On coslices and commas of locally finitely presentable categories

TL;DR

This paper characterizes finitely presented objects in coslice and comma categories of locally finitely presentable categories, providing explicit descriptions and showing that certain comma categories remain locally finitely presentable.

Contribution

It offers explicit descriptions of generators of finitely presented objects in coslice and comma categories, and proves that the 2-category LFP has comma objects computed in Cat.

Findings

Explicit description of finitely presented objects in coslice categories.

Proof that comma categories under certain morphisms are locally finitely presentable.

Demonstration that the 2-category LFP has comma objects in Cat.

Abstract

We give an explicit description of the generator of finitely presented objects of the coslice of a locally finitely presentable category under a given object, as consisting of all pushouts of finitely presented maps under this object. Then we prove that the comma category under the direct image part of a morphism of locally finitely presentable category is still locally finitely presentable, and we give again an explicit description of its generator of finitely presented objects. We finally deduce that 2-category $\LFP$ has comma objects computed in $\Cat$.