Characterizing Artin stacks (Q641891)

The author studies properties of morphisms of stacks in the context of the homotopy theory of presheaves of groupoids on a small site \(\mathcal{C}\). In this work, stacks are understood as fibrant objects in the local model structure on the category of presheaves of groupoids on \(\mathcal{C}\). Stacks are the presheaves which satisfy the analogue of the sheaf condition when the homotopy theory of groupoids is taken into account. Using the homotopy invariance of the properties of being a representable morphism, representable in algebraic spaces, and of being a cover, the author obtains homotopy theoretic characterizations of algebraic stacks as those which are equivalent to simplicial objects in \(\mathcal{C}\) satisfying certain analogues of the Kan conditions. The author defines \(n\)-algebraic stacks and provides a characterization of these in terms of simplicial objects. As a consequence, one gets that, for presheaves of groupoids, \(n\)-algebraic is equivalent to \(3\)-algebraic for \(n\geq 3\).