Talk:superstack

Defn

From ADV. THEOR. MATH. PHYS. Volume 23, Number 2, 345–402, 2019 Moduli and Periods of Supersymmetric Curves Giulio Codogni and Filippo Viviani

Definition 3.3 (Superstack).

A stack in groupoids over S for the ´etale topology (or simply a superstack) is a category fibered in groupoids M over S that satisfies the following two conditions

1. Every descent datum (for the ´etale topology) is effective. 364 G. Codogni and F. Viviani
2. For any S ∈ S, and any pair of objects η and η 0 in M(S), the contravariant functor (called the isomorphism functor between η and η 0 ) IsomS(η, η0 ) : Hom(−, S) → Sets that associate to every f : T → S the set of isomorphisms in M(T) between f ∗ (η) and f ∗ (η 0 ) is a sheaf in the ´etale topology, i.e. for every surjective ´etale morphism π : X → Y of complex superspaces over S the diagram IsomS(η, η0 )(Y ) m /IsomS(η, η0 )(X) m2 / m1 / IsomS(η, η0 )(X ×Y X) is exact, where m := IsomS(η, η0 )(π) and mi := IsomS(η, η0 )(πi) with πi : X ×Y X → X is the projection onto the i-th factor.

Rich Farmbrough, 18:02, 23 October 2024 (UTC).