Descriptive Set Theory in the Category of Represented Spaces

We propose to extend descriptive set theory (DST) beyond its traditional setting of Polish spaces to the represented spaces. There, we can reformulate DST in terms of endofunctors on the categories of represented spaces and computable or continuous functions. In particular, this approach satisfies the demand for a uniform approach to both classic and effective DST -- computability follows naturally from the setting, rather than having to be explicitly demanded. The previous endeavour to extend DST to the Quasi-Polish spaces is subsumed by this work. In several cases the category-theoretic setting enables new, very succinct proofs, and sheds a new light on why certain results are true. The framework lets us make formal some natural questions not easily approachable by traditional methods.