PS

Fubini theorems via ends

Fubini theorem.1

 \mathcal{V}-functor:

  •  T : \mathcal{A} ^{\operatorname{op}} \otimes \mathcal{B} ^{\operatorname{op}} \otimes \mathcal{A} \otimes \mathcal{B} \to (\mathcal{V})

について

  1.  \displaystyle\int _ {A,B} T(A,B,A,B) \cong \displaystyle\int _ A \displaystyle\int _ B T(A,B,A,B)
  2.  \displaystyle\int _ {A,B} T(A,B,A,B) \cong \displaystyle\int _ B \displaystyle\int _ A T(A,B,A,B)

証明

1.について(2.も同様)・・・

左辺から右辺:

f:id:mbps:20150802095207p:plain

f:id:mbps:20150802095215p:plain

右辺から左辺:

f:id:mbps:20150802095302p:plain

f:id:mbps:20150802095307p:plain

Fubini theorem.2

  •  \displaystyle\int _ A \displaystyle\int _ B T(A,B,A,B) \cong \displaystyle\int _ B \displaystyle\int _ A T(A,B,A,B)

証明

f:id:mbps:20150802095313p:plain

f:id:mbps:20150802095318p:plain

参考文献