  For a complete and cocomplete category $\mathcal{C}$ with a well-behaved
class of `projectives' $\bar{\mathcal{P}}$, we construct a model structure on
the category $s\mathcal{C}$ of simplicial objects in $\mathcal{C}$ where the
weak equivalences, fibrations and cofibrations are defined in terms of
$\bar{\mathcal{P}}$. This holds in particular when $\mathcal{C}$ is
$\mathcal{U}$, the category of compactly generated, weakly Hausdorff spaces,
and $\bar{\mathcal{P}}$ is the class of compact Hausdorff spaces.
  We also construct a new model structure on $\mathcal{U}$ itself, where the
cofibrant spaces are generalisations of CW-complexes allowing spaces, rather
than sets, of $n$-cells to be attached. The singular simplicial complex and
geometric realisation functors give a Quillen adjunction between these model
structures.
  For a space in $\mathcal{U}$, these structures allow the definition of
homotopy group objects in the exact completion of $\mathcal{U}$, which are
invariant under weak equivalence and have a lot of the nice properties usually
expected of homotopy groups. There is a long exact sequence of homotopy group
objects arising from a fibre sequence in $\mathcal{U}$.
  Working along similar lines, we study homological algebra in categories of
internal modules in $\mathcal{U}$, getting in particular a
Lyndon--Hochschild--Serre spectral sequence for extensions of topological
groups in $\mathcal{U}$.
