K(pi,1) conjecture for picture-group classifying spaces

Establish whether the classifying space of the cspan class="math-inline"e\tauc/span-cluster morphism category of a finite-dimensional algebra is a K(\pi,1)-space for the associated picture group.

Background

The paper explains that the \tau-cluster morphism category encodes all \tau-tilting reductions of a finite-dimensional algebra and is closely related to picture groups. The classifying space of this category is expected to provide a topological model for the corresponding picture group, but the K(\pi,1) property is presented as conjectural rather than established.

The paper develops picture categories for 0-Auslander extriangulated categories and proves several structural and functorial results, including criteria involving faithful group functors. These results are motivated in part by the unresolved K(\pi,1) problem, but the paper does not prove the conjecture itself.