There are many examples.
You might like Rousseau's reduction of many complex variables to a single one.
In general, topos theory allows one to avoid the axiom of choice and classical logic. For instance, see our concrete treatment of parts of the theory of Banach algebras.
These ideas of course have their roots in the work of Banaschewski and Mulvey of finding a proper treatment of sheaves of C-algebras/ C-bundles by considering a single C*-algebra in a sheaf topos. This work is also used in the Bohr toposes mentioned above.