General rectification for algebras over G-operads

Establish a rectification theorem in full generality for algebras over G-operads that compares the homotopy theory of algebras over a G-operad with the algebras over the associated G-∞-operad for finite groups G.

Background

Rectification results are central in comparing algebraic structures defined by operads with their ∞-categorical counterparts. In non-equivariant contexts, such rectification is known under Σ-cofibrancy assumptions, but many equivariant operads of interest (e.g., N∞-operads) are not Σ-cofibrant.

This paper proves preservation of sifted homotopy colimits for algebras over operads with free symmetric group actions, a step used elsewhere to obtain rectification for N∞-operads. Nonetheless, the existence of a rectification theorem for algebras over arbitrary G-operads remains unresolved.