Characterizing model structures on finite posets
Kristen Mazur, Angélica M. Osorno, Constanze Roitzheim, Rekha Santhanam, Danika Van Niel, Valentina Zapata Castro
TL;DR
This work classifies model category structures on finite lattices by translating the data into wide decomposable subcategories of weak equivalences $\mathsf{W}$ and transfer systems $T\subseteq \mathsf{W}$. It proves that model structures correspond to pairs $(\mathsf{W},T)$ with $\mathsf{W}$ decomposable and $^{\boxslash}T\cap \mathsf{W}$ a cotransfer system, and dual statements with cotransfer/cofibration data; crucially, it provides explicit constructions and lattice-theoretic descriptions of all such structures, including finite lattices like $[n]$, $[2]^{\!*n}$, and $N_5$. The paper also develops general criteria (e.g., allaboutw and AFW-is-lattice) to determine when a given subcategory can be the weak equivalences and how many compatible acyclic fibrations exist, yielding concrete enumerations for key examples. Overall, the results establish a practical, combinatorial approach linking transfer systems, weak factorization systems, and model structures to equivariant homotopy theory and its combinatorial manifestations on finite posets. The work broadens the toolkit for understanding homotopy-theoretic structures in discrete settings and highlights new connections between abstract homotopy theory and equivariant topology through transfer/cotransfer formalisms.
Abstract
Transfer systems on finite posets have recently been gaining traction as a key ingredient in equivariant homotopy theory. Additionally, they also naturally occur in the data of a model structure. We give a complete characterization of all model category structures on a finite lattice, using transfer systems as our main tool, resulting in new connections between abstract homotopy theory and equivariant methods.