There are many examples. 

You might like Rousseau's [reduction](http://link.springer.com/content/pdf/10.1007/BFb0061838.pdf) 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](http://www.logicandanalysis.org/index.php/jla/article/view/84/33) 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](http://www.maths.sussex.ac.uk/Staff/CJM/research/pdf/globgelf.pdf) 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.