• 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

• Let $W$

§ Reflective localization

§ Accessible Reflective localization