AI 中文总结
该研究利用范畴上的子对象函子构建广义拓扑与测度论框架，开发两种关联演算，推导相关表示定理并提出n-范畴推广猜想，为范畴间理论提供新方法。
AI 中文摘要
我们引入一种系统方法，利用任意范畴上的子对象函子，构建广义拓扑与测度论概念及推理的框架。具体而言，我们开发了两种关联演算：1. 一种源于小性与稠密性概念的四重演算，它推广了拓扑稠密性、测度论可忽略性等概念；2. 一种推广拓扑与测度论推理程序的二重演算。所得理论的独特之处在于其范畴间特性：其定义不仅可用于描述任意范畴中子对象的小性与稠密性、按态射处理这些性质的方式对态射进行分类，还可用于按函子与自然变换转换这些性质的方式对其进行分类，这与极限理论类似，但具有适用于每个实例的简单直观的概念意义。我们提供了来自数学多个领域的此类概念的示例，并在添加假设使其更接近经典设定的过程中概述了它们在整个范畴论中的展开，我们通过推广开集与闭集的拓扑对偶性并概述范畴测度论的可能框架来完成这一过程。这种相互作用的核心是子对象2-函子与上有界预序的2-范畴，因此在文本的最后部分，我们研究了它们的功能。具体而言，我们推导了该2-函子与2-范畴的表示定理，并提出了将这些定理推广到n-范畴的猜想。
英文摘要
We introduce a systematic way to use the subobject functor on any category to set up an framework for generalized topological and measure-theoretic notions and inferences. Concretely, we develop two connected calculi: 1. a fourfold calculus derived from notions of smallness and density that generalizes, among other things, topological density and measure-theoretic negligibility, 2. a twofold calculus that generalizes topological and measure-theoretic inference procedures. The resulting theory is distinguished in that it is inter-categorical in character: its definitions can be applied not just to describe smallness and density of subobjects in any category and classify morphisms by how they handle them, but also functors and natural transformations by how they transform them, somewhat like the theory of limits but with a simple and intuitive conceptual meaning that applies to each incarnation. We provide examples of these notions from many areas of mathematics and outline their unfolding throughout category theory as we add assumptions that bring them closer to classical settings, which we complete by generalizing the topological duality of open and closed subsets and outlining a possible framework for a categorical measure theory. Central to this interaction is the subobject 2-functor and the 2-category of upward-bounded preorders, so in the last part of the text we provide a study of their function. Concretely we derive representation theorems for this 2-functor and 2-category and a conjecture about how to generalize these theorems to n-categories.
Comments127 pages, 41 figures, part of my Ph.D. thesis
