Arboreal Objects and Their Homotopy Theory

Abstract

We construct a category $\OrdFor$ as an arboreal extension of $\hat{I}{”}_{\mathrm{epi}}\subseteq \hat{I}”$, whose morphisms are ordered forests composed by grafting. We define a full functor $π\colon \OrdFor\toΔ_{\mathrm{epi}}^{op}$ 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.