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

## TL;DR

This paper characterizes the Wadge degrees of Borel functions from Baire space to better quasi orderings, revealing their structure through transfinite nests of well-founded trees, thus advancing the understanding of their descriptive complexity.

## Contribution

It provides a complete description of the Wadge degrees of Borel functions into BQOs, using transfinite nests of labeled well-founded trees for representation.

## Key findings

- Wadge degrees of $oldsymbol{ riangle}^0_{1+\xi}$-measurable functions are characterized.
- Representation of degrees via countable joins of transfinite nests.
- Complete structural description of Wadge degrees for Borel functions into BQOs.

## 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.