A monadicity theorem for higher algebraic structures (1712.00555v5)

Abstract: We develop a theory of Eilenberg-Moore objects in any $(\infty,2)$-category and extend Barr-Beck's monadicity theorem to any $(\infty,2)$-category that admits Eilenberg-Moore objects. We apply this result to various $(\infty,2)$-categories of higher algebraic structures to obtain a monadicity theorem for enriched $\infty$-categories, $\infty$-operads and higher Segal objects. Analyzing Eilenberg-Moore objects in the $(\infty,2)$-categories of symmetric monoidal $\infty$-categories and lax (oplax) symmetric monoidal functors we extend classical results about lax (oplax) symmetric monoidal monads to the $\infty$-categorical world: we construct two tensor products for algebras over lax (oplax) symmetric monoidal monads, respectively: the first one extends the relative tensor product for modules over an $\mathbb{E}{\mathrm{k}}$-algebra for $2 \leq \mathrm{k} \leq \infty$, the second one generalizes the object-wise tensor product for $\mathbb{E}{\mathrm{k}}$-algebras in any $\mathbb{E}_{\mathrm{k}}$-monoidal $\infty$-category. As an application we construct a tensor product for algebras over any Hopf $\infty$-operad. As another application we prove that for every cartesian presentably symmetric monoidal $\infty$-category $\mathcal{V}$, presentably symmetric monoidal $\mathcal{V}$-enriched $\infty$-category $\mathcal{C}$ and $\mathcal{C}$-enriched $\infty$-operad $\mathcal{O}$ the $\infty$-category of $\mathcal{O}$ -algebras in $\mathcal{C}$ is $\mathcal{V}$-enriched.