Thomason-Type Model Structures on Simplicial Complexes and Graphs
Emilio Minichiello
TL;DR
The paper develops Thomason-type right-transferred model structures on simplicial complexes and graphs, showing that Matsushita's model structure on loop graphs Gr_ℓ factors through two intermediate right-transferred models on simplicial complexes and reflexive graphs, with all resulting adjunctions being Quillen equivalences. It establishes that these structures are cofibrantly generated and proper, characterizes key cofibrant objects (e.g., forests, double subdivisions, flag complexes), and proves that the intermediate adjunctions induce Quillen equivalences, linking to Thomason’s work on Cat and related frameworks. It also analyzes derived mapping spaces, demonstrating that the underlying ∞-category of the Matsushita theory on Gr_ℓ is not cartesian closed and relating derived Hom to clique-complex-based internal homs under fibrant conditions. Together, these results provide a unified, transferable framework for ×-homotopy theory on graphs via Thomason-type models and their graph-theoretic analogues, enabling robust homotopical analysis of graph constructions and Hom-complexes.
Abstract
In this paper we show that the Matsushita model structure on loop graphs, which is right-transferred from the Kan-Quillen model structure on simplicial sets, factors through two other right-transferred model structures on simplicial complexes and reflexive graphs. We show that each Quillen adjunction between these right-transferred model categories is a Quillen equivalence. These model structures are analogous to the Thomason model structure on small categories, and we prove that they are all cofibrantly generated and proper. Furthermore we show that all cofibrant simplicial complexes are flag complexes, and all forests are cofibrant.