A note on images of cover relations

TL;DR

This paper investigates how images of cover relations in a category extend to functor categories, providing explicit constructions and conditions under which properties like algebraic cartesian closedness are preserved.

Contribution

It establishes conditions for the existence of images in functor categories based on componentwise images and provides explicit constructions using limits and images from the base category.

Findings

Images in functor categories can be constructed componentwise under certain completeness assumptions.

Properties like algebraic cartesian closedness and normalizers are preserved in functor categories for finite index categories.

Explicit methods for constructing centralizers and normalizers in functor categories are provided.

Abstract

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.