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Arboreal Objects and Their Homotopy Theory

Atabey Kaygun

Abstract

We construct a category $\OrdFor$ as an arboreal extension of $Δ_{\mathrm{epi}}\subseteqΔ$, 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.

Arboreal Objects and Their Homotopy Theory

Abstract

We construct a category as an arboreal extension of , whose morphisms are ordered forests composed by grafting. We define a full functor extracting the semisimplicial shadow. For every complete category , this induces a fully faithful functor from semisimplicial objects in to -valued presheaves on , 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.
Paper Structure (9 sections, 14 theorems, 33 equations)

This paper contains 9 sections, 14 theorems, 33 equations.

Table of Contents

  1. Ordered Quivers, Forests, And Trees
  2. Ordered forests and trees
  3. The decomposition theorem
  4. The Cospan Category $\mathbf{OrdFor}$
  5. Height-one Forests
  6. Homotopy Theory of Arboreal Objects
  7. Equivalence of homotopy localizations
  8. Automatic right-induced model structures
  9. Quillen equivalence

Key Result

Theorem 1.3

Let $(P,\leq_P,\preceq_P)$ be an ordered forest, and write $\max(P,\leq_P)=\{m_0,\dots,m_r\}$ with $m_0\preceq_P\cdots\preceq_P m_r$. Set $T_i:=L_P(m_i)$. Then: In particular, every ordered forest decomposes canonically into an ordered disjoint union of ordered trees.

Theorems & Definitions (44)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • proof
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • ...and 34 more