Effective Wadge hierarchy in computable quasi-Polish spaces

We define and study an effective version of the Wadge hierarchy in computable quasi-Polish spaces which include most spaces of interest for computable analysis. Along with hierarchies of sets we study hierarchies of $k$-partitions which are interesting on their own. We show that levels of such hierarchies are preserved by the computable effectively open surjections, that if the effective Hausdorff-Kuratowski theorem holds in the Baire space then it holds in every computable quasi-Polish space, and we extend the effective Hausdorff theorem to $k$-partitions.