Extensions of Picard 2-Stacks and the cohomology groups Ext^i of length 3 complexes

TL;DR

This paper develops a 3-category framework for extensions of Picard 2-stacks and provides a geometric interpretation of Ext^i cohomology groups for length 3 complexes of abelian sheaves, linking algebraic and geometric perspectives.

Contribution

It introduces a 3-category of extensions of Picard 2-stacks and describes Ext^i groups geometrically via these extensions, using a triequivalence with complexes of sheaves.

Findings

Parametrization of extension classes in the 3-category of Picard 2-stacks.

Geometric description of Ext^i groups for length 3 complexes.

Use of calculus of fractions in the algebraic framework.

Abstract

The aim of this paper is to define and study the 3-category of extensions of Picard 2-stacks over a site S and to furnish a geometrical description of the cohomology groups Ext^i of length 3 complexes of abelian sheaves. More precisely, our main Theorem furnishes (1) a parametrization of the equivalence classes of objects, 1-arrows, 2-arrows, and 3-arrows of the 3-category of extensions of Picard 2-stacks by the cohomology groups Ext^i, and (2) a geometrical description of the cohomology groups Ext^i of length 3 complexes of abelian sheaves via extensions of Picard 2-stacks. To this end, we use the triequivalence between the 3-category of Picard 2-stacks and the tricategory T^[-2,0](S) of length 3 complexes of abelian sheaves over S introduced by the second author in arXiv:0906.2393, and we define the notion of extension in this tricategory T^[-2,0](S), getting a pure algebraic analogue of the 3-category of extensions of Picard 2-stacks. The calculus of fractions that we use to define extensions in the tricategory T^[-2,0](S) plays a central role in the proof of our Main Theorem.