固定对象集的小范畴范畴中的作用、半直积和交叉半模
Actions, semidirect products and crossed semimodules in the category of small categories with a fixed set of objects
AI总结：
将幺半群的作用和半直积推广到以对象集为纤维的范畴纤维上，证明Schreier点与作用之间的伴随在幺半群情形下等价，并建立Schreier内范畴与交叉半模之间的伴随。
摘要：
我们将幺半群的作用和半直积的经典概念推广到纤维化 $\mathcal{O}\colon\mathbf{Cat}\rightarrow\mathbf{Set}$ 的纤维上，其中 $\mathcal{O}$ 将小范畴 $\mathbb{X}$ 映射到其对象集 $X_0=ob(\mathbb{X})$。我们证明了幺半群 $Y$ 的幺半群作用与 $Y$ 上的Schreier分裂扩张之间的等价性（该等价性众所周知地推广了群的作用与分裂扩张之间的等价性）是纤维 $\mathcal{O}^{-1}(B)$ 中Schreier点与作用之间的更广泛伴随的一个实例。这个伴随当且仅当 $B=1$（即对于幺半群范畴 $\mathbf{Mon}$）时是等价。类似地，我们证明了在纤维 $\mathcal{O}^{-1}(B)$ 中，Schreier内范畴与 $\mathcal{O}^{-1}(B)$ 中的交叉半模范畴之间存在一个伴随（在幺半群情形下，该伴随由Patchkoria得到已知等价）。后者是通过将 $\mathbf{Mon}$ 中的交叉半模概念翻译到 $\mathcal{O}^{-1}(B)$ 中定义的。最后，我们证明，通过适当定义交叉模，最后一个伴随在 $\mathcal{O}$ 的纤维中给出了交叉模与Schreier内群胚之间的等价。
英文摘要：
We generalize to the fibres of the fibration $\mathcal{O}\colon\mathbf{Cat}\rightarrow\mathbf{Set},$ defined by mapping a small category $\mathbb{X}$ to its set of objects $X_0=ob(\mathbb{X}),$ the classical notions of action and semidirect product of monoids. We prove that the equivalence between monoid actions of a monoid $Y$ and Schreier split extensions on $Y,$ which is well known to generalize the equivalence between actions and split extensions for groups, is an instance of a broader adjunction between Schreier points and actions in the fibres $\mathcal{O}^{-1}(B).$ This adjunction is an equivalence if and only if $B=1,$ i.e., for the category $\mathbf{Mon}$ of monoids. Similarly, we prove that there is an adjunction (which, in the case of monoids, results in a known equivalence due to Patchkoria) between Schreier internal categories in the fibres $\mathcal{O}^{-1}(B)$ and the category of crossed semimodules in $\mathcal{O}^{-1}(B).$ The latter are defined by translating in $\mathcal{O}^{-1}(B)$ the notion of crossed semimodule in $\mathbf{Mon}.$ Eventually, we prove that, by defining crossed modules appropriately, this last adjunction yields an equivalence between crossed modules and Schreier internal groupoids in the fibres of $\mathcal{O}.$
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