Representability and autoequivalence groups

TL;DR

This paper establishes a duality between the bounded homotopy and derived categories of a finite dimensional algebra, and explores its implications for autoequivalence groups using 2-categorical duality.

Contribution

It introduces a novel 2-categorical duality linking homotopy and derived categories, and applies this to analyze triangle autoequivalence groups.

Findings

Duality between homotopy and derived categories established.

2-categorical duality applied to autoequivalence groups.

Results analogous to those in derived categories of sheaves.

Abstract

For a finite dimensional algebra $A$, we prove that the bounded homotopy category of projective $A$-modules and the bounded derived category of $A$-modules are dual to each other via certain categories of locally-finite cohomological functors. The duality gives rise to a $2$-categorical duality between certain strict $2$-categories involving the bounded homotopy categories and bounded derived categories, respectively. We apply the $2$-categorical duality to the study of triangle autoequivalence groups. These results are analogous to the ones in [M.R. Ballard, {\em Derived categories of sheaves on singular schemes with an application to reconstruction}, Adv. Math. {\bf 227} (2011), 895--919].