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 stabilise as well. This
generalises 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 various examples including 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.
