Arboreal Objects and Their Homotopy Theory

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.