stack
generalization (and categorification) of a sheaf; a fibered category that admits effective descent
sheaf
collection of objects associated to subsets of a space in a manner admitting gluing and restriction
algebraic stack
stack in groupoids over the étale site of schemes such that the diagonal map is representable and that there exists a smooth surjection from a scheme
differentiable stack
stack in groupoids on the site of differentiable manifolds with a representable submersion (atlas) from a manifold