  In "On o-minimal homotopy groups", o-minimal homotopy was developed for the
definable category, proving o-minimal versions of the Hurewicz theorems and the
Whitehead theorem. Here, we extend these results to the category of locally
definable spaces, for which we introduce homology and homotopy functors. We
also study the concept of connectedness in V-definable groups -- which are
examples of locally definable spaces. We show that the various concepts of
connectedness associated to these groups, which have appeared in the
literature, are non-equivalent.
