[论文解读] Positive and negative extensions in extriangulated categories

Mikhail Gorsky, Hiroyuki Nakaoka+1 more|arXiv (Cornell University)|Mar 23, 2021

Algebraic structures and combinatorial models参考文献 28被引用 5

一句话总结

本文通过共端（coends）建立了一类在扩张范畴中更高阶正向与负向扩张的 Yoneda 型理论，证明当存在足够多的投射或内射对象时，这些扩张与经典定义（通过投射/内射解析式定义）一致。本文将负向扩张定义为 Hom 的左导出函子，在适当条件下证明其构成普遍 δ-函子，并通过 ∞-范畴增强方法，在拓扑扩张范畴中证明正向与负向扩张的平衡性。

ABSTRACT

We initiate the study of derived functors in the setting of extriangulated categories. By using coends, we adapt Yoneda's theory of higher extensions to this framework. We show that, when there are enough projectives or enough injectives, thus defined extensions agree with the ones defined earlier via projective or injective resolutions. For categories with enough projective or enough injective morphisms, we prove that these are right derived functors of the $\operatorname{Hom}$-bifunctor in either argument. Since $\operatorname{Hom}$ is only half-exact in each argument, it is natural to expect "negative extensions", i.e. its left derived functors, to exist and not necessarily vanish. We define negative extensions with respect to the first and to the second argument and show that they give rise to universal $δ$-functors, when there are enough projective or injective morphisms, respectively. In general, they are not balanced. However, for topological extriangulated categories, the existence of a balanced version of negative extensions follows from combining the work of Klemenc on exact $\infty$-categories with results of the second and third authors. We discuss various criteria under which one has the balance of the above bifunctors on the functorial or on the numerical levels. This happens, in particular, in the cases of exact or triangulated categories, and also in the case of the category of $n$-term complexes with projective components over a finite-dimensional algebra. Given a connected sequence of functors on an extriangulated category $(\mathcal{C},\mathbb{E},\mathfrak{s})$, we determine the maximal relative extriangulated structure, with respect to which the sequence is a $δ$-functor. We also find several equivalent criteria for the existence of enough projective or injective morphisms in a given extriangulated category.

研究动机与目标

• 发展一种扩张范畴中更高阶扩张的 Yoneda 型理论，统一经典范畴与三角范畴的理论。
• 将负向扩张定义为 Hom 双函子的左导出函子，其不因 Hom 的半正合性而必然消失。
• 建立正向与负向扩张平衡的条件，特别是在由精确 ∞-范畴导出的拓扑扩张范畴中。
• 通过范畴准则刻画扩张范畴中存在足够多投射或内射态射的条件。

提出的方法

• 使用共端定义扩张范畴中的更高阶正向扩张，推广 Yoneda 的原始构造。
• 证明这些基于共端的扩张在两个变量中均满足长正合序列，从而构成 δ-函子。
• 通过 C-对偶与投射亏格化定义负向扩张，证明当存在足够多的投射或内射态射时，其构成普遍 δ-函子。
• 在存在足够多投射或内射态射的条件下，建立基于共端的扩张与通过投射/内射解析式定义的扩张之间的等价性。
• 应用 Klemenc 关于精确 ∞-范畴的稳定外壳结果，证明拓扑扩张范畴中存在负向扩张的平衡版本。
• 利用缺陷范畴及其性质分析扩张结构，并证明缺陷函子的反射性质。

实验结果

研究问题

• RQ1 在不依赖嵌入到三角范畴的条件下，如何在扩张范畴中定义更高阶扩张？
• RQ2 负向扩张（即 Hom 的左导出函子）在何种条件下存在并构成普遍 δ-函子？
• RQ3 在扩张范畴中，正向与负向扩张在函子意义和数值意义下何时达到平衡？
• RQ4 何种范畴准则可确保给定扩张范畴中存在足够多的投射或内射态射？
• RQ5 基于共端的扩张与通过投射/内射解析式定义的经典定义之间有何关系？

主要发现

• 当存在足够多的投射或内射对象时，基于共端的正向扩张定义与通过投射或内射解析式定义的经典定义一致。
• 正向扩张在两个变量中均满足长正合序列，且通过该构造可明确定义扩张范畴的正向整体维数。
• 负向扩张作为 Hom 的左导出函子存在，并在存在足够多投射或内射态射时构成普遍 δ-函子。
• 在拓扑扩张范畴（即具有精确 ∞-范畴增强的范畴）中，负向扩张存在平衡版本，其由 Klemenc 的稳定外壳构造导出。
• 若原始范畴具有足够多的投射态射，则其缺陷范畴具有足够多的投射对象。
• 为缺陷函子建立了反射性质，表明每个有限表示函子均可表示为两个缺陷函子之间态射的余核。