  We consider the effect of $t$-structures on the Tannaka duality theory for dg
categories developed in our previous paper. We associate non-negative dg
coalgebras $C$ to dg functors on the hearts of $t$-structures, and relate dg
$C$-comodules to the original dg category. We give several applications for
pro-algebraic homotopy types associated to various cohomology theories, and for
motivic Galois groups.
