How to build a Hopf algebra

Abstract: We construct a functor that inputs a retract in an $(\infty,3)$-category satisfying some adjunctibility conditions and outputs a Hopf algebra in a braided monoidal $(\infty,1)$-category. Provided the braided monoidal category is presentable, any Hopf algebra can be obtained in this way. Our functor specializes to - and provides a higher-categorical explanation for - the Tannakian reconstruction of a Hopf algebra from a monoidal category with duals and a fiber functor. Towards this end, we review and develop the lax (aka Gray) tensor product $\otimes$ of $(\infty,\infty)$-categories, and we analyze the "lax smash product" of pointed $(\infty,\infty)$-categories. We compute the lax-$\wedge$-square of the "walking adjunction" and show that its $3$-localization corepresents retracts with some adjunctibilty conditions, whereas the $3$-localization of the "walking monad" corepresents bialgebras. In these terms, our functor is restriction along the lax-$\wedge$-square of the inclusion ${\text{walking monad}} \to {\text{walking adjunction}}$. After $3$-localization, we show that this restriction inverts a certain shear map, proving the existence of an antipode. We discuss generalizations of this construction to Hopf monads, analyze additional adjunctibility conditions and their interplay with integrals and cointegrals, and finally explain how variants of classical Tannakian reconstruction fit into our scheme.