nLab free completion




For a small category SS, just as the presheaf category Set S opSet^{S^{op}} is the free cocompletion of SS, by formal duality its opposite category (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 “opposite Yoneda embeddingS(Set S) opS\to (Set^S)^{op} via a continuous functor (Set S) opC(Set^S)^{op}\to C.


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.

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