John L. MacDonald

Soft adjunction between 2-categories

Abstract Each of the usual three equivalent forms for definition of adjoint functor admits various generalizations to the 2-category case. By replacing the notion of natural isomorphism between hom sets in the 1-category case by natural adjunction between hom categories in the 2-category case we obtain an appropriate generalization which is shown to have an equivalent couniversal 2-cell form as well as an equivalent equational form. Various dualities and uniqueness properties are pointed out. These ideas are illustrated in the example of the 2-category Adj as well as in a series of examples to be described later.

Limits in categories of relations and limit-colimit commutation

The category A of relations in an Abelian category A is isomorphic to its own dual. This entails that direct limit in A can be computed as inverse limits and vice- versa. This, together with the fact that limits of the same type commute, suggests the use of the category A to obtain criteria for limit-colimit commutation in A. The first section consists of an account of the necessary introductory paraphernalia. The second section is devoted to cofinal functors and their applications to the theory of limits and relative limits. In the third section conditions are given for the existence of certain limits and relative limits in the category A of relations. In section four we show that in A limits can be computed as colimits and vice-versa. Under certain circumstances the limit of the colimit functor of a bifunctor F with range A is just the limit in A of a suitable functor. A key point for applications is given by a theorem giving conditions for othe commutativity of limits and relative limits in A. In section five we apply the foregoing to the problem of commuting of limits with colimits in A.

COHOMOLOGY OPERATIONS IN A CATEGORY

The purpose of this paper is to make clear that the concepts of cohomology and cohomology operations, as well as homotopy operations, are very natural ones and in fact arise with respect to any object or set of objects in a category which has enough structure for the definitions to make sense. This means, for example, that given an object A in a category .dwe can use the object A to form a sequence of algebraic “pictures” or algebraic higher order structure of the category :/taken relative to A. We confine ourselves here to the general construction of primary and secondary operations, carried out in the first two sections. It should be noted, however, that this is