On the equivalence of Brantner's and Chu--Haugseng's approaches to enriched $\infty$-operads

TL;DR

This paper proves the equivalence of two models of enriched ∞-operads by Brantner and Chu--Haugseng, unifying their approaches and implications for algebraic structures.

Contribution

It establishes the equivalence of two models of enriched ∞-operads and their associated monoidal ∞-categories, linking their constructions and results.

Findings

Models of enriched ∞-operads are equivalent.

Equivalence of monoidal ∞-categories of symmetric sequences.

Results like algebra notions and Koszul duality are also equivalent.

Abstract

We prove that two models of (monochromatic) enriched $\infty$-operads, due to Brantner and Chu--Haugseng, are equivalent. We show this as a consequence of the equivalence of two models of monoidal $\infty$-categories of symmetric sequences and the composition product, due to Brantner and Haugseng. As a consequence, constructions and results formulated in either framework, such as notions of algebra and Koszul duality, are also shown to be equivalent.