How to build a Hopf algebra
Theo Johnson-Freyd, David Reutter

TL;DR
This paper constructs a functor linking retracts in higher categories to Hopf algebras, providing a higher-categorical perspective on Tannakian reconstruction and exploring the structure of related algebraic objects.
Contribution
It introduces a functor from retracts in (,3)-categories to Hopf algebras in braided monoidal (,1)-categories, generalizing Tannakian reconstruction using higher category theory.
Findings
Constructs a functor from higher categorical retracts to Hopf algebras.
Shows the functor specializes to classical Tannakian reconstruction.
Analyzes the lax smash product and corepresenting objects for algebraic structures.
Abstract
We construct a functor that inputs a retract in an -category satisfying some adjunctibility conditions and outputs a Hopf algebra in a braided monoidal -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 of -categories, and we analyze the "lax smash product" of pointed -categories. We compute the lax--square of the "walking adjunction" and show that its -localization corepresents retracts with some adjunctibilty conditions, whereas the -localization of the lax--square of the "walking…
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