阿贝尔范畴与三角范畴的2-范畴的精确性
Exactness of the 2-categories of abelian and triangulated categories
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中文总结 AI 辅助
本文引入$2$-同调范畴概念，证明双正合、三角、带点、加性及阿贝尔范畴的$2$-范畴均为$2$-同调，并给出统一商构造准则。
中文摘要 AI 辅助
我们引入了一个$2$-同调范畴的概念，该概念以Grandis意义下的带点同调范畴为模型，使用双零对象，其零$1$-胞腔是hom-范畴中的零对象，用以表述$2$维带点性。我们首先直接证明，推广了Puppe精确范畴的双正合同调范畴的$2$-范畴是$2$-同调的，其中态射为保持所有核与余核的函子，以及任意自然变换。其正规子范畴是饱和厚子范畴，其精确商通过类似于Puppe精确范畴已知的完全三箭头分数演算构造。然后我们调整此证明以证明三角范畴的$2$-范畴也是$2$-同调的；此处，我们使用众所周知的Verdier分数而非三元分数。抽象这些证明的共同商结构得出一个一般准则，利用该准则我们进一步确立了带点、加性及阿贝尔范畴的$2$-范畴也是$2$-同调的。该准则也适用于在固定交换半环上富集的、具有零对象的半模范畴，以及其具有有限双积的全子$2$-范畴。它们的商是线性同余商。将富集与Puppe精确性结合给出更多例子，其商是精确线性局部化，包括作为有限双积情形的线性阿贝尔范畴。在阿贝尔与三角情形中，正规子范畴与商分别为Serre子范畴与Serre商，以及厚三角子范畴与Verdier局部化。
英文摘要
We introduce a notion of $2$-homological category modelled on pointed homological categories in the sense of Grandis, using bizero objects whose null $1$-cells are zero objects in the hom-categories for formulating $2$-dimensional pointedness. We first prove directly that the $2$-category of di-exact homological categories, which generalize Puppe exact categories, functors preserving all kernels and cokernels, and arbitrary natural transformations is $2$-homological. Its normal subcategories are saturated thick subcategories, and its exact quotients are constructed by a complete three-arrow fraction calculus similar to the one known for Puppe exact categories. We then adapt this proof to prove that the $2$-category of triangulated categories is also $2$-homological; here, instead of the ternary fractions we use the well-known Verdier fractions. Abstracting the common quotient structure of these proofs yields a general criterion, using which we further establish that the $2$-categories of pointed, additive and abelian categories are also $2$-homological. The criterion also applies to categories enriched in semimodules over a fixed commutative rig, with a zero object, and to their full sub-$2$-category with finite biproducts. Their quotients are linear congruence quotients. Combining enrichment with Puppe exactness gives further examples whose quotients are exact linear localizations, including linear abelian categories as the finite-biproduct case. In the abelian and triangulated cases the normal subcategories and quotients are, respectively, Serre subcategories and Serre quotients, and thick triangulated subcategories and Verdier localizations.
发表机构
- Stellenbosch University（斯坦伦布什大学）
- National Institute for Theoretical and Computational Sciences (NITheCS)（国家理论与计算科学研究所）
- University of Tartu（塔尔图大学）
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