Monoidal envelopes and Grothendieck construction for dendroidal Segal objects

TL;DR

This paper develops a framework for constructing monoidal envelopes and cocartesian fibrations of $ abla$-operads within dendroidal Segal spaces, extending Lurie's straightening/unstraightening theory to this setting.

Contribution

It introduces a novel construction of monoidal envelopes for $ abla$-operads using dendroidal spaces and generalizes Lurie's fibrations and straightening/unstraightening equivalence.

Findings

Constructed monoidal envelopes in dendroidal Segal spaces.

Defined cocartesian fibrations of $ abla$-operads.

Extended straightening/unstraightening to dendroidal spaces.

Abstract

We propose a construction of the monoidal envelope of $\infty$-operads in the model of Segal dendroidal spaces, and use it to define cocartesian fibrations of such. We achieve this by viewing the dendroidal category as a "plus construction" of the category of pointed finite sets, and work in the more general language of algebraic patterns for Segal conditions. Finally, we rephrase Lurie's definition of cartesian structures as exhibiting the categorical fibrations coming from envelopes, and deduce a straightening/unstraightening equivalence for dendroidal spaces.