A. Carboni

Introduction to extensive and distributive categories

Abstract In recent years, there has been considerable discussion as to the appropriate definition of distributive categories. Three definitions which have had some support are: (1) A category with finite sums and products such that the canonical map δ: A × B + A × C → A ×( B + C ) is an isomorphism (Walters). (2) A category with finite sums and products such that the canonical functor +: A / A × A / B → A /( A + B ) is an equivalence (Monro). (3) A category with finite sums and finite limits such that the canonical functor + of (2) is an equivalence (Lawvere and Schanuel). There has been some confusion as to which of these was the natural notion to consider. This resulted from the fact that there are actually two elementary notions being combined in the above three definitions. The first, to which we give the name distributivity , is exactly that of (1). The second notion, which we shall call extensivity , is that of a category with finite sums for which the canonical functor + of definitions (2) and (3) is an equivalence. Extensivity, although it implies the existence of certain pullbacks, is essentially a property of having well-behaved sums. It is the existence of these pullbacks which has caused the confusion. The connections between definition (1) and definitions (2) and (3) are that any extensive category with products is distributive in the first sense, and that any category satisfying (3) satisfies (1) locally. The purpose of this paper is to present some basic facts about extensive and distributive categories, and to discuss the relationships between the two notions.

Variation through enrichment

This paper continues the authors’ various works [3,4,12,14] on categories enriched in bicategories. We treat the elements of the theory again, here from a more algebraic (logical) and less geometric viewpoint. For a bicategory ~1 we first develop V -matrices before passing on to ti -modules, an approach tihich allows a simple proof of the cocompleteness of the 2-category ti -Cat of

Regular and exact completions

1 -categories. When 5 has precisely one object (and so is a monoidal category) the main results are in works of Bknabou [21p Lawvere [6], and Wolff

Some free constructions in realizability and proof theory

The regular and exact completions of categories with weak limits are proved to exist and to be determined by an appropriate universal property. Several examples are discussed, and in particular the class of examples given by categories monadic over a power of Set: any such a category is in fact the exact completion of the full subcategory of free algebras. Applications to Grothendieck toposes and geometric morphisms, and to epireflective hulls are also discussed

Connected limits, familial representability and Artin glueing

Abstract Some old and new constructions of free categories with good properties (regularity, exactness, etc.) are investigated, consistently showing their role in proof theory and in realizability theory, and in particular in the construction of the “effective topos” of M. Hyland. The subject of “small complete categories” is discussed, with a proposed new definition of what “complete” should mean for a full reflective subcategory of a topos.

On Localization and Stabilization for Factorization Systems

We consider the following two properties of a functor F from a presheaf topos to the category of sets: (a) F preserves connected limits, and (b) the Artin glueing of F is again a presheaf topos. We show that these two properties are in fact equivalent. In the process, we develop a general technique for associating categorical properties of a category obtained by Artin glueing with preservation properties of the functor along which the glueing takes place. We also give a syntactic characterization of those monads on Set whose functor parts have the above properties, and whose units and multiplications are cartesian natural transformations.

A Categorical Approach to Realizability and Polymorphic Types

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 free exact category on a left exact one

A categorical calculus of relations is used to derive a unified setting for higher order logic and polymorphic lambda calculus.

Type theory via exact categories

We give an explicit one step description of the free (Barr) exact category on a left exact one. As an application we give a new two step construction of the free abelian category on an additive one.

Bicategories of spans and relations

Partial equivalence relations (and categories of these) are a standard tool in semantics of type theories and programming languages, since they often provide a cartesian closed category with extended definability. Using the theory of exact categories, we give a category-theoretic explanation of why the construction of a category of partial equivalence relations often produces a cartesian closed category. We show how several familiar examples of categories of partial equivalence relations fit into the general framework.