A syntactic approach to Borel functions: some extensions of Louveau’s theorem

Louveau showed that if a Borel set in a Polish space happens to be in a Borel Wadge class \(\Gamma \), then its \(\Gamma \)-code can be obtained from its Borel code in a hyperarithmetical manner. We extend Louveau’s theorem to Borel functions: If a Borel function on a Polish space happens to be a \( \underset{\widetilde{}}{\varvec{\Sigma }}\hbox {}_t\)-function, then one can find its \( \underset{\widetilde{}}{\varvec{\Sigma }}\hbox {}_t\)-code hyperarithmetically relative to its Borel code. More generally, we prove extension-type, domination-type, and decomposition-type variants of Louveau’s theorem for Borel functions.