  We describe the E-infinity algebra structure on the complex of singular
cochains of a topological space, in the context of sheaf theory. As a first
application, for any algebraic variety we define a weight filtration compatible
with its E-infinity structure. This naturally extends the theory of mixed Hodge
structures in rational homotopy to p-adic homotopy theory. The spectral
sequence associated to the weight filtration gives a new family of
multiplicative algebraic invariants of the varieties for any coefficient ring,
carrying Steenrod operations. As a second application, we promote Deligne's
intersection complex computing intersection cohomology, to a sheaf carrying
E-infinity structures. This allows for a natural interpretation of the Steenrod
operations defined on the intersection cohomology of any topological
pseudomanifold.
