Higher Dimensional Homological Algebra

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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.