AI 中文总结
本文引入R-全 Schreier 内部范畴概念，关联其方向并证明该关联为保积上纤维化函子，进而为函子纤维连通分支配备交换幺半群结构，借助等价关系用交叉 Schreier 扩张描述该结构。
AI 中文摘要
我们引入了R-全 Schreier 内部范畴的概念，它是非球面阿贝尔群胚概念的幺半群类比。我们为每个R-全 Schreier 内部范畴关联一个方向，该方向是具有交换且可消去核的 Schreier 分裂扩张。我们证明该关联是函子，且该函子是保积上纤维化。借助这些性质，我们为该函子纤维的连通分支配备了典范交换幺半群结构。利用 Schreier 内部范畴与幺半群的交叉半模之间的等价关系，我们通过交叉 Schreier 扩张描述了这些交换幺半群。
英文摘要
We introduce the notion of R-full Schreier internal category, which is the monoid analogue of the notion of aspherical abelian groupoid. We associate with every R-full Schreier internal category a direction, which is a Schreier split extension with commutative and cancellative kernel. We show that this association is functorial and that such functor is a product preserving cofibration. Thanks to these properties, we equip the connected components of the fibres of such functor with canonical commutative monoid structures. Using the equivalence between Schreier internal categories and crossed semimodules of monoids, we describe these commutative monoids in terms of crossed Schreier extensions.
