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.



nonstandard set theories
It seems that Lars Brünjes and Christian Serpé have a whole program for introducing non standard mathematics in algebraic geometry; they wish to play with non standard contructions, seen internally and externally (the interest of this game consists precisely to look at the non-classical logic (or internal) point of view and at the classical (or external) one at the same time). For instance, for an infinite prime number $P$, the ring $\mathbf{Z}/P\mathbf{Z}$ behaves internally like a finite field, while externally, it is a field of characteristic zero which contains an algebraic closure of $\mathbf{Q}$. Non-standard constructions can often be interpreted in a precise way as standard ones using ultraproducts and ultrafilters. Their purpose is to develop all the tools of classical algebraic geometry (homotopical and homological algebra, stacks, étale cohomology, algebraic K-theory, higher Chow groups...) in a non-standard way, in order to prove facts in the classical setting. Most of their papers can be found [here](http://wwwmath.uni-muenster.de/u/serpe/arbeiten-en.html) (their papers contains more precise ideas on the possible interpretations and explanations). Brünjes and Serpé see non-standard mathematics as an enlargement of standard mathematics, and their work deals a lot in making this precise. However, they seem to have quite few concrete problems in mind. For instance, they have found sufficient conditions on cohomology classes to be algebraic in a very classical sense (see arXiv:0901.4853).


