Homotopy coherent Hopf Algebras

From the (co)homology of a topological group with coefficients in a field to the (dual) Steenrod algebra, certain algebraic invariants that show up in topology are equipped with a compatible comultiplication that upgrades them to Hopf algebras. One of the main insights of homotopy theory is that algebraic invariants are truncations of objects equipped with homotopy coherent algebraic structures (such as ring spectra), so one might hope that the additional "coalgebraic" structures (and their interactions with algebraic ones) can also be lifted to homotopy coherent ones.

In this talk, we will give an overview of the theory of such homotopy coherent Hopf algebras and (co)modules over them, which in particular provides a unifying point of view on concepts such as (naive) equivariant homotopy theory and Thom spectra.