Operations on Étale Sheaves of Sets

Abstract: Rydh showed in 2011 that any unramified morphism ƒof algebraic spaces (algebraic stacks) has a canonical and universal factorization through an algebraic space (algebraic stack) called the étale envelope of ƒ, where the first morphism is a closed immersion and the second is étale. We show that when ƒ is étale then the étale envelope can be described by applying the left adjoint of the pullback of ƒ to the constant sheaf defined by a pointed set with two elements. When ƒ is a monomorphism locally of finite type we have a similar construction using the direct image with proper support.