---
date: 2022-03-26 21:44
modification date: Saturday 26th March 2022 21:44:37
title: "free cocompletion"
aliases: [free cocompletion]

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Last modified date: <%+ tp.file.last_modified_date() %>


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- Tags 
	- #higher-algebra 
- Refs:
	- #todo/add-references
- Links:
	- #todo/create-links 

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# free cocompletion

The Yoneda embedding $\cat{C}\to \cat{P_C} \da \Presh(\cat C, \Set)$ has a universal property: for any complete category $\cat D$, $\Fun(\cat C, \cat D) \homotopic \Fun^L(\cat {P_C}, \cat D)$ where the latter are colimit-preserving functors. This embedding is left exact and preserves limits.