Justin Young has a paper on the brace bar-cobar duality between hopf algebras and $E\_2$-algebras: <https://arxiv.org/pdf/1309.2820.pdf>


I was wondering if anybody knows of a nice relationship between the categories of left modules over a Hopf Algebra and left modules over the underlying e1 structure of an $E\_2$ algebra.


In my mind, one should be able to use the $E\_2$ structure to turn left modules into bi-modules in a systematic way and then use that the category of bi-modules is monoidal.


We know that the category of left modules over a Hopf algebra is monoidal. How are these monoidal categories related?


Also, a big related question I have is: can we pick an $E\_3$-algebra related to the Hopf/$E\_2$-algebra such that the category of left modules is braided and gives knot invariants similar to the Jones polynomial and other braided category invariants (from tangles)?


(Note: if any clarification is needed in my question please let me know and I will edit it.)



Modules over Hopf Algebras and $E_2$-algebras
Let $A$ be a brace algebra and $B$ the Koszul dual bialgebra. There is a natural adjunction
$$
\Omega\colon \mathrm{CoMod}\_B\rightleftarrows \mathrm{LMod}\_A
$$
where, for instance, the functor $\mathrm{LMod}\_A\rightarrow\mathrm{CoMod}\_B$ sends $M\mapsto M\otimes B$ equipped with some standard differential.


For $\mathrm{LMod}\_A$ the natural notion of weak equivalence is that of quasi-isomorphism and in $\mathrm{CoMod}\_B$ a map $C\_1\rightarrow C\_2$ is a weak equivalence if $\Omega C\_1\rightarrow \Omega C\_2$ is a quasi-isomorphism. The localization of $\mathrm{LMod}\_A$ with respect to quasi-isomorphisms is the derived category of $A$-modules and the localization of $\mathrm{CoMod}\_B$ with respect to $\Omega$-quasi-isomorphisms is the coderived category of $B$-comodules. The above adjunction induces an adjoint equivalence between the corresponding localizations (see <https://arxiv.org/abs/0905.2621> for details on all of these constructions).


$\mathrm{CoMod}\_B$ indeed has a natural monoidal structure. However, $\mathrm{LMod}\_A$ does not have any natural monoidal structure (there is no strict way to turn left modules over a brace algebra into right modules). There is, however, a monoidal structure on the derived category of $A$-modules. It is obtained using the Dunn--Lurie additivity equivalence $\mathcal{A}\mathrm{lg}\_{\mathbb{E}\_2}\cong \mathcal{A}\mathrm{lg}(\mathcal{A}\mathrm{lg})$.


Brace bar-cobar duality gives an adjunction $\mathrm{Alg}(\mathrm{CoAlg})\rightleftarrows \mathrm{Alg}\_{\mathrm{Brace}}$. After inverting corresponding weak equivalences you obtain another equivalence of $\infty$-categories $\mathcal{A}\mathrm{lg}(\mathcal{A}\mathrm{lg})\cong \mathcal{A}\mathrm{lg}\_{\mathbb{E}\_2}$. As far as I know, nobody has compared Dunn--Lurie additivity for $\mathbb{E}\_2$ with brace bar-cobar duality. Assuming such a comparison, the monoidal structure on the coderived category of $B$-comodules will indeed coincide with the monoidal structure on the derived category of $A$-modules.


