双极空间
Bipolar spaces
摘要：
研究了范畴A上的离散纤维化和离散光纤维化同时包含在A上的范畴范畴中的一些基本性质;特别地，考虑了后者在前者中的反射和共反射，以及负补算子，其应用于离散纤维化，给出了一个函子的值在离散opfibrations（反之亦然），这原来是经典的，因为强对位法成立。这种分析是在一个适当的概念框架内进行的，该框架包括类似的“两极”情况，其中“共同数字”即产品的组成部分发挥着关键作用;例如，负补算子的经典性对应于离散光学纤维化的事实，（或一般来说“封闭部分”）通过具有离散纤维化（“开放部分”）中的形状的共形来适当地分析，即，后者对于前者是“共适的”，反之亦然。在此背景下，我们提出了一个非常自然的“原子”定义，并证明了在上述情形下，原子反射范畴是A的柯西完备化。
英文摘要：
Some basic features of the simultaneous inclusion of discrete fibrations and discrete opfibrations on a category A in the category of categories over A are studied; in particular, the reflections and the coreflections of the latter in the former are considered, along with a negation-complement operator which, applied to a discrete fibration, gives a functor with values in discrete opfibrations (and vice versa) and which turns out to be classical, in that the strong contraposition law holds. Such an analysis is developed in an appropriate conceptual frame that encompasses similar "bipolar" situations and in which a key role is played by "cofigures", that is components of products; e.g. the classicity of the negation-complement operator corresponds to the fact that discrete opfibrations (or in general "closed parts") are properly analyzed by cofigures with shape in discrete fibrations ("open parts"), that is, that the latter are "coadequate" for the former, and vice versa. In this context, a very natural definition of "atom" is proposed and it is shown that, in the above situation, the category of atoms reflections is the Cauchy completion of A.
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。
