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fix(Section 3.11): typo (A -> Set) (#336)
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ya-poo authored Jun 20, 2024
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6 changes: 3 additions & 3 deletions src/content/3.11/kan-extensions.tex
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Expand Up @@ -336,13 +336,13 @@ \section{Kan Extensions as Ends}
We can use the product-exponential adjunction:
\[\int_a \int_i \Set(\cat{A}(K i, a),\ (F' a)^{D i})\]
The exponential is isomorphic to the corresponding hom-set:
\[\int_a \int_i \Set(\cat{A}(K i, a),\ \cat{A}(D i, F' a))\]
\[\int_a \int_i \Set(\cat{A}(K i, a),\ \Set(D i, F' a))\]
There is a theorem called the Fubini theorem that allows us to swap the
two ends:
\[\int_i \int_a \Set(\cat{A}(K i, a),\ A(D i, F' a))\]
\[\int_i \int_a \Set(\cat{A}(K i, a),\ \Set(D i, F' a))\]
The inner end represents the set of natural transformations between two
functors, so we can use the Yoneda lemma:
\[\int_i \cat{A}(D i, F' (K i))\]
\[\int_i \Set(D i, F' (K i))\]
This is indeed the set of natural transformations that forms the right
hand side of the adjunction we set out to prove:
\[[\cat{I}, \Set](D, F' \circ K)\]
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