nLab
free completion
Contents
Context
Category theory
Limits and colimits
limits and colimits

1-Categorical
limit and colimit

limits and colimits by example

commutativity of limits and colimits

small limit

filtered colimit

sifted colimit

connected limit , wide pullback

preserved limit , reflected limit , created limit

product , fiber product , base change , coproduct , pullback , pushout , cobase change , equalizer , coequalizer , join , meet , terminal object , initial object , direct product , direct sum

finite limit

Kan extension

weighted limit

end and coend

fibered limit

2-Categorical
(∞,1)-Categorical
Model-categorical
Contents
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 $(Set^S)^{op}\to C$ .

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 .

Last revised on January 21, 2024 at 04:57:39.
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