Borel and analytic sets in locales

Abstract

We systematically develop analogs of basic concepts from classical descriptive set theory in the context of pointless topology. Our starting point is to take the elements of the free complete Boolean algebra generated by the frame $\mathcal{O}(X)$ of opens to be the "$\infty$-Borel sets" in a locale $X$. We show that several known results in locale theory may be interpreted in this framework as direct analogs of classical descriptive set-theoretic facts, including e.g., the Lusin separation, Lusin-Suslin, and Baire category theorems for locales; we also prove several extensions of these results. We give a detailed analysis of various notions of image, and prove that a continuous map need not have an $\infty$-Borel image. We introduce the category of "analytic $\infty$-Borel locales" as the regular completion under images of the unary site of locales and $\infty$-Borel maps, and prove analogs of several classical results about analytic sets, such as a boundedness theorem for well-founded analytic relations. We also consider the "positive $\infty$-Borel sets" of a locale, formed from opens without using $\neg$. We in fact work throughout with $\kappa$-copresented $\kappa$-locales and $\kappa$-Borel sets for arbitrary regular $\omega_1 \le \kappa \le \infty$, thereby incorporating the classical context as the special case $\kappa = \omega_1$. The basis for the aforementioned localic results is a detailed study of various known and new methods for presenting $\kappa$-frames, $\kappa$-Boolean algebras, etc. In particular, we introduce a new type of "two-sided posite" for presenting $(\kappa, \kappa)$-frames $A$ (i.e., both $A$ and $A^{op}$ are $\kappa$-frames), and use this to prove a general $\kappa$-ary interpolation theorem for $(\kappa, \kappa)$-frames, which dualizes to the aforementioned separation theorems.