# Pairwise Compatibility for 2-Simple Minded Collections

## TL;DR

This paper develops an algorithm to determine when a set of bricks forms a 2-simple minded collection in $ au$-tilting finite algebras, with applications to gentle algebras and classifying spaces.

## Contribution

It extends mutation definitions to semibrick pairs and characterizes 2-simple minded collections via pairwise compatibility for certain gentle algebras.

## Key findings

- Algorithm for checking semibrick pairs in 2-simple minded collections.
- Characterization of collections in gentle algebras with quivers of degree at most 2.
- Classifying space of the $ au$-cluster morphism category is an Eilenberg-MacLane space under certain conditions.

## Abstract

In $\tau$-tilting theory, it is often difficult to determine when a set of bricks forms a 2-simple minded collection. The aim of this paper is to determine when a set of bricks is contained in a 2-simple minded collection for a $\tau$-tilting finite algebra. We begin by extending the definition of mutation from 2-simple minded collections to more general sets of bricks (which we call semibrick pairs). This gives us an algorithm to check if a semibrick pair is contained in a 2-simple minded collection. We then use this algorithm to show that the 2-simple minded collections of a $\tau$-tilting finite gentle algebra (whose quiver contains no loops or 2-cycles) are given by pairwise compatibility conditions if and only if every vertex in the corresponding quiver has degree at most 2. As an application, we show that the classifying space of the $\tau$-cluster morphism category of a $\tau$-tilting finite gentle algebra (whose quiver contains no loops or 2-cycles) is an Eilenberg- MacLane space if every vertex in the corresponding quiver has degree at most 2.