On the structure of the Wadge degrees of BQO-valued Borel functions

Abstract: In this article, we give a full description of the Wadge degrees of Borel functions from $\omega^\omega$ to a better quasi ordering $\mathcal{Q}$. More precisely, for any countable ordinal $\xi$, we show that the Wadge degrees of $\mathbf{\Delta}^0_{1+\xi}$-measurable functions $\omega^\omega\to\mathcal{Q}$ can be represented by countable joins of the $\xi$-th transfinite nests of $\mathcal{Q}$-labeled well-founded trees.