Cauchy completion

The Cauchy completion of an object $X$ is a universal way to make it “Cauchy complete”. There are different (overlapping) meanings of “Cauchy complete”, corresponding to different notions of “Cauchy completion”:

1. The Cauchy completion of a space is a complete space.

2. The Cauchy completion of a category is the Karoubi envelope of the category.

3. The Cauchy completion of an enriched category is a Cauchy complete category.

4. The Cauchy completion of an (infinity,1)-category is an idempotent complete (infinity,1)-category.