In higher category theory
Limits and colimits
limits and colimits
limit and colimit
limits and colimits by example
commutativity of limits and colimits
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
end and coend
The Yoneda extension of a functor is extension along the Yoneda embedding of its domain category to a functor
The Yoneda extension exhibits the presheaf category as the free cocompletion of .
For a small category and a functor, its Yoneda extension
is the left Kan extension of along the Yoneda embedding :
Often it is of interest to Yoneda extend not itself, but the composition to get a functor entirely between presheaf categories
This is in fact a left adjoint to the direct image or restriction functor which maps ; see restriction and extension of sheaves.
Recalling the general formula for the left Kan extension of a functor through a functor
one finds for the Yoneda extension the formula
(Recall the notation for the comma category whose objects are pairs .
For the full extension this yields
Here the first step is from above, the second uses that colimits in presheaf categories are computed objectwise and the last one is again using the Yoneda lemma.
The restriction of the Yoneda extension to coincides with the original functor: .
The Yoneda extension commutes with small colimits in in that for a diagram, we have .
Moreover, is defined up to isomorphism by these two properties.