Armin Frei

Extending localization functors

We prove that, under suitable restrictions, an idempotent monad t defined on a full subcategory A of a category C can be extended to an idempotent monad T on C in a universal (terminal) way. Our result applies in particular to the case when t is P-localization of nilpotent groups (where P denotes a set of primes) and C is the category of all groups. The corresponding monad T on C is, in a certain precise sense, the best idempotent approximation to the usual Zp-completion of groups; it turns out to be a strict epimorphic image of Bousfields HZp-localization.

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.