Left and right Bousfield localization on lattices

TL;DR

The paper examines how left and right Bousfield localization modify the transfer-system data that encode model-category structures on finite lattices. It delivers an explicit, constructive framework: right localization at a short indecomposable morphism $f$ yields $R_f(W,AF)=(R_f(W),\langle AF\cup \Gamma_f\rangle)$ with Golden Arrows $\,\Gamma_f$ capturing needed new acyclic fibrations, while left localization has a dual via Copper Arrows $\kappa_f$ governing new acyclic cofibrations. It also analyzes saturation, showing when model structures with saturated acyclic fibrations can be obtained from trivial ones through localizations, and it provides exact combinatorial constructions (via smaller-to-larger grid bijections and HMOO) describing how saturated transfer systems extend across grids. Together, these results give concrete, finite-combinatorial control over localization phenomena for model structures on posets, linking equivariant homotopy-theory insights to practical descriptions of model-categorical data on lattices.

Abstract

The key information of a model category structure on a poset is encoded in a transfer system, which is a combinatorial gadget, originally introduced to investigate homotopy coherence structures in equivariant homotopy theory. We describe how a transfer system associated with in a model structure on a lattice is affected by left and right Bousfield localization and provide a minimal generating system of morphisms which are responsible for the change in model structure. This leads to new concrete insights into the behavior of model categories on posets in general.

Figures (33)

• Figure 1: Transfer Systems on $[1] \times [1]$
• Figure 2: Right localization at the arrow $i \rightarrow j$ in $[n]$. The shaded areas depict the weak equivalence classes.
• Figure 3: Example of a right localization at the morphisms $f$. The shaded regions denote the respective weak equivalence sets.
• Figure 4: The star-shaped nodes are the elements of $S(f)$ when right localizing along $f$, while the shaded regions depict the weak equivalences before localization.
• Figure 5: Localizations on the $[1] \times [1]$ grid. The blue arrows denote right localization, and the grey arrows a left localization.

Theorems & Definitions (49)

• Theorem
• Theorem
• Definition 1.1
• Definition 1.2
• Definition 1.3
• Lemma 1.4
• Definition 1.5
• Definition 1.6
• Example 1.8
• Proposition 1.9