论文标题
封面关系图像的注释
A note on images of cover relations
论文作者
论文摘要
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} $在$ \ sqsubset $引起的$ \ mathbb c^{\ mathbb i} $上的预覆盖关系的图像，一旦$ g $的每个组成部分都与$ \ sqssubset $相对于$ g $的每个组成部分。然后，我们将其应用于表明，如果一个指向类别$ \ mathbb {c} $是：（i）代数笛卡尔封闭； （ii）可访问的精确原始和动作；或（iii）允许正常化的人，那么每个函数类别$ \ mathbb {c}^{\ mathbb i} $带有$ \ mathbb {i} $ firite是正确的。此外，我们的结果使用基础类别中的限制和图像提供了函子类别中图像的明确结构。特别是，它们可用于在基础类别中使用限制和集中仪或正常化器（分别）使用限制和正常化的函数类别中的中心化构造和正常化的构造。
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.
