Equivalence of Operads over Symmetric Monoidal Categories

TL;DR

This paper establishes conditions under which Quillen equivalences between symmetric monoidal categories extend to their operad categories, broadening the understanding of operad equivalences in homotopical algebra.

Contribution

It proves that under certain conditions, Quillen equivalences between symmetric monoidal categories induce Quillen equivalences between their operad categories, generalizing previous results.

Findings

Quillen equivalences extend to operad categories under weak monoidal adjunctions.

The result generalizes Schwede-Shipley's work from monoids to operads.

Operad categories inherit model structures compatible with base categories.

Abstract

In this paper, we study conditions for extending Quillen model category properties , between two symmetric monoidal categories, to their associated category of symmetric sequences and of operads. Given a Quillen equivalence $\lambda: \mathcal{C}=Ch_{\Bbbk,t} \leftrightarrows \mathcal{D}: R,$ so that $\mathcal{D}$ is any symmetric monoidal category and the adjoint pair $(\lambda, R)$ is weak monoidal, we prove that the categories of connected operads $Op_\mathcal{C}$ and $Op_\mathcal{D}$ are Quillen equivalent. This expands an analogous result of Schwede-Shipley(\cite{SS03}) when we replace these categories of operads with the sub-categories of $\mathcal{C}$-Monoid and $\mathcal{D}$-monoid.