Higher Dimensional Homological Algebra

经典同调代数起源于大约60年前的代数拓扑学。从那时起，它已经成长为一种理论，应用于数学的许多领域，包括组合学、几何学和表示论。经典的同调代数通常被称为交换范畴和三角范畴理论。高维同调代数是近十年来的一个新发展。它是n-交换范畴和(n+2)-角化范畴的理论，其中n是正整数。在这些类别中，以前由1-扩展所扮演的角色被n-扩展所取代。请注意，在n=1的情况下，给出了普通的阿贝尔范畴和三角范畴，因此是经典的同调代数。我们称其为(n+2)-角化范畴，因为n=1的情况给出了三角化范畴。高维同调代数目前非常活跃。它在代数几何、组合学和有限维代数的表示理论中都有应用。许多杰出的数学家(Iyama、Keller、Reiten)做出了重大贡献。一些新出现的组合结构，如高维循环多面体，对于同调代数和表示理论来说是新的。本项目将提供目前高维同调代数中缺失的三个关键项：倾斜对象、高维派生范畴和高维模型范畴。我们将在高维同调代数中定义倾斜对象，并展示支撑倾斜对象、挠类、中间t-结构和非交叉划分之间的Ingalls-Thomas双射的一个版本。这将加强和阐明与组合数学的联系。我们将定义高维派生范畴和高维模型范畴。这将为高维倾斜理论提供合适的背景，并为高维同调代数提供一个全面的框架。

Classic homological algebra had its origin some 60 years ago in algebraic topology. Since then, it has grown to a theory with applications to many areas of mathematics, including combinatorics, geometry, and representation theory. Classic homological algebra is often phrased as the theory of abelian and triangulated categories.Higher dimensional homological algebra is a new development of the last decade. It is the theory of n-abelian categories and (n+2)-angulated categories, where n is a positive integer. In these categories, the role previously played by 1-extensions is taken over by n-extensions. Note that the case n=1 gives ordinary abelian and triangulated categories, hence classic homological algebra. We refer to (n+2)-angulated categories because the case n=1 gives triangulated categories. Higher dimensional homological algebra is currently very active. It has applications to algebraic geometry, combinatorics, and the representation theory of finite dimensional algebras. There are substantial contributions from a number of strong mathematicians (Iyama, Keller, Reiten). Some of the combinatorial structures which appear, like higher dimensional cyclic polytopes, are novel to homological algebra and representation theory.This project will provide three key items currently missing in higher dimensional homological algebra: Tilting objects, higher dimensional derived categories, and higher dimensional model categories. We will define tilting objects in higher dimensional homological algebra and show a version of the Ingalls-Thomas bijections between support tilting objects, torsion classes, intermediate t-structures, and non-crossing partitions. This will enhance and illuminate the links to combinatorics. We will define higher dimensional derived categories and higher dimensional model categories. This will provide the right context for higher dimensional tilting theory, and give a comprehensive framework for higher dimensional homological algebra.