Cofree Objects in The Centralizer and The Center Categories

TL;DR

This paper investigates the properties of centralizer and center categories in monoidal categories, focusing on cocompleteness, co-wellpoweredness, generators, and cofree objects of comonoids, providing conditions for inheritance of these properties.

Contribution

It offers explicit conditions for when colimits, cocompleteness, co-wellpoweredness, and generators are inherited in centralizer and center categories, and studies cofree comonoids in these contexts.

Findings

Conditions for inheritance of cocompleteness and generators.

Explicit descriptions of cofree comonoids.

Analysis of properties in centralizer and center categories.

Abstract

We study cocompleteness, co-wellpoweredness and generators in the centralizer category of an object or morphism in a monoidal category, and the center or the weak center of a monoidal category. We explicitly give some answers for when colimits, cocompleteness, co-wellpoweredness and generators in these monoidal categories can be inherited from their base monidal categories. Most importantly, we investigate cofree objects of comonoids in these monoidal categories.