22 (198O) of this Bulletin , had an encyclopaedic goal, he now takes the opportunity to answer … Since the author's article “A unified treatment of transfinite constructions …”, in Volume 22 (198O) of this Bulletin , had an encyclopaedic goal, he now takes the opportunity to answer two further questions raised since that article was submitted. The lesser of these asks whether the only pointed endofunctors for which every action is an isomorphism are the well-pointed ones, at least when the endofunctor is cocontinuous; a counter-example provides a negative answer. The more important question concerns the reflexion from the comma-category T / A into the category of algebras for the pointed endofunctor T of A , and the algebra-reflexion sequence which converges to this reflexion; and asks for simplified descriptions in the special case where T is cocontinuous . We give closed formulas in this case, both for the reflexion and for the sequence which converges to it. The reader may wonder why we care about the approximating sequence when we have a closed formula for the reflexion; the answer is that, in certain applications, we need to separate the roles of finite colimits and filtered ones.

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On the essentially-algebraic theory generated by a sketch

1982-08-01

G. M. Kelly

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Basic concepts of enriched category theory

1982-01-01

G. M. Kelly

Coherence for compact closed categories

1980-12-01

G. M. Kelly , Miguel L. Laplaza

A unified treatment of transfinite constructions for free algebras, free monoids, colimits, associated sheaves, and so on

1980-08-01

G. M. Kelly

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Examples of non-monadic structures on categories

1980-07-01

G. M. Kelly

Algebraic categories with few monoidal biclosed structures or none

1980-05-01

F. Foltz , C. Lair , G. M. Kelly

A notion of limit for enriched categories

1975-02-01

Francis Borceux , G. M. Kelly

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On clubs and doctrines

1974-01-01

G. M. Kelly

Categories of continuous functors, I

1972-09-01

Peter Freyd , G. M. Kelly

Interlacing methods and large indecomposables

1970-12-01

Spencer E. Dickson , G. M. Kelly

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On topological quotient maps preserved by pullbacks or products

1970-05-01

Brian J. Day , G. M. Kelly

Monomorphisms, Epimorphisms, and Pull-Backs

1969-02-01

G. M. Kelly

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Adjunction for enriched categories

1969-01-01

G. M. Kelly

Enriched functor categories

1969-01-01

Brian J. Day , G. M. Kelly

A generalization of the functorial calculus

1966-05-01

Samuel Eilenberg , G. M. Kelly

Chain maps inducing zero homology maps

1965-10-01

G. M. Kelly

Tensor products in categories

1965-03-01

G. M. Kelly

A lemma in homological algebra

1965-01-01

G. M. Kelly

In the development of homological algebra, one has to prove at some point that, in defining the derived functors of ⊗ and of Hom, it makes no difference whether we … In the development of homological algebra, one has to prove at some point that, in defining the derived functors of ⊗ and of Hom, it makes no difference whether we resolve both variables or only one of them. Taking ⊗ ( = ⊗ R ) as a typical example, what has to be proved is (A) If the complex F is projective, or even flat, as a right R-module, and if f: P → A is a projective resolution of the left R-module A, then is an isomorphism .

On MacLane's conditions for coherence of natural associativities, commutativities, etc.

1964-12-01

G. M. Kelly

Complete functors in homology II. The exact homology sequence

1964-10-01

G. M. Kelly

Complete functors in homology I. Chain maps and endomorphisms

1964-10-01

G. M. Kelly

Coauthor	Papers Together
Stephen Lack	5
John Power	3
A. Carboni	3
Francis Borceux	2
Brian J. Day	2
Richard J. Wood	2
Geun Bin IM	2
Vincent Schmitt	2
M. Hébert	2
Charles Cassidy	2
R. F. C. Walters	2
Miguel L. Laplaza	1
Robert Blackwell	1
George Janelidze	1
Ross Street	1
F. Foltz	1
I. J. Le Creurer	1
Samuel Eilenberg	1
Spencer E. Dickson	1
Gregory J. Bird	1
Dominic Verity	1
C. Lair	1
Fabio Rossi	1
Robert Paré	1
Peter Freyd	1
Maria Cristina Pedicchio	1
T. G. Room	1
E. Faro	1

Basic concepts of enriched category theory

1982-01-01

G. M. Kelly

Monomorphisms, Epimorphisms, and Pull-Backs

1969-02-01

G. M. Kelly

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Categories of continuous functors, I

1972-09-01

Peter Freyd , G. M. Kelly

A unified treatment of transfinite constructions for free algebras, free monoids, colimits, associated sheaves, and so on

1980-08-01

G. M. Kelly

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FUNCTORIAL SEMANTICS OF ALGEBRAIC THEORIES

1963-11-01

F. William Lawvere

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On closed categories of functors

1970-01-01

Brian Day

A generalization of the functorial calculus

1966-05-01

Samuel Eilenberg , G. M. Kelly

Categories for the Working Mathematician

1971-01-01

Saunders Mac Lane

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Introduction to bicategories

1967-01-01

Jean Bénabou

A reflection theorem for closed categories

1972-04-01

Brian Day

Elementary observations on 2-categorical limits

1989-04-01

G. M. Kelly

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Kan extensions in Enriched Category Theory

1970-01-01

Eduardo J. Dubuc

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Two-dimensional monad theory

1989-07-01

Robert Blackwell , G. M. Kelly , John Power

Metric spaces, generalized logic, and closed categories

1973-12-01

F. William Lawvere

Reflective subcategories, localizations and factorizationa systems

1985-06-01

Charles Cassidy , M. Hébert , G. M. Kelly

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Calculus of Fractions and Homotopy Theory

1967-01-01

Peter Gabriel , Michel Zisman

Adjunction for enriched categories

1969-01-01

G. M. Kelly

On topological quotient maps preserved by pullbacks or products

1970-05-01

Brian J. Day , G. M. Kelly

Relative functor categories and categories of algebras

1969-01-01

Marta Bunge

The formal theory of monads II

2002-11-01

Stephen Lack , Ross Street

Enriched functor categories

1969-01-01

Brian J. Day , G. M. Kelly

Fibrations and Yoneda's lemma in a 2-category

1974-01-01

Ross Street

Morita equivalences of enriched categories

1974-01-01

Harald Lindner

Foundations of Algebraic Topology

1952-12-31

Samuel Eilenberg , Norman Steenrod

The need for an axiomatic treatment of homology and cohomology theory has long been felt by topologists. Professors Eilenberg and Steenrod present here for the first time an axiomatization of … The need for an axiomatic treatment of homology and cohomology theory has long been felt by topologists. Professors Eilenberg and Steenrod present here for the first time an axiomatization of the complete transition from topology to algebra. Originally published in 1952. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

General Functorial Semantics, I

1972-04-01

John Isbell

Rings of Quotients

1975-01-01

Bo Stenström

On limit-preserving functors

1968-12-01

John F. Kennison

BY 3. F. IENNISON Following Lambek [2] we shall use the suggestive term "infimum" for the generalized inverse limit of Kan. "Supremum" is defined dually.In the infimum (supremum) is known … BY 3. F. IENNISON Following Lambek [2] we shall use the suggestive term "infimum" for the generalized inverse limit of Kan. "Supremum" is defined dually.In the infimum (supremum) is known as a "left root" ("right root" ).The terms "inf-complete" and "inf-preserving" are used in the obvious way.If is a small category then [, Ens] shall denote the category of all variant) functors from to the category Ens of sets.[, Ens]inf shall be the full subcategory of inf-preserving functors.The theorem below answers an open question raised in the introduction to

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Limits indexed by category-valued 2-functors

1976-06-01

Ross Street

An interpolation theorem for adjoint functors

1970-08-01

S. A. Huq

0. Introduction. In this paper we present a category theoretic generalization of the construction of the tensor algebra or symmetric algebra of a module which proceeds by representing the module … 0. Introduction. In this paper we present a category theoretic generalization of the construction of the tensor algebra or symmetric algebra of a module which proceeds by representing the module as the quotient of the free module on the underlying set of the given module by its module of relations, then obtaining the tensor algebra as the quotient of the free algebra on the underlying set of the module modulo, the ideal generated by the module of relations, thus giving a uniform construction of many universal gadgets used in algebra. We use the notation of MacLane [1].

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Cartesian bicategories I

1987-11-01

A. Carboni , R. F. C. Walters

Coherence for a closed functor

1972-01-01

Geoffrey Lewis

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Tensor products in categories

1965-03-01

G. M. Kelly

Monads for which structures are adjoint to units

1995-10-01

Anders Kock

Structures algébriques dans les catégories

1968-01-01

Jean Bénabou

On closed categories of functors II

1974-01-01

Brian Day

Colimits of algebras revisited

1977-12-01

Jiřı́ Adámek

It has “been open for some time whether, given an algebraic theory (triple, monad) Π in a cocomplete category K , also the category K Π of Π-algebras must be … It has “been open for some time whether, given an algebraic theory (triple, monad) Π in a cocomplete category K , also the category K Π of Π-algebras must be cocomplete. We solve this in the negative by exhibiting a free algebraic theory Π in the category Gra of graphs such that Gra Π is not cocomplete. Further, we improve somewhat the well-known colimit theorem of Barr and Linton by showing that the base category need not be complete.

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On clubs and doctrines

1974-01-01

G. M. Kelly

Constructions of factorization systems in categories

1977-01-01

A. K. Bousfield

Adjunctions whose counits are coequalizers, and presentations of finitary enriched monads

1993-10-01

G. M. Kelly , John Power

On locales of localizations

1987-01-01

Francis Borceux , G. M. Kelly

Cocompleteness over coverings

1985-10-01

Renato Betti

Abstract For enriched categories the correct notion of limit involves indexing by a module. This paper studies the question of cocompletion for a given set of indexing modules. As well … Abstract For enriched categories the correct notion of limit involves indexing by a module. This paper studies the question of cocompletion for a given set of indexing modules. As well as providing a simplified treatment of cocompleteness for ordinary categories, associated sheaves and associated stacks are also included as cocompletion processes for appropriate bases. In fact the saturation of a general set of indexing modules has properties which justify our use of the term “covering” for members of the saturation.

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Variation through enrichment

1983-08-01

Renato Betti , A. Carboni , Ross Street +1 more

Yoneda structures on 2-categories

1978-02-01

Ross Street , R. F. C. Walters

Order ideals in categories

1986-10-01

A. Carboni , Ross Street

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Zur Homotopietheorie der Kettenkomplexe

1960-08-01

Albrecht Dold

Closed categories generated by commutative monads

1971-11-01

Anders Kock

The notion of commutative monad was defined by the author in [4]. The content of the present paper may briefly be stated: The category of algebras for a commutative monad … The notion of commutative monad was defined by the author in [4]. The content of the present paper may briefly be stated: The category of algebras for a commutative monad can in a canonical way be made into a closed category, the two adjoint functors connecting the category of algebras with the base category are in a canonical way closed functors, and the front- and end-adjunctions are closed transformations. (The terms ‘Closed Category’ etc. are from the paper [2] by Eilenberg and Kelly). In particular, the monad itself is a ‘closed monad’; this fact was also proved in [4].

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Algebra valued functors in general and tensor products in particular

1966-01-01

Peter Freyd

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