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 O. Jussila. There are also applications to acyclic coverings and to
mappings.
