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 S , just as the presheaf category Set S op Set^{S^{op}} is the free cocompletion of S S , by formal duality the opposite category ( Set S ) op (Set^S)^{op} of the category of presheaves on S op S^{op} is the free completion of S S .
This means that any functor S → C S\to C where C C is complete factors uniquely (up to isomorphism ) through the “opposite Yoneda embedding ” S → ( Set S ) op S\to (Set^S)^{op} via a continuous functor ( Set S ) op → C (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 2 -category a ‘free completion’. See discussion at completion .
Last revised on January 21, 2024 at 04:57:39.
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