关于覆盖关系象的一个注记
A note on images of cover relations
摘要：
对于范畴$\mathbb{C}$、小范畴$\mathbb{I}$和$\mathbb C$上的预覆盖关系$\sqsubset$，在$\mathbb C$上的完备性假设下，证明了：当$g$的每个分量允许关于$\sqsubset$的象时，函子范畴$\mathbb {C}^{\mathbb I}$中的态射$g: B\to C$允许关于$\sqsubset$诱导的$\mathbb C^{\mathbb I}$上的预覆盖关系的象.然后，我们应用这个证明，如果一个指向范畴$\mathbb{C}$是：（i）代数Carnitarian封闭的;（ii）精确原模和行动可达的;或（iii）承认正规化，那么同样的是每个函子范畴$\mathbb{C}^{\mathbb I}$与$\mathbb{I}$有限。此外，我们的结果给出了明确的构造图像函子范畴中使用的限制和图像的基础类别。特别是，它们可以用来给出函子范畴中的中心化子和规范化子的显式构造，它们分别使用底层范畴中的极限和中心化子或规范化子。
英文摘要：
For a category $\mathbb{C}$, a small category $\mathbb{I}$, and a pre-cover relation $\sqsubset$ on $\mathbb C$ we prove, under certain completeness assumptions on $\mathbb C$, that a morphism $g: B\to C$ in the functor category $\mathbb {C}^{\mathbb I}$ admits an image with respect to the pre-cover relation on $\mathbb C^{\mathbb I}$ induced by $\sqsubset$ as soon as each component of $g$ admits an image with respect to $\sqsubset$. We then apply this to show that if a pointed category $\mathbb{C}$ is: (i) algebraically cartesian closed; (ii) exact protomodular and action accessible; or (iii) admits normalizers, then the same is true of each functor category $\mathbb{C}^{\mathbb I}$ with $\mathbb{I}$ finite. In addition, our results give explicit constructions of images in functor categories using limits and images in the underlying category. In particular, they can be used to give explicit constructions of both centralizers and normalizers in functor categories using limits and centralizers or normalizers (respectively) in the underlying category.
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