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定理3
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定理3

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書籍「層・圏・トポス」内の定理の証明。

書籍「層・圏・トポス」内の定理の証明。

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  • 1. P53 3 f' g' f g pullback g mono f' mono f' g' f g
  • 2. 3 (1) f' mono f'○k1 = f'○k2 k1 k2 k1 f' k2 g' f g
  • 3. 3 (2) k1 f'○k1 g'○k2 f'○k1 ○○ ○○ ○ ○ f' g'○k1 ○○ g' f g
  • 4. 3 (3) pullback e f'○k1 e g'○e = g'○k1 f' g'○k1 f'○e = f'○k1 g' f g
  • 5. 3 (4) k1 (3) e f'○k1 e k1 k1 f' g'○k1 = g'○k1 g'○k1 g' f f'○k1 = f'○k1 g
  • 6. 3 (5) k2 (3) e f'○k1 e k2 k2 f' g'○k2 = g'○k1 g'○k1 g' f f'○k2 = f'○k1 g
  • 7. 3 (6) f'○k2 = f'○k1 k1 k2 k1 f' ((1) ) k2 g' f g
  • 8. 3 (7) g mono ○ ○ ○ ○ ○ ○ ○ ○ f'○k1 ○ ○ ○ ○ g'○k1 ○ ○ f' ○ ○ g'○k2 g' f mono g g'○k1 = g'○k2 ∴
  • 9. 3 (8) k2 (3) e f'○k1 e k1,k2 f' k1 = k2 g'○k1 g' f g
  • 10. 3 (9) k1 k2 f'○k1 = f'○k2 k1 = k2 f' mono k1 f' Q.E.D. k2 g' f g