Algebras for enriched $\infty$-operads

Abstract

Using the description of enriched $\infty$-operads as associative algebras in symmetric sequences, we define algebras for enriched $\infty$-operads as certain modules in symmetric sequences. For $\mathbf{V}$ a symmetric monoidal model category and $\mathbf{O}$ a $\Sigma$-cofibrant operad in $\mathbf{V}$ for which the model structure on $\mathbf{V}$ can be lifted to one on $\mathbf{O}$-algebras, we then prove that strict algebras in $\mathbf{V}$ are equivalent to $\infty$-categorical algebras in the symmetric monoidal $\infty$-category associated to $\mathbf{V}$. We also show that for an $\infty$-operad $\mathcal{O}$ enriched in a suitable closed symmetric monoidal $\infty$-category $\mathcal{V}$, we can equivalently describe $\mathcal{O}$-algebras in $\mathcal{V}$ as morphisms of $\infty$-operads from $\mathcal{O}$ to a self-enrichment of $\mathcal{V}$.