Given a graded -module over an -algebra in spaces, we construct an
augmented semi-simplicial space up to higher coherent homotopy over it, called
its canonical resolution, whose graded connectivity yields homological
stability for the graded pieces of the module with respect to constant and
abelian coefficients. We furthermore introduce a notion of coefficient systems
of finite degree in this context and show that, without further assumptions,
the corresponding twisted homology groups stabilize as well. This generalizes a
framework of Randal-Williams and Wahl for families of discrete groups.
In many examples, the canonical resolution recovers geometric resolutions
with known connectivity bounds. As a consequence, we derive new twisted
homological stability results for e.g. moduli spaces of high-dimensional
manifolds, unordered configuration spaces of manifolds with labels in a
fibration, and moduli spaces of manifolds equipped with unordered embedded
discs. This in turn implies representation stability for the ordered variants
of the latter examples.