nLab free completion

1. Idea

For a small category $S$, just as the presheaf category $Set^{S^{op}}$ is the free cocompletion of $S$, by formal duality the opposite category $(Set^S)^{op}$ of the category of presheaves on $S^{op}$ is the free completion of $S$.

This means that any functor $S\to C$ where $C$ is complete factors uniquely (up to isomorphism) through the “opposite Yoneda embedding” $S\to (Set^S)^{op}$ via a continuous functor $(Set^S)^{op}\to C$.

2. Properties

The operation of completion is a 2-monad which is colax idempotent but not (even weakly) idempotent. In analogy with the general concept of completion, we might call the operation of any colax idempotent monad on a $2$-category a ‘free completion’. See discussion at completion.

3. Related concepts

• free cocompletion

• free bicompletion

• free construction

• free coproduct completion

• idempotent completion, Karoubi envelope, Cauchy completion