非标准分析作为函子、局部、迭代
Non Standard Analysis as a Functor, as Local, as Iterated
摘要：
本说明有几个目的。首先，它将非标准分析描述为一个函子，即一个将任何集合A映射到其非标准元的集合 *A的函子 *。这个函子，从集范畴到它自身，被假定为在有限集合的满子范畴到它自身上的等价，并且保持有限的投射极限（等价地，保持有限的乘积和均衡器）。其次，“局部”非标准分析被引入作为一种结构，我称之为lim-rim，特别是精确lim-rims。这些之间的相互作用，和超过滤器和ultrapowers，也基数关系和概念取决于基数，如饱和度和我所谓的“禁闭”和“精确度”，进行了研究。特别是，人们构建非标准的分析，与“好”种lim-rims。在这些人可以说，*A -“所有可能的限制从A的附加”-发挥类似的作用，代数闭包的领域-“的附加所有根的多项式”。然后，在与后者相同的精神下，只要有足够的一般连续统假设，就有直到同构的唯一性，也有普遍性和同质性。*A的基数将类似于$2^{2^{|A|}}$，并且具有高饱和度。此外，我们注意到函子 * 可以再次应用于 *A，给出 **A，*A，等等。特别是，我专注于 *A到 **A的两种不同的嵌入，并证明了它们的一些性质，以及一些应用。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
