Non Standard Analysis as a Functor, as Local, as Iterated

Abstract: This note has several aims. Firstly, it portrays a non-standard analysis as a functor, namely a functor * that maps any set A to the set A of its non-standard elements. That functor, from the category of sets to itself, is postulated to be an equivalence on the full subcategory of finite sets onto itself and to preserve finite projective limits (equivalently, to preserve finite products and equalizers). Secondly, "Local" non-standard analysis is introduced as a structure which I call lim-rim, in particular exact lim-rims. The interplay between these, and ultrafilters and ultrapowers, and also cardinality relations and notions depending on a cardinality such as saturation and what I call "confinement" and "exactness", are investigated. In particular, one constructs non-standard analyses, with "good" kinds of lim-rims. In these one may say that *A - "the adjunction of all possible limits from A" - plays a role analogous to that of the algebraic closure of a field - "the adjunction of all roots of polynomials". Then in the same spirit as with the latter, one has uniqueness up to isomorphism, and also universality and homogeneity, provided one has enough General Continuum Hypothesis. The cardinality of *A will be something like $2^{2^{|A|}}$, and one has a high degree of saturation. Also, one notes that the functor * can be applied again to *A, giving **A, **A, and so forth. In particular, I focus on the two different embeddings of *A into **A and prove some of their properties, with some applications.