free completion

For a small category SS, just as the presheaf category Set S opSet^{S^{op}} is the free cocompletion of SS, by duality we can conclude that (Set S) op(Set^S)^{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 “dual Yoneda embedding” S(Set S) opS\to (Set^S)^{op} via a continuous functor (Set S) opC(Set^S)^{op}\to C.

This operation 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 February 10, 2016 at 18:50:56. See the history of this page for a list of all contributions to it.