Rectification of Deligne's mixed Hodge structures

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.