Sheafification functors and Tannaka's reconstruction

We introduce "sheafification" functors from categories of (lax monoidal) linear functors to cat- egories of quasi-coherent sheaves (of algebras) of stacks. They generalize the homogeneous sheafification of graded modules for projective schemes and have applications in the theory of non-abelian Galois covers and of Cox rings and homogeneous sheafification functors. Moreover, using this theory, we prove a non-neutral form of Tannaka's reconstruction, extending the classical correspondence between torsors and strong monoidal functors.