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We describe the E-infinity algebra structure on the complex of singularcochains of a topological space, in the context of sheaf theory. As a firstapplication, for any algebraic variety we define a weight filtration compatiblewith its E-infinity structure. This naturally extends the theory of mixed Hodgestructures in rational homotopy to p-adic homotopy theory. The spectralsequence associated to the weight filtration gives a new family ofmultiplicative algebraic invariants of the varieties for any coefficient ring,carrying Steenrod operations. As a second application, we promote Deligne'sintersection complex computing intersection cohomology, to a sheaf carryingE-infinity structures. This allows for a natural interpretation of the Steenrodoperations defined on the intersection cohomology of any topologicalpseudomanifold.

Sheaves of E-infinity algebras and applications to algebraic varieties and singular spaces