Pairwise Compatibility for the 2-Simple Minded Collections of Gentle Algebras
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 gentle algebra. We begin by extending the definition of mutation from 2-simple minded collections to more general sets of bricks (called semibrick pairs) and showing that a semibrick pair is contained in a 2-simple minded collection if and only if it is mutation compatible. We then use this result 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.