A comparison of definitions of equivariant trees

TL;DR

This paper compares different categorical models of trees, including dendroidal and equivariant trees, by demonstrating their equivalence through Grothendieck constructions, enhancing understanding of their structural relationships.

Contribution

It establishes that various categories of trees can be modeled via Grothendieck constructions, unifying dendroidal, G-equivariant, and genuine equivariant tree categories.

Findings

Categories of trees are modeled by Grothendieck constructions.

Equivalence of dendroidal and equivariant tree categories.

Supports recent work on genuine equivariant operads.

Abstract

We show that various categories of trees can be modeled by Grothendieck constructions on categories of trees with a fixed set of leaves. We prove this result for the dendroidal category $\Omega$, the category $\Omega^G$ of trees with a $G$-action for a finite group $G$, and finally for the category of genuine equivariant trees $\Omega_G$ that has played an important role in recent work on genuine equivariant operads.