$\tau$-cluster morphism categories of factor algebras

TL;DR

This paper introduces a lattice-theoretic framework for the $ au$-cluster morphism category of a finite-dimensional algebra, establishing functors between categories via lattice congruences and characterizing their properties.

Contribution

It provides a novel combinatorial approach to understanding $ au$-cluster morphism categories through torsion class lattices and explores functorial relationships induced by ideals.

Findings

The category is defined via the lattice of torsion classes.

A functor $F_I$ relates categories of an algebra and its quotient.

Characterization of when $F_I$ is full and faithful.

Abstract

We take a novel lattice-theoretic approach to the $\tau$-cluster morphism category $\mathfrak{T}(A)$ of a finite-dimensional algebra $A$ and define the category via the lattice of torsion classes $\mathrm{tors } A$. Using the lattice congruence induced by an ideal $I$ of $A$ we establish a functor $F_I: \mathfrak{T}(A) \to \mathfrak{T}(A/I)$. If $\mathrm{tors } A$ is finite, $F_I$ is a regular epimorphism in the category of small categories and we characterise when $F_I$ is full and faithful. The construction is purely combinatorial, meaning that the lattice of torsion classes determines the $\tau$-cluster morphism category up to equivalence.