We show that for a given exact category, there exists a bijection between semibricks (pairwise Hom-orthogonal set of bricks) and length wide subcategories (exact extension-closed length abelian subcategories). In particular, we show that a length exact category is abelian if and only if simple objects form a semibrick, that is, Schur's lemma holds.

中文翻译：

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精确范畴的 Schur 引理意味着阿贝尔

我们表明，对于给定的精确类别，在半砖（成对 Hom 正交组砖）和长度宽子类别（精确扩展闭合长度阿贝尔子类别）之间存在双射。特别地，我们证明长度精确范畴是阿贝尔范畴当且仅当简单对象形成半砖，即 Schur 引理成立。