指向范畴的保守函子
Conservative functors to pointed categories
- University of Milan（米兰大学）
- Scuola Universitaria Superiore IUSS Pavia（帕维亚高等大学学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文引入预指向范畴概念，研究指向范畴的保守函子，推广经典引理并建立普遍指向化构造。
AI中文摘要：
范畴代数中的许多结果从根本上依赖于指向性，然而许多数学上感兴趣的范畴并非指向的。基于G. Janelidze最近引入的理想正合范畴理论的思想，我们研究了指向背景下的构造和结果在多大程度上可以推广到允许到指向范畴的合适遗忘函子的范畴。我们引入了预指向范畴的概念，将其描述为允许到指向lex范畴的保守右伴随函子的范畴，并给出了这一概念的内在刻画。然后，我们研究并刻画了给定函子的目标是指向原模、同调或正规范畴的情形，并在这些设定中建立了经典结果的推广——包括短五引理、九引理和诺特定理的同构定理——以及一个行为良好的对象理想概念。我们还证明了指向lex范畴的2-范畴在具有初始对象的lex范畴的2-范畴中是2-反射的，反射由在初始对象上的切片给出，从而提供了一个普遍的“指向化”。
英文摘要：
Many results in categorical algebra rely fundamentally on pointedness, yet numerous categories of mathematical interest are not pointed. Building on ideas from the theory of ideally exact categories, recently introduced by G. Janelidze, we investigate the extent to which constructions and results from pointed contexts can be extended to categories admitting suitable forgetful functors to pointed categories. We introduce the notion of a propointed category, described as a category admitting a conservative right-adjoint functor to a pointed lex category, and give an intrinsic characterisation of this notion. We then investigate and characterise the cases in which the target of the given functor is pointed protomodular, homological, or normal, and establish within these settings generalisations of classical results - including the short five lemma, the nine lemma, and Noether's isomorphism theorems - as well as a well-behaved notion of ideal of an object. We also prove that the 2-category of pointed lex categories is 2-reflective in the 2-category of lex categories with an initial object, the reflection being given by the slice over the initial object, which thus provides a universal 'pointification'.
