Algebras for enriched ∞-operads

Haugseng, Rune

doi:10.2140/agt.2025.25.3789

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Algebras for enriched $\infty$-operads

Rune Haugseng

Algebraic & Geometric Topology 25 (2025) 3789–3811

DOI: 10.2140/agt.2025.25.3789

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 $V$ a symmetric monoidal model category and $O$ a $Σ$ -cofibrant operad in $V$ for which the model structure on $V$ can be lifted to one on $O$ -algebras, we then prove that strict algebras in $V$ are equivalent to $\infty$ -categorical algebras in the symmetric monoidal $\infty$ -category associated to $V$ . We also show that for an $\infty$ -operad $𝒪$ enriched in a suitable closed symmetric monoidal $\infty$ -category $𝒱$ , we can equivalently describe $𝒪$ -algebras in $𝒱$ as morphisms of $\infty$ -operads from $𝒪$ to a self-enrichment of $𝒱$ .

Keywords

operads, $\infty$-operads, operad algebras

Mathematical Subject Classification 2010

Primary: 18D50, 55U35

References

Publication

Received: 14 November 2019

Revised: 10 September 2024

Accepted: 23 September 2024

Published: 29 October 2025

Authors

Rune Haugseng
Department of Mathematical Sciences Norwegian University of Science and Technology Trondheim Norway

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Volume 25, issue 7 (2025)