Entry tags:
Σ⊣*⊣Π
Given a topos ℰ and an internal functor
These two functors,
Meaning, for a copresheaf
(I mean, I'm still stuffing Kan extensions into my brain, and this picture is what makes the picture clearer)
P.S. Ха! Оказалось, что это теорема, и доказывается через Йонеду.
Theorem 2.6. If the Kan extensions exist for all F, then LanK− and RanK− are
respectively the left and right adjoints to the functor − ○ K which is precomposition with K.
f: C → D in ℰ between two internal categories, we have three adjoint functors, Σf⊣f*⊣Πf.These two functors,
Σf and Πf, can be viewed as giving left and right Kan extensions for diagrams (copresheaves) of ℰC along f. Meaning, for a copresheaf
X: C → ℰ,Σf(X) = Lanf(X)Πf(X) = Ranf(X)(I mean, I'm still stuffing Kan extensions into my brain, and this picture is what makes the picture clearer)
P.S. Ха! Оказалось, что это теорема, и доказывается через Йонеду.
Theorem 2.6. If the Kan extensions exist for all F, then LanK− and RanK− are
respectively the left and right adjoints to the functor − ○ K which is precomposition with K.