Tannaka duality for enhanced triangulated categories II: $t$-structures and homotopy types

TL;DR

This paper explores how $t$-structures influence Tannaka duality in dg categories, linking dg coalgebras to homotopy types and motivic Galois groups, with applications in algebraic topology and motives.

Contribution

It extends Tannaka duality to enhanced triangulated categories with $t$-structures, introducing dg coalgebras associated to dg functors and applying this to homotopy types and motivic Galois groups.

Findings

Established a correspondence between dg $C$-comodules and dg categories.

Applied the theory to pro-algebraic homotopy types from cohomology theories.

Derived implications for motivic Galois groups.

Abstract

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.