发表机构
Stellenbosch University; National Institute for Theoretical and Computational Sciences (NITheCS); Institut de Recherche en Mathématique et Physique, Université catholique de Louvain; Vrije Universiteit Brussel（斯泰伦博斯大学; 国家理论与计算科学研究所; 天主教鲁汶大学数学与物理研究学院; 布鲁塞尔自由大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文在2-范畴中证明蛇引理，通过引入2-核、2-余核及di-正合性等概念，建立2-正合六项序列，并展示三个模型实例，推广了经典蛇引理。
AI 中文摘要
我们在2-范畴中证明了蛇引理。在一个具有强双零对象的2-范畴中，我们发展了2-核与2-余核、2-单态与2-满态（作为完全忠实和完全余忠实的1-胞腔）、短2-正合列以及正规像分解，并证明了二维正规短五引理。随后，我们引入了反正规对的dinversion以及同调自对偶2-范畴的概念，该概念等价地由同调的自对偶性、纯蛇引理和第三同构性质刻画。二维di-正合性蕴含同调自对偶性。我们的主要结果是在2-di-正合2-范畴中的蛇引理：具有正规纵向态射的2-正合行阶梯诱导一个2-正合六项序列，其连接1-胞腔关于阶梯是2-自然的。Di-正合性可以替换为两个非自对偶的假设：dinversion保持正规性，以及正规2-满态可复合。在过渡到局部离散2-范畴时，所有结果都特化为其经典对应物。最后，我们展示了三个模型：一个包含每个诺特概形$X$的$\mathsf{Coh}(X)$的阿贝尔范畴的2-di-正合2-范畴；完全模格的局部有序2-范畴，其中2-di-正合性等价于戴德金换位原理；以及希尔伯特格的2-范畴，它并非2-di-正合但满足非自对偶假设，依据Mackey关于闭子空间对的定理。
英文摘要
We prove a Snake Lemma for 2-categories. Working in a 2-category with a strong bizero object, we develop 2-kernels and 2-cokernels, 2-monomorphisms and 2-epimorphisms as fully faithful and cofully faithful 1-cells, short 2-exact sequences and normal image factorisations, and we prove a two-dimensional Normal Short Five Lemma. We then introduce the dinversion of an antinormal pair and the notion of a homologically self-dual 2-category, characterised equally by the self-duality of homology, by a Pure Snake Lemma and by a Third Isomorphism Property. Two-dimensional di-exactness implies homological self-duality. Our main result is the Snake Lemma in a 2-di-exact 2-category: a ladder of 2-exact rows with normal verticals induces a 2-exact six-term sequence, with a connecting 1-cell that is 2-natural in the ladder. Di-exactness can be traded for two hypotheses that are not self-dual: that dinversion preserve normality, and that normal 2-epimorphisms compose. Everything specialises, on passing to a locally discrete 2-category, to its classical counterpart. We close by exhibiting three models: a 2-di-exact 2-category of abelian categories containing $\mathsf{Coh}(X)$ for every noetherian scheme $X$; the locally ordered 2-category of complete modular lattices, in which 2-di-exactness amounts to Dedekind's transposition principle; and the 2-category of Hilbert lattices, which is not 2-di-exact but satisfies the non-self-dual hypotheses, by a theorem of Mackey on pairs of closed subspaces.
Comments62 pages. Lean 4 formalisation: https://github.com/tvdlinde/snake-lean Blueprint: https://tvdlinde.github.io/snake-lean/
