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

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 ${\mathbf{\Delta }}_{1+\xi }^0$-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.