Length 3 Complexes of Abelian Sheaves and Picard 2-Stacks

TL;DR

This paper establishes a deep equivalence between a tricategory of length 3 complexes of abelian sheaves and a 3-category of Picard 2-stacks, generalizing classical results in algebraic geometry.

Contribution

It introduces a new tricategory of complexes of abelian sheaves and proves its triequivalence with the 3-category of Picard 2-stacks, extending Deligne's classical theorem.

Findings

Categories are triequivalent as tricategories.

Generalizes Deligne's result on Picard stacks.

Provides a new framework for understanding complex sheaf structures.

Abstract

We define a tricategory T of length 3 complexes of abelian sheaves, whose hom-bigroupoids consist of weak morphisms of such complexes. We also define a 3-category 2PIC(S) of Picard 2-stacks, whose hom-2-groupoids consist of additive 2-functors. We prove that these categories are triequivalent as tricategories. As a consequence we obtain a generalization of Deligne's analogous result about Picard stacks in SGA4, Exp. XVIII.