About

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, first introduced by Iyama. It is the theory of d-abelian categories, as defined by Jasso [2], and (d+2)-angulated categories, as defined by Geiss, Keller, and Oppermann [1], where d>0 is an integer. In these categories, the role previously played by 1-extensions is taken over by d-extensions. Note that the case d=1 gives ordinary abelian and triangulated categories, hence classic homological algebra. We refer to (d+2)-angulated categories because the case d=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. Some of the combinatorial structures which appear, like higher dimensional cyclic polytopes in the work by Oppermann and Thomas [3], are novel to homological algebra and representation theory.

References:

    1. C. Geiss, B. Keller, and S. Oppermann, n-angulated categories, J. Reine Angew. Math. 675 (2013), 101-120
    2. G. Jasso, n-abelian and n-exact categories, Math. Z. 283 (2016), 703-759
    3. S. Oppermann and H. Thomas, Higher-dimensional cluster combinatorics and representation theory, J. Eur. Math. Soc. (JEMS) 14 (2012), 1679-1737