  Holomorphic chains on a Riemann surface arise naturally as fixed points of
the natural C*-action on the moduli space of Higgs bundles. In this paper we
associate a new quiver bundle to the Hom-complex of two chains, and prove that
stability of the chains implies stability of this new quiver bundle. Our
approach uses the Hitchin-Kobayashi correspondence for quiver bundles.
Moreover, we use our result to give a new proof of a key lemma on chains (due
to \'Alvarez-C\'onsul, Garc\'ia-Prada and Schmitt), which has been important in
the study of Higgs bundle moduli; this proof relies on stability and thus
avoids the direct use of the chain vortex equations.
