  We promote Beilinson's triangulated equivalence between the bounded derived
category of rational polarizable mixed Hodge structures and the derived
category of rational polarizable mixed Hodge complexes to an equivalence of
symmetric monoidal quasi-categories. We use this equivalence to construct a
presheaf of commutative differential graded algebras in the ind-completion of
the category of rational mixed Hodge structures which computes Deligne's mixed
Hodge structure on the rational Betti cohomology of $\mathbf{C}$-schemes of
finite type. This leads to a preshea--in the quasi-categorical sense--of
$\mathbf{E}_{\infty}$-algebras computing integral mixed Hodge structures.
