Abstract In τ-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 τ-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 τ-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 τ-cluster morphism category of a τ-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.

中文翻译：

---

2-简单思想集合的成对兼容性

摘要 在 τ 倾斜理论中，通常很难确定一组积木何时形成 2-简单思想集合。本文的目的是确定一组积木何时包含在 τ 倾斜有限代数的 2-simple mind 集合中。我们首先将变异的定义从 2 个简单的集合扩展到更一般的砖集（我们称之为半砖对）。这给了我们一个算法来检查一个半砖对是否包含在一个 2-simple 思想的集合中。然后，我们使用该算法来证明 τ 倾斜有限温和代数（其箭袋不包含循环或 2 循环）的 2-简单思想集合由成对兼容性条件给出当且仅当相应箭袋中的每个顶点具有学位最多 2. 作为一个应用程序，