## § Locally Presentable Category

- A category is locally presentable iff it has a set $S$ of objects such that every object is a colimit over these objects. This definition is correct upto size issues.
- A locally presentable category is a reflective localization $C \to Psh(S)$ of a category of presheaves over $S$. Since $Psh(S)$ is the free cocompletion, and localization imposes relations, this lets us write a category in terms of generators and relations.

- Formally, $C$ :
- 1. is locally small
- 2. has all small colimits
- 3.

#### § Localization

#### § Reflective localization

#### § Accessible Reflective localization