Generation of Grothendieck topologies, provability and operations on subtoposes

Abstract: After reviewing the multiple roles of toposes - as generalized topological spaces, as universal invariants, as categorical analogues of the set-theoretic universe, and as semantic environments for first-order theories - we recall the notion of subtopos and its dual expression: both in terms of Grothendieck topologies and in terms of first-order logic. We emphasize the significance of this duality, which enables the translation of provability problems in first-order logic into problems concerning the generation of Grothendieck topologies. We also introduce the natural geometric operations, both inner and outer, on subtoposes. Building on these foundations, we present a new formulation of the duality between Grothendieck topologies and subtoposes, as well as the duality between topologies and closedness properties of subpresheaves. This presentation relies on general categorical principles and aims to clarify the structural relationships involved. We then provide two general formulas for the Grothendieck topology generated by an arbitrary family of sieves or covering families of morphisms. In addition, we refine the constructive procedures that translate logical provability into topology generation, highlighting their role in bridging logic and geometry within the topos-theoretic framework. Finally, we study the inner geometric operations on subtoposes: union, intersection, and difference, along with the outer adjoint operations of pushforward and pullback along topos morphisms. We prove that pullback operations preserve not only arbitrary intersections but also finite unions of subtoposes, and that pullbacks along locally connected morphisms even preserve arbitrary unions.