Some theories are unnaturally complicated when described in the language of sets. Using the languages of other topoi can help to make them look simpler, and therefore to be simpler to work with. In a sense, relativization is the search for the natural habitat of a mathematical theory, where it can be ‘its true self’ and thrive. Stochastic calculus is one such a theory: topoi of sheaves over suitably defined sites make the theory tame and natural-looking.
My goal was to construct Ito's and Stratonovich's integrals in this way.