Algebras for enriched $\infty$-operads
Rune Haugseng
TL;DR
The paper develops a cohesive framework for algebras over enriched $\infty$-operads by modeling enriched $\infty$-operads as associative algebras in symmetric sequences and defining algebras for these operads as (right) modules in a double $\infty$-category of symmetric collections. A rectification theorem connects strict operad algebras in a symmetric monoidal model category to $\infty$-categorical algebras in the localized setting, while endomorphism $\infty$-operads and a self-enrichment $\overline{\mathcal{V}}$ provide a universal, morphism-based description of $\mathcal{O}$-algebras in $\mathcal{V}$ via maps $\mathcal{O} \to \overline{\mathcal{V}}$. The work also clarifies the relationship between $\infty$-categorical algebras and model-categorical algebras under suitable flatness/admissibility/ cofibrancy hypotheses, and constructs endomorphism operads that capture universal algebraic structures in enriched settings. Together, these results unify classical and higher-categorical perspectives on algebras over enriched $\infty$-operads and enable robust comparisons across foundational frameworks.
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 $Σ$-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 $\mathcal{V}$.