A Picard category \(P\) is a groupoid with the extra compatible structure of a symmetric monoidal category. If \(\mathbf{S}\) is a site, a Picard stack \(\mathcal{P}\) is a stack of groupoids with a functor, \(\otimes :\mathcal{P}\times \mathcal{P}\to \mathcal{P}\), such that, for each object \(U\) of \(\mathbf{S}\), \(\mathcal{P}(U)\) is a Picard category. The now classical theory of P. Deligne [Lect. Notes Math. 305, 481–587 (1973; Zbl 0259.14006)] shows that any 2-term complex of abelian sheaves on \(\mathcal{S}\) gives a pre-stack whose stackification is a Picard stack, and that that construction gives a bi-equivalence of bicategories, \[ 1st:T^{[-1,0]}(\mathbf{S})\to {\mathrm{Pic}}(\mathbf{S}), \] where \(T^{[-1,0]}(\mathbf{S})\) is the bicategory of such complexes whose 1- and 2-morphisms are, essentially, given by ‘fractions’, or ‘butterflies’ in the sense of E. Aldrovandi and B. Noohi [Adv. Math. 221, No. 3, 687–773 (2009; Zbl 1179.18007)].
The second of the present authors extended this result, see [A. E. Tatar, Adv. Math. 226, No. 1, 62–110 (2011; Zbl 1227.14023)], to a tri-equivalence, \(2st\), between length-3 complexes of sheaves on \(\mathbf{S}\), defining a tri-category, \(T^{[-2,0]}(\mathbf{S})\), and a natural tri-category of Picard 2-stacks. A Picard 2-stack, \(\mathbb{P}\), on \(\mathbf{S}\) is a 2-stack of 2-groupoids equipped with a morphism \(\otimes :\mathbb{P}\times \mathbb{P}\to \mathbb{P}\) of 2-stacks such that each \(\mathbb{P}(U)\) is a Picard 2-category, in a fairly obvious sense. These form a tri-category \(2{\mathrm{Picard}}(\mathbf{S})\). (The corresponding category, after dividing out by ‘modifications’ will be denoted \(2{\mathrm{Picard}}^{\flat\flat}(\mathbf{S})\) and the above tri-equivalence induces an (ordinary) equivalence of categories \[ 2st^{\flat\flat}:\mathcal{D}^{[-2,0]}(\mathbf{S})\to 2{\mathrm{Picard}}^{\flat\flat}(\mathbf{S}) \] between the derived category of length-3 complexes of sheaves and that category of Picard 2-stacks. (The functor quasi-inverse to \(2st^{\flat\flat}\) will be denoted \([\quad]^{\flat\flat}\), below.)
In this paper, the authors define and study a natural notion of extension of Picard 2-stacks and morphisms between them of various types, and also some groups, \(\mathcal{E}xt^1(\mathbb{A},\mathbb{B})\), naturally associated to the resulting 3-category, \(\mathcal{E}xt(\mathbb{A},\mathbb{B})\), of such extensions. One of the main theorems is that if \(\mathbb{A}\) and \(\mathbb{B}\) are two Picard 2-stacks, then, for \(i = 1,0,-1,-2\), there is an isomorphism of groups \[ \mathcal{E}xt^i(\mathbb{A},\mathbb{B})\cong \mathrm{Ext}^i([\mathbb{A}]^{\flat\flat},[\mathbb{B}]^{\flat\flat}) \cong \mathcal{D}(\mathbf{S})([\mathbb{A}]^{\flat\flat},[\mathbb{B}]^{\flat\flat}[i]). \] A crucial role is played in the proof by the calculus of fractions that is used to define extensions in the tri-category, \(T^{[-2,0]}(\mathbf{S})\).