Length 3 Complexes of Abelian Sheaves and Picard 2-Stacks

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.

💡 Research Summary

The paper establishes a precise correspondence between two high‑level categorical structures: length‑3 complexes of abelian sheaves and Picard 2‑stacks. The authors begin by fixing a site (S) and considering complexes (C^\bullet : A^0 \to A^1 \to A^2) of abelian sheaves concentrated in degrees 0, 1, 2. Morphisms between such complexes are taken to be “weak morphisms”: a chain map together with a chain homotopy, and higher homotopies, organized into a hom‑groupoid. By endowing these hom‑groupoids with vertical and horizontal composition, they assemble a tricategory (\mathcal{T}) whose objects are the length‑3 complexes, 1‑cells are weak morphisms, 2‑cells are homotopies, and 3‑cells are higher homotopies. The coherence axioms for a tricategory (associators, unitors, and the pentagon‑type constraints) are verified explicitly.

In parallel, the authors define a 3‑category (\mathbf{2PIC}(S)) whose objects are Picard 2‑stacks on the same site. A Picard 2‑stack is a stack of groupoids equipped with a symmetric monoidal structure that is strictly associative and unital up to coherent isomorphism, and whose underlying groupoid is a Picard groupoid (i.e., a strictly commutative group object in the 2‑category of stacks). Additive 2‑functors between Picard 2‑stacks serve as 1‑cells, natural transformations as 2‑cells, and modifications as 3‑cells. Again, the authors spell out the composition laws and verify the required coherence conditions (triangular and pentagonal identities) for a genuine 3‑category.

The central result is a triequivalence \