Arboreal Objects and Their Homotopy Theory

TL;DR

This paper introduces a new categorical framework called $ ext{OrdFor}$ for arboreal objects, establishing their homotopy theory and relating it to semisimplicial objects via model category equivalences.

Contribution

It constructs the category $ ext{OrdFor}$ as an arboreal extension of $ riangle_{ ext{epi}}$, and shows its homotopy theory aligns with that of semisimplicial objects through model structures.

Findings

The functor $ ext{OrdFor} o riangle_{ ext{epi}}^{op}$ extracts semisimplicial shadows.

Weak equivalences of arboreal objects are detected by the right adjoint, preserving localizations.

The model structure on arboreal objects is Quillen equivalent to the Reedy model structure on semisimplicial objects.

Abstract

We construct a category $\OrdFor$ as an arboreal extension of $\Delta_{\mathrm{epi}}\subseteq\Delta$, whose morphisms are ordered forests composed by grafting. We define a full functor $\pi\colon \OrdFor\to\Delta_{\mathrm{epi}}^{op}$ extracting the semisimplicial shadow. For every complete category $\mathcal C$, this induces a fully faithful functor from semisimplicial objects in $\mathcal C$ to $\mathcal C$-valued presheaves on $\OrdFor$, with right adjoint given by right Kan extension. We show that if weak equivalences of arboreal objects are detected by this right adjoint, then their Gabriel--Zisman localization is equivalent to that of semisimplicial objects. For bicomplete cofibrantly generated model categories, under the usual acyclicity hypothesis for right-induced transfer, the corresponding model structure on arboreal objects is Quillen equivalent to the Reedy model structure on semisimplicial objects.