# Colimit-Dense Subcategories

## TL;DR

This paper characterizes locally presentable categories via colimit-dense subcategories and generators, establishing new conditions under Vop{
}enka's Principle and providing examples in Set and Vec categories.

## Contribution

It introduces a new characterization of locally presentable categories using colimit-dense subcategories and generators, extending the understanding of their structure.

## Key findings

- A cocomplete category is locally presentable iff it has a colimit dense subcategory and a generator of presentable objects.
- A 3-element set is colimit-dense in Set^{op}.
- Spaces of countable dimension are colimit-dense in Vec^{op}.

## Abstract

Among cocomplete categories, the locally presentable ones can be defined as those with a strong generator consisting of presentable objects. Assuming Vop{\v{e}}nka's Principle, we prove that a cocomplete category is locally presentable iff it has a colimit dense subcategory and a generator consisting of presentable objects. We further show that a $3$-element set is colimit-dense in $\Set^{\op}$, and spaces of countable dimension are colimit-dense in $\Vec^{op}$.