  We consider the notion of metric spaces being locally Lipschitz contractible
introduced by Yamaguchi, and a category of metric spaces satisfying this
condition. Many objects in metric geometry including CAT-spaces and Alexandrov
spaces, belong to this category. We consider the homology of integral currents
with compact support in a metric space, introduced by Ambrosio and Kirchheim,
and prove that it and the usual integral singular homology are isomorphic on
the category. The proof of it is based on the theory of cosheaves. A method to
compare the homologies associated to cosheaves is also proved in this paper.
