Compact Corigid Objects in Triangulated Categories and Co-t-structures

TL;DR

This paper introduces the concept of co-t-structures induced by compact corigid objects in triangulated categories, extending the framework of t-structures and their hearts to a dual setting.

Contribution

It defines co-t-structures associated with compact corigid objects and proves their properties, including the equivalence of the coheart to module categories, thus extending existing theory.

Findings

Co-t-structures are induced by compact corigid objects.

The coheart of a co-t-structure is equivalent to a module category.

This work extends the theory of t-structures to a dual setting.

Abstract

In the work of Hoshino, Kato and Miyachi, the authors look at t-structures induced by a compact object, C, of a triangulated category, T, which is rigid in the sense of Iyama and Yoshino. Hoshino, Kato and Miyachi show that such an object yields a non-degenerate t-structure on T whose heart es equivalent to Mod(End(C)^op). Rigid objects in a triangulated category can be thought of as behaving like chain differential graded algebras (DGAs). Analogously, looking at objects which behave like cochain DGAs naturally gives the dual notion of a corigid object. Here, we see that a compact corigid object, S, of a triangulated category, T, induces a structure similar to a t-structure which we shall call a co-t-structure. We also show that the coheart of this non-degenerate co-t-structure is equivalent to Mod(End(S)^op), and hence an abelian subcategory of T.