  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 two
sufficient conditions. We prove that both conditions are satisfied for finite
fans in $\mathbb{R}^2$ unless one particular identification occurs. As a
consequence, the classifying space of the $\tau$-cluster morphism category of
any $\tau$-tilting finite algebra of rank 2 is a $K(\pi,1)$ space for its
picture group. As an application of the lattice structure, we show an analogous
result holds for the Brauer cycle algebra of rank 3. In the final section we
also offer a new algebraic proof of the relationship between an algebra and its
$g$-vector fan.

Representation Theory