  We give a moduli-theoretic proof of the classical theorem of Gabriel, stating
that a scheme can be reconstructed from the abelian category of quasi-coherent
sheaves over it. The methods employed are elementary and allow us to extend the
theorem to (quasi-compact and separated) algebraic spaces. Using more advanced
technology (and assuming flatness) we also give a proof of the folklore result
that the group of autoequivalences of the category of quasi-coherent sheaves
consists of automorphisms of the underlying space and twists by line bundles.
We apply our strategy to prove analogous statements for categories of sheaves
twisted by a Gm-gerbe. Our methods allow us to treat even gerbes not coming
from a Brauer class. As a pleasant consequence, we deduce a Morita theory for
sheaves of abelian categories.
