Robert Paré

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.

Indexed categories and their applications

Abstract families and the adjoint functor theorems.- V-indexed categories.- Algebraic theories in toposes.- Coequalizers in algebras for an internal type.

Families parametrized by coalgebras

Since Hopf [H] first introduced the coalgebra structure on the homology ring of a grouplike manifold, making it into what is now called a Hopf algebra, coalgebras have been appearing more and more frequently in many branches of mathematics, in particular in algebraic topology, homological algebra and in algebraic geometry. This prompted fundamental work on the structure of coalgebras themselves, a good part of which is outlined in the influential article of Milnor and Moore [M-M], in Sweedler’s book [S] and in articles mentioned later in this text. At first glance, coalgebras are strange objects. Although they are defined in algebraic terms, our algebraic intuition breaks down when trying to understand them. They really are more geometric than algebraic. This is somewhat explained by the partial duality that exists between algebras and coalgebras (with scalars from a field). The dual of a coalgebra is an algebra and although the dualizing functor is not an equivalence of categories, it does have an adjoint on the right ( )” [S, p. 1091. These two functors restrict to a contravariant equivalence between finite dimensional coalgebras and finite dimensional algebras. The category of cocommutative coalgebras is similar in many respects to the opposite of the category of commutative k-algebras, and this is the same as the category of afftne schemes over k. So, it is not surprising that cocommutative coalgebras are geometric entities. In fact, they correspond to formal schemes [C, T]. Due to this partial duality between coalgebras and algebras, many definitions in the theory of coalgebras were suggested by the corresponding concepts for algebras (such as the cotensor 0 <. of [M-M]), and the statements of many theorems are inspired by the corresponding results for algebras. But in many respects, the category of cocommutative coalgebras is much better than the category of algebras. It is Cartesian closed, i.e., there is a coalgehra of morphisms from one coalgebra to another, with the appropriate universal property (see [ML, p. 953 for the definition of “cartesian closed”). This makes Coalg into a monoidal category [K, p. 213 and

Monoidal categories with natural numbers object

The notion of a natural numbers object in a monoidal category is defined and it is shown that the theory of primitive recursive functions can be developed. This is done by considering the category of cocommutative comonoids which is cartesian, and where the theory of natural numbers objects is well developed. A number of examples illustrate the usefulness of the concept.

On a commutator theorem of robert c. thompson

A very short proof of the following special case of R. C. Thompsons Theorem is presented. If A is an n × n mairix over a field with more than n elements and if det A = 1. then Ais a multiplicative commutator.

What is a free double category like

Abstract We give a geometric description of the free double category generated by a double reflexive graph. Its cells are homotopy classes of colourings of certain rectangular complexes in the plane. A number of examples illustrate the wide variety of combinatorial properties of the plane this touches.

Undecidability of the free adjoint construction

In this paper we discuss some aspects of categories obtained by freely adding right adjoints to all arrows in a category. We will give a description of the arrows and 2-cells in such a category and show how the equivalence relation on the 2-cells for an appropriately chosen category CA can be used to simulate a 2-register abacus A, so that deciding whether two 2-cells with different representatives are equal becomes equivalent to solving the halting problem for the abacus. In particular, this implies that (in general) equality of 2-cells in such categories is undecidable.

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

Abstract In the preceding article “The closure of a class of colimits” by M.H. Albert and G.M. Kelly, the authors were unable, except in one special case, to answer the following question. Let an indexing type ψ for colimits be such that every category admitting all ∅-indexed colimits for ∅ in some class Φ also admits ψ-indexed colimits; need a functor which preserves all Φ-indexed colimits also preserve ψ-indexed ones? Without settling the question, the present authors give two more positive results in this direction, including an affirmative answer when Φ-indexed colimits are conical.

Mealy Morphisms of Enriched Categories

We define and study the properties of a notion of morphism of enriched categories, intermediate between strong functor and profunctor. Suggested by bicategorical considerations, it turns out to be a generalization of Mealy machine, well-known since the 1950’s in the theory of computation. When the base category is closed we construct a classifying category for Mealy morphisms, as we call them. This is also seen to give the free tensor completion of an enriched category.

Wobbly double functors

Abstract We introduce a weakened notion of double functor, which we call wobbly, and which arises naturally in the study of double adjoints. We then show how horizontal invariance can be used to lift results about wobbly double functors to genuine ones.