G.M. Kelly

Coherence for compact closed categories

It gives us great pleasure to honour, on the occasion of his seventieth birthday, our friend and mentor Saunders MacLane; and to acknowledge that our interest in coherence problems stems from his influence, beginning with his fundamental paper [13], and continued through personal contacts-whose value to us, and whose warmth, much transcend what we can express in a formal dedication. Our purpose is to give an explicit description of the free compact closed category on a given category. A compact closed category is a symmetric monoidal one whose internal-horn [A, C] has the form CBA’. Before giving examples of these we analyze and simplify the definition. A monoidal category d with tensor product @ and unit object I can be regarded as a bicategory B with a single O-cell, the l-cells of B being the objects of ti with @ as their composition, and the 2-cells of B being the morphisms of do’. We can therefore speak of a lefr adjoinf of an object A of .9, meaning thereby an object A * of .I together with a “unit” map & : I-+.4 @A * and a “counit” eA : A * @A -I satisfying the usual “triangular equations”, namely that each of the composites

Two-dimensional monad theory

Abstract We consider a 2-monad T with rank on a complete and cocomplete 2-category, and write T-Alg for the 2-category given the T-algebras, the morphisms preserving the structure to within coherent isomorphisms, and the appropriate 2-cells; T-Algs is the sub-2-category obtained by taking the strict morphisms. We show that T-Alg admits pseudo-limits and certain other limits, and that the inclusion 2-functor T-Algs → T-Alg has a left adjoint. We introduce the notion of flexible algebra, and use it to prove that T-Alg admits all bicolimits and that the 2-functor T-Alg → S-Alg induced by a monad-map S → T admits a left biadjoint.

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

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.

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

Abstract A right adjoint functor is said to be of descent type if the counit of the adjunction is pointwise a coequalizer. Building on the results of Tholens doctoral thesis, we give necessary and sufficient conditions for a composite to be of descent type when each factor is so. We apply this to show that every finitary monad on a locally-finitely-presentable enriched category A admits a presentation in terms of basic operations and equations between derived operations, the arties here being the finitely-presentable objects of A .

Elementary observations on 2-categorical limits

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.

Some remarks on Maltsev and Goursat categories

Our aim is to analyze and to publicize two interesting properties — well known in universal algebra for varieties — that a regular category, and in particular an exact category, may possess: theMaltsev property, asserting the permutabilitySR=RS of equivalence relations on any object, and the weakerGoursat property, asserting only thatSRS=RSR. We investigate these properties, give various equivalent forms of them, and develop some of their useful consequences.

Galois theory and a general notion of central extension

Abstract We propose a theory of central extensions for universal algebras, and more generally for objects in an exact category C , centrality being defined relatively to an “admissible” full subcategory X of C . This includes not only the classical notions of central extensions for groups and for algebras, but also their generalization by Frohlich to a pair consisting of a variety C of ω-groups and a subvariety X . Our notion of central extension is adapted to the generalized Galois theory developed by the first author, the use of which enables us to classify completely the central extensions of a given object B, in terms of the actions of an “internal Galois pregroupoid”.

On Localization and Stabilization for Factorization Systems

If (ε, M)is a factorization system on a category C, we define new classes of maps as follows: a map f:A→B is in ε′ if each of its pullbacks lies in ε(that is, if it is stably in ε), and is in M* if some pullback of it along an effective descent map lies in M(that is, if it is locally in M). We find necessary and sufficient conditions for (ε′, M*) to be another factorization system, and show that a number of interesting factorization systems arise in this way. We further make the connexion with Galois theory, where M*is the class of coverings; and include self-contained modern accounts of factorization systems, descent theory, and Galois theory.

The closure of a class of colimits

We consider V-categories where V is a symmetric monoidal closed category, and we write φ * T for the colimit of T : K → A indexed by φ : Kop → V, where K is small. Let Φ be a class of such indexing types (K, φ), and write Φ ∗ for the class of indexing types (J, ψ) such that every Φ-cocomplete A is ψ-cocomplete and every Φ-cocontinuous functor is ψ-cocontinuous. We show that ψ ∈ [Jop, V] lies in Φ ∗ if and only if it lies in the Φ-colimit closure of J in [Jop, V], and characterize those Φ for which Φ ∗ = Φ.

Flexible limits for 2-categories

Many important 2-categories — such as Lex, Fib/B, elementary toposes and logical morphisms, the dual of Grothendieck toposes and geometric morphisms, locally-presentable categories and left adjoints, the dual of this last, and the Makkai-Pare 2-category of accessible categories and accessible functors — fail to be complete, lacking even equalizers. These examples do in fact admit all bilimits — those weakenings of the limit notion that represent not by an isomorphism but only by an equivalence — but much more is true: they admit important classes of honest limits, including products, cotensor products, comma objects, Eilenberg-Moore objects, descent objects, inserters, equifiers, inverters, lax limits, pseudo limits, and idempotent-splitting. We introduce the class of flexible limits, which includes all of the above and is, in the technical sense, a closed class. Note that such honest limits, when they exist, have many advantages over bilimits: they are unique to within isomorphism, and their universal properties are both stronger and more convenient to use, a whole level of coherent families of invertible 2-cells being avoided.