nLab free completion

Contents

Context

Category theory

Limits and colimits

Contents

Idea

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

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

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 22-category a ‘free completion’. See discussion at completion.

Last revised on January 21, 2024 at 04:57:39. See the history of this page for a list of all contributions to it.