Co-t-structures: The First Decade

TL;DR

This survey reviews a decade of research on co-t-structures, highlighting their differences from t-structures, their philosophical interpretations, and their relevance in various triangulated categories.

Contribution

It provides a comprehensive overview of co-t-structures, emphasizing their conceptual distinctions from t-structures and their applications in derived and homotopy categories.

Findings

Co-t-structures are analogous to 'hard' truncation in homotopy categories.

Bounded co-t-structures lead to silting subcategories.

Triangulated categories tend to favor either t-structures or co-t-structures.

Abstract

Co-t-structures were introduced about ten years ago as a type of mirror image of t-structures. Like t-structures, they permit to divide an object in a triangulated category T into a "left part" and a "right part", but there are crucial differences. For instance, a bounded t-structure gives rise to an abelian subcategory of T, while a bounded co-t-structure gives rise to a so-called silting subcategory. This brief survey will emphasise three philosophical points. First, bounded t-structures are akin to the canonical example of "soft" truncation of complexes in the derived category. Secondly, bounded co-t-structures are akin to the canonical example of "hard" truncation of complexes in the homotopy category. Thirdly, a triangulated category T may be skewed towards t-structures or co-t-structures, in the sense that one type of structure is more useful than the other for studying T. In particular, we think of derived categories as skewed towards t-structures, and of homotopy categories as skewed towards co-t-structures.