Rectification of dendroidal left fibrations

TL;DR

This paper constructs and proves a Quillen equivalence between dendroidal sets over a nerve of a discrete operad and simplicial operad algebras, extending known results and connecting to operadic straightening.

Contribution

It establishes a new Quillen equivalence for dendroidal sets and simplicial operad algebras, generalizing previous work and providing an independent proof for specific cases.

Findings

Constructed an adjunction between dendroidal sets over a nerve and simplicial P-algebras.

Proved the adjunction is a Quillen equivalence when P is Σ-free.

Connected the equivalence to operadic straightening and monoidal Quillen model categories.

Abstract

For a discrete colored operad $P$, we construct an adjunction between the category of dendroidal sets over the nerve of $P$ and the category of simplicial $P$-algebras, and prove that when $P$ is $\Sigma$-free it establishes a Quillen equivalence with respect to the covariant model structure on the former category and the projective model structure on the latter. When $P=A$ is a discrete category, this recovers a Quillen equivalence previously established by Heuts-Moerdijk, of which we provide an independent proof. To prove the constructed adjunction is a Quillen equivalence, we show that the left adjoint presents a previously established operadic straightening equivalence between $\infty$-categories. This involves proving that, for a discrete symmetric monoidal category $A$, the Heuts-Moerdijk equivalence is a monoidal equivalence of monoidal Quillen model categories.