Does anyone know of good references for nonstandard set theories and their applications to various branches of mathematics like category theory, algebra, geometry, etc.?

Edit: What I mean by "nonstandard set theory" is a formalization of the naive notion of sets that allows direct arguments about certain intuitive notions like infinitesimals and infinite integers without recourse to model theoretic constructions like ultrafilters and ultraproducts. Infinitesimal number is the usual application I keep seeing but I'm sure there must be other applications and that's the intent of my question. I'm not sure if this is precise enough.

Does anyone know of good references for nonstandard set theories and their applications to various branches of mathematics like category theory, algebra, geometry, etc.?