PUB - Publikationen an der Universität Bielefeld

This paper takes a novel lattice theoretic approach to the tau-cluster morphism category & Tfr;(A) of a finite-dimensional algebra A. Defining the category via the lattice of torsion classes tors A of mod A enables the construction of a functor F-I : & Tfr;(A) -> & Tfr;(A/I) for any ideal I of A. This functor lifts the "quotient relationship" between A and A/I to their tau-cluster morphism categories, revealing new structural insights into these categories. If tors A is finite, many categorical properties of F-I are characterised. Moreover, in this case, the construction of & Tfr;(A) is purely combinatorial, so that the lattice of torsion classes determines the tau-cluster morphism category up to equivalence. This leads to many new examples of algebras for which the classifying space & Bernoullis;& Tfr; (A) is a K(pi, 1) space. (c) 2026 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons.org/licenses/by/4.0/).