Cosheaves and homology

Abstract

In this paper we initiate a study of the theory of cosheaves of modules. We are interested mainly in those facts which are not encompassed by the known theories of sheaves with values in general categories. A central result is the establishment of the existence of a reasonably large subcategory of the category of precosheaves and a reflector from this to the subcategory of cosheaves. The general theory is applied to the study of the Čech, singular, and Borel-Moore homology theories. The main applications establish that the Čech and Borel-Moore homology theories coincide on locally compact and paracompact clc∞ spaces and that the Čech and singular theories coincide on paracompact HLC spaces. These isomorphisms are established for locally constant coefficients. For constant coefficients the latter result was originally established by Mardešić and the former by 0. Jussila. There are also applications to acyclic coverings and to mappings. © 1968 by Pacific Journal of Mathematics.