Structure theory for pro-objects from an arbitrary category

Abstract

The focus of this paper is a well-known lemma: A continuous group homomorphism between profinite groups which has dense image must be an open surjection. Its importance lies in the common occurrence of infinite inversely directed systems of finite groups. A limit, in the category of groups, is rarely useful. Instead, experience reveals that an inverse limit in the topologized sense completely characterizes the system. Principles like the above lemma empower this model. The author needs “good” inverse limits for a broad clas of categories. We develop an idea from the 1960s: Given a categoryn C, expand it with purely formal projective limits. Although definition of such a category is elementary, the approach has difficulties. In the past, crucial manipulations refused generalization. Our thesis is that, by tightening hypotheses, the method creates a category valuable to analysis. In this paper, we 1. review and modify the categorical formulation, 2. prove a structure theory for pullbacks and inversely directed systems, and 3. generalize the indicated lemma.