P53             3

f' g' f   g         pullback       g mono
  f' mono                f'


              g'               ...
3             (1)
f'   mono
f'○k1 = f'○k2       k1   k2

                    k1              f'

                    k2
  ...
3           (2)
               k1
       f'○k1   g'○k2
                                      f'○k1
○○
○○
○  ○             ...
3               (3)
               pullback
                   e
                                                f'○k1
   ...
3           (4)

     k1     (3)           e

                                    f'○k1
e    k1
                          ...
3           (5)

     k2     (3)           e

                                    f'○k1
e    k2
                          ...
3          (6)
            f'○k2 = f'○k1

       k1   k2
                            k1         f'
((1)             )
    ...
3               (7)
                           g mono
 ○     ○           ○   ○        ○   ○
     ○     ○
                 ...
3             (8)
k2   (3)           e


                                   f'○k1
           e               k1,k2

      ...
3              (9)
                                 k1   k2
f'○k1 = f'○k2   k1 = k2
                       f' mono
       ...
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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。

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  • 定理3

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