Generic absoluteness and boolean names for elements of a Polish space

It is common knowledge in the set theory community that there exists a duality relating the commutative C∗-algebras with the family of B -names for complex numbers in a boolean valued model for set theory VB. Several aspects of this correlation have been considered in works of the late 1970s and early 1980s, for example by Takeuti (Two Applications of Logic to Mathematics. Iwanami Shoten, Publishers, Tokyo; Princeton University Press, Princeton, Kanô Memorial Lectures, vol 3. Publications of the Mathematical Society of Japan, No. 13, 1978) and Fourman et al. (eds.) (Applications of sheaves. In: Lecture Notes in Mathematics, vol 753. Springer, Berlin, 1979), and by Jech (Trans Am Math Soc 289(1):133-162, 1985). Generalizing Jech's results, we extend this duality so as to be able to describe the family of boolean names for elements of any given Polish space Y (such as the complex numbers) in a boolean valued model for set theory VB as a space C+ (X, Y) consisting of functions f whose domain X is the Stone space of B, and whose range is contained in Y modulo a meager set. We also outline how this duality can be combined with generic absoluteness results in order to analyze, by means of forcing arguments, the theory of C+ (X, Y). © 2017 Unione Matematica Italiana.