  The intuitionistic implication and hence the notion of function space in
constructive disciplines is both non-geometric and impredicative. In this paper
we try to solve both of these problems by first introducing weak exponential
objects as a formalization for predicative function spaces and then by
proposing modal spaces as a way to introduce a natural family of geometric
predicative implications based on the interplay between the concepts of time
and space. This combination then leads to a brand new family of modal
propositional logics with predicative implications and then to topological
semantics for these logics and some weak modal and sub-intuitionistic logics,
as well. Finally, we will lift these notions and the corresponding relations to
a higher and more structured level of modal topoi and modal type theory.
