Strongly graded groupoids and strongly graded Steinberg algebras

We study strongly graded groupoids, which are topological groupoids $\mathcal G$equippedwithacontinuous,surjectivefunctor$\kappa: \mathcal G \to \Gamma$,toadiscretegroup$\Gamma$,suchthat$\kappa^{-1}(\gamma)\kappa^{-1}(\delta) = \kappa^{-1}(\gamma \delta)$,forall$\gamma, \delta \in \Gamma$\mathrm{.}Weintroducethecategoryofgraded$\mathcal G$-sheaves,andproveananalogueofDad{e}^{\mathrm{\prime }}sTheorem:$\mathcal G$ is strongly graded if and only if every graded $\mathcal{G}$-sheaf is induced by a $\mathcal G_{\epsilon}$-sheaf. The Steinberg algebra of a graded ample groupoid is graded, and we prove that the algebra is strongly graded if and only if the groupoid is. Applying this result, we obtain a complete graphical characterisation of strongly graded Leavitt path and Kumjian-Pask algebras.