New approaches to a homotopical problem in representation theory

Let K be a field. The central object of this thesis is the τ-cluster morphism category W(A) of a finite-dimensional K-algebra A. This category encodes the information of all possible τ-tilting reductions in mod(A) and encompasses many objects considered in representation theory, for example (semi-)bricks, (τ-)tilting modules and (τ-)exceptional sequences. When A is a hereditary algebra, the category W(A) is well-understood and its classifying space BW(A) is a K(π,1) space in all representation finite and some tame cases. By definition, this means that the fundamental group is the only nontrivial homotopy group. This is insightful, because the fundamental group of this space, known as the picture group, is closely connected to maximal green sequences, a central object in the theory of cluster algebras. The guiding question of this thesis is based on the conjecture that the classifying space of the τ-cluster morphism category is a K(π,1) space for all τ-tilting finite algebras. It is known that the classifying space of W(A) is a cube complex. Thus, the conditions developed by Gromov for cube complexes to be nonpositively curved may be lifted to three conditions which together imply that BW(A) is a K(π,1) space. The focus of this thesis lies on one of these conditions: the existence of a faithful functor from W(A) to a group considered as a groupoid with one object. In fact, given this condition BW(A) is nonpositively curved if and only if it satisfies the other two conditions. Such a faithful functor to a group is conjectured to exist for all finite-dimensional algebras which are τ-tilting finite. However, few families of algebras satisfying this condition have been found so far. The first part of this thesis builds on a recently introduced geometric viewpoint of W(A) by Schroll-Tattar-Treffinger-Williams. This geometric approach is developed further to obtain a new family of algebras admitting faithful functors to groups. This is achieved by relaxing a condition in the geometric definition. In this way, a category is defined for any simplicial polyhedral fan with an admissible partition, so that the collection of these categories, for a given fan, forms a lattice. For the g-vector fan of an algebra A, the category W(A) is an element of this lattice. If the g-vector fan of A is a finite hyperplane arrangement, it is shown that W(A) admits a faithful functor to a group by using the theory of hyperplane arrangements in convex geometry. In the second part of this thesis, a lattice theoretic approach to W(A) is introduced. As a first step, the τ-cluster morphism category is defined using the lattice of torsion classes of mod(A). Whenever A is τ-tilting finite, this definition is purely combinatorial, so that the lattice of torsion classes determines W(A) up to equivalence. Moreover, the lattice of torsion classes of a τ-tilting finite algebra is isomorphic to that of infinitely many others. Thus, this result extends the families of algebras whose τ-cluster morphism categories admit faithful functors to groups. Let I be an ideal of A. The lattice theoretic approach provides a framework for constructing a functor F from W(A) to W(A/I) and various properties of this functor are investigated. In particular, if A is τ-tilting finite, the functor is a regular epimorphism in the category of small categories. The final chapter contains another application of the lattice theoretic approach. Let L:K be a MacLane separable field extension. After further developing the behaviour of τ-tilting theory under base field extension, a faithful functor G from W(A) to the τ-cluster morphism of the algebra A_L, obtained from A by tensoring by L over K, is constructed using lattice theory. The existence of G leads to the discovery of new families of algebras whose τ-cluster morphism categories admit faithful functors to groups. Thus, this thesis contributes substantially to understanding the relationship between τ-cluster morphism categories of different algebras as well as to answering the question of when their classifying spaces are K(π,1) spaces.