The category of a partitioned fan

Abstract

In this paper, we introduce the notion of an admissible partition of a simplicial polyhedral fan and define the category of a partitioned fan as a generalisation of the $\tau$-cluster morphism category of a finite-dimensional algebra. This establishes a complete lattice of categories around the $\tau$-cluster morphism category, which is closely tied to the fan structure. We prove that the classifying spaces of these categories are cube complexes, which reduces the process of determining if they are $K(\pi,1)$ spaces to three sufficient conditions. We characterise when these conditions are satisfied for fans in $\mathbb{R}^2$ and prove that the first one, the existence of a certain faithful functor, is satisfied for hyperplane arrangements whose normal vectors lie in the positive orthant. As a consequence we obtain a new infinite class of algebras for which the $\tau$-cluster morphism category admits a faithful functor and for which the cube complexes are $K(\pi,1)$ spaces. In the final section we also offer a new algebraic proof of the relationship between an algebra and its $g$-vector fan.