Higher Dimensional Homological Algebra

经典同态代数起源于60多年前的代数拓扑学。从那时起，它已经发展成为一种应用于许多数学领域的理论，包括组合学、几何和表示理论。经典同调代数通常被描述为阿贝尔和三角范畴的理论。高维同调代数是近十年来的一个新发展。它是关于n-阿贝尔范畴和(n+2)-角范畴的理论，其中n是一个正整数。在这些类别中，先前由1扩展所扮演的角色被n扩展所取代。注意，n=1的情况给出了普通的阿贝尔和三角范畴，因此是经典的同调代数。我们提到（n+2）角化的类别，因为情况n=1给出了三角化的类别。高维同调代数目前非常活跃。它应用于代数几何、组合学和有限维代数的表示理论。有许多优秀的数学家（Iyama, Keller, Reiten）做出了实质性的贡献。出现的一些组合结构，如高维循环多面体，在同调代数和表示理论中是新颖的。该项目将提供目前在高维同调代数中缺失的三个关键项：倾斜对象、高维派生类别和高维模型类别。我们将在高维同调代数中定义倾斜对象，并展示支持倾斜对象、扭转类、中间t结构和非交叉分区之间的英格尔-托马斯双射的一个版本。这将加强和阐明与组合学的联系。我们将定义高维派生类别和高维模型类别。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.