Rectification of dendroidal left fibrations
Francesca Pratali
TL;DR
This work constructs and analyzes an adjunction between dendroidal sets over the dendroidal nerve of a discrete operad $P$ and simplicial $P$-algebras, proving a Quillen equivalence when $P$ is $ ext{$oldsymbol{ Sigma}$}$-free and establishing a monoidal rectification in the symmetric monoidal discrete category case. The left adjoint $ ho_!^P$ is built via left Kan extension from a base diagram, and its right adjoint $ ho_P^*$ is the relative dendroidal nerve; together they yield a Quillen adjunction between the covariant model structure on $ extbf{dSets}/ ext{$bmathcal{N}$}_d P$ and the projective model structure on $ extsf{Alg}_P( extbf{sSets})$. By relating the derived functor of $ ho_!^P$ to the operadic straightening functor $ extsf{St}^P$ in the $ extbf{$ ext{∞}$-categorical}$ setting, the paper provides a unified pathway from dendroidal left fibrations to $ extsf{P}$-algebras via straightening/unstraightening. In the special case $P=A$ is a discrete (symmetric) monoidal category, the results recover and independently prove the Heuts–Moerdijk rectification and monoidal straightening, with a detailed fiberwise interpretation and explicit formulas for the derived functors. The framework generalizes to non-$oldsymbol{ Sigma}$-free discrete operads, where the Quillen equivalence may fail, highlighting the delicate role of $oldsymbol{ Sigma}$-freeness in presenting the operadic straightening.”,
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 $Σ$-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.