Given a Hopf-algebra $H$ (over a commutative ring), it is a classical fact that its category of (left) modules is monoidal, even if $H$ is not commutative. Given two left modules $M$ and $N$, we can form a new left module structure on $M\otimes N$ via the structure map $$H\otimes M\otimes N\overset{\Delta\otimes 1\otimes 1}\to H\otimes H\otimes M\otimes N\overset{1\otimes\tau\otimes 1}\to H\otimes M\otimes H \otimes N\to M\otimes N.$$

When working in a derived setting (let's assume $H$ is an object in a symmetric monoidal quasicategory $\mathscr{C}$), things can be slightly more complicated, and we should probably have $H$ with monoidal structure and comonoidal structure given by some operads like the little $n$-disk operads $\mathbb{E}_n.$ It's basically formal when working in quasicategories to say that $H$ is an $\mathbb{E}_n$-algebra with a compatible $\mathbb{E}_m$-coalgebra structure, making it into an $\mathbb{E}_n/\mathbb{E}_m$-bialgebra in $\mathscr{C}$. We just say that $H$ is an $\mathbb{E}_n$-algebra object in the quasicategory of $\mathbb{E}_m$-coalgebra objects in $\mathscr{C}$.

It's known that, in general, given an $\mathbb{E}_n$-algebra, the category of left modules over it is $\mathbb{E}_{n-1}$-monoidal. This is why, for instance, left modules over a noncommutative ring (i.e. an $\mathbb{E}_1$-algebra) are not monoidal at all. So my question is, to what extent can we perform the above trick in a "derived" way? Obviously it does not suffice to simply write down the structure map, since we need a whole lot of coherent data to write down a module structure now, but is there some other way to do it?

A good example would be, I think, the example of an $n$-fold loop space $X$. Any space, via the diagonal map, is an $\mathbb{E}_\infty$-coalgebra. In fact there's an equivalence of quasicategories $CoAlg_{\mathbb{E}_\infty}(Top)\simeq Top$. So an $n$-fold loop space is definitely an $\mathbb{E}_n$-algebra in $\mathbb{E}_\infty$-coalgebras in $Top$. So, is the category of modules in $Top$ over $X$ somehow "more monoidal than it should be?" In general, how well does this type of thing work?

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