AI 中文总结
研究克利福德半群与幺半格罗滕迪克构造的关系，通过将格罗滕迪克构造应用于特定函子得到克利福德半群范畴，还在克利福德幺半群范畴构造分解系统，并得出逆半环与特定宽松幺半函子的对应。
AI 中文摘要
已知克利福德半群对应于从半格到群范畴的函子。我们证明这种对应是幺半格罗滕迪克构造的一个实例。此外，将格罗滕迪克构造应用于将半格\(L\)映射到函子范畴\([L, Grp]\)的函子，得到所有克利福德半群的范畴。我们用此在克利福德幺半群范畴上构造了许多分解系统。最后，我们证明了关于在幺半纤维化中取幺半群的一个一般结果，并将其应用于给出逆半环与从幂等半环到阿贝尔群范畴的宽松幺半函子之间的对应。
英文摘要
We show that the monoidal Grothendieck construction can be applied to recover the known structure theorem for Clifford monoids, which states that they correspond to functors from a semilattice into the category of groups. Furthermore, we capture the category of Clifford monoids itself as a Grothendieck construction of the functor sending a semilattice L to the functor category [L, Grp] and use this to construct a number of factorisation systems on this category. Finally, we prove a general result on taking monoids in a monoidal fibration and apply it to establish a correspondence between inverse semirings and lax monoidal functors from an idempotent semiring into the category of abelian groups.
Comments32 pages, a number of minor improvements
