Calabi-Yau Category

A Calabi-Yau category, in the triangulated sense, is a -linear triangulated category whose hom-sets are finite-dimensional vector spaces over a field and whose Serre functor is isomorphic to an integer power of its shift as a triangle functor. Here is an integer, and the triangle structure on is the one compatible with its Serre duality (Keller 2008, corrected author version, pp. 7-8).

More precisely, is weakly -Calabi-Yau when

as -linear functors. Equivalently, there are bifunctorial -linear isomorphisms

where the star denotes the dual vector space over . The stronger -Calabi-Yau condition requires an isomorphism of triangle functors. With coherent shift powers, the standard triangle structure on has shift comparison . The comparison between and must respect these shift isomorphisms, rather than merely be a natural isomorphism of the underlying functors (Keller 2008, corrected author version, p. 8).