nLab
free completion
Contents
Context
Category theory
Limits and colimits
limits and colimits
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
Model-categorical
Contents
1. 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 .
2. 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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