PS

Enriched isomorphism まとめ

記法

Ends

  •  \mathcal{V} _ 0(K, \textstyle\int _ A T(A,A) ) \cong \operatorname{\mathcal{V}-nat} _ A (K, T(A,A) )
    •  T : \mathcal{A} ^ {\operatorname{op}} \otimes \mathcal{A} \to (\mathcal{V})

Fubini

  •  \textstyle\int _ A \textstyle\int _ B T(A,B,A,B) \cong \textstyle\int _ {A,B} T(A,B,A,B) \cong \textstyle\int _ B \textstyle\int _ A T(A,B,A,B)
    •  T : \mathcal{A} ^{\operatorname{op}} \otimes \mathcal{B} ^{\operatorname{op}} \otimes \mathcal{A} \otimes \mathcal{B} \to (\mathcal{V})

Yoneda bijections

  •  \operatorname{\mathcal{V}-nat} _ A(\mathcal{A}(K,A), FA) \cong \mathcal{V} _ 0(I, FK)
    •  F : \mathcal{A} \to (\mathcal{V})
  •  \operatorname{\mathcal{V}-nat} _ A(\mathcal{A}(KB,A),F(B,A) ) \cong \mathcal{V} _ 0(I,F(B,KB) )
    •  F : \mathcal{B} ^ {\operatorname{op}} \otimes \mathcal{A} \to (\mathcal{V})
    •  K : \mathcal{B} \to \mathcal{A}

(Co)representations

  •  \mathcal{A}(\operatorname{rep} _ {A} FA, K) \cong FK
    •  F : \mathcal{A} \to (\mathcal{V})
  •  \mathcal{A}(K, \operatorname{corep} _ {A} F A) \cong FK
    •  F : \mathcal{A} ^ {\operatorname{op}} \to (\mathcal{V})

Yoneda

  •  \textstyle\int _ A \lbrack \mathcal{A}(K,A), FA \rbrack \cong FK
    •  F : \mathcal{A} \to (\mathcal{V})

Limits

  •  \textstyle\lim _ K ^ {FK} GK := \operatorname{corep} _ {B} \textstyle\int_ K \lbrack FK, \mathcal{B}(B, GK) \rbrack
  •  \mathcal{B}(B, \textstyle\lim _ K ^ {FK} GK) \cong \textstyle\int_ K \lbrack FK, \mathcal{B}(B, GK) \rbrack
    •  F : \mathcal{K} \to (\mathcal{V})
    •  G : \mathcal{K} \to \mathcal{B}

Colimits

  •  \operatorname{colim} _ K ^ {FK} GK := \operatorname{rep} _ {B} \textstyle\int_ K \lbrack FK, \mathcal{B}(GK, B) \rbrack
  •  \mathcal{B}(\operatorname{colim} _ K ^ {FK} GK, B) \cong \textstyle\int _ K \lbrack FK, \mathcal{B}(GK, B) \rbrack
    •  F : \mathcal{K} ^ {\operatorname{op}} \to (\mathcal{V})
    •  G : \mathcal{K} \to \mathcal{B}

Limits in self-enriched categories

  •  \operatorname{lim} _ K ^ {FK} GK \cong \textstyle\int _ K \lbrack FK, GK \rbrack
    •  F, G : \mathcal{K} \to (\mathcal{V})

Yoneda via (co)limits

  •  \operatorname{lim} _ A ^ { \mathcal{K}(K,A) } GA \cong GK \cong \operatorname{colim} _ A ^ { \mathcal{K}(A,K) } GA
    •  G : \mathcal{K} \to \mathcal{B}

Fubini via (co)limits

  •  \operatorname{lim} _ K ^ {FK} \lim _ A ^ {HA} P(K,A) \cong \operatorname{\lim} _ {K,A} ^ { FK \otimes HA } P(K,A) \cong \operatorname{lim} _ A ^ {HA} \operatorname{lim} _ K ^ {FK} P(K,A)
    •  F : \mathcal{K} \to (\mathcal{V})
    •  H : \mathcal{A} \to (\mathcal{V})
    •  P : \mathcal{K} \otimes \mathcal{A} \to \mathcal{B}
  •  \operatorname{colim} _ K ^ {FK} \operatorname{colim} _ A ^ {HA} P(K,A) \cong \operatorname{colim} _ {K,A} ^ { FK \otimes HA } P(K,A) \cong \operatorname{colim} _ A ^ {HA} \operatorname{colim} _ K ^ {FK} P(K,A)
    •  F : \mathcal{K} ^ {\operatorname{op}} \to (\mathcal{V})
    •  H : \mathcal{A} ^ {\operatorname{op}} \to (\mathcal{V})
    •  P : \mathcal{K} \otimes \mathcal{A} \to \mathcal{B}

Powers

  •  \lbrack X, C \rbrack := \operatorname{corep}\lbrack X, \mathcal{B}(B,C) \rbrack
  •  \mathcal{B}(B, \lbrack X, C \rbrack) \cong \lbrack X, \mathcal{B}(B,C) \rbrack
    •  X \in \mathcal{V} _ 0
    •  C \in \mathcal{B}

Copowers

  •  X  \otimes C := \operatorname{rep}\lbrack X, \mathcal{B}(C,B) \rbrack
  •  \mathcal{B}(X \otimes C, B) \cong \lbrack X, \mathcal{B}(C,B) \rbrack
    •  X \in \mathcal{V} _ 0
    •  C \in \mathcal{B}

Continuity in weights

  •  \operatorname{lim} _ A ^ { (\operatorname{colim} _ K ^ {FK} GK)A } TA \cong \operatorname{lim} _ {K,A} ^ { (GK)A } FK \pitchfork TA \cong \operatorname{lim} _ K ^{FK} \operatorname{lim} _ A ^ {(GK)A} TA
    •  F : \mathcal{K} ^ {\operatorname{op}} \to (\mathcal{V})
    •  G : \mathcal{K} \to \lbrack \mathcal{A}, (\mathcal{V}) \rbrack
    •  T : \mathcal{A} \to \mathcal{B}
  •  \operatorname{colim} _ A ^ { (\operatorname{colim} _ K ^ {FK} GK)A } TA \cong \operatorname{colim} _ {K,A} ^ { (GK)A } FK \otimes TA \cong \operatorname{colim} _ K ^{FK} \operatorname{colim} _ A ^ {(GK)A} TA
    •  F : \mathcal{K} ^ {\operatorname{op}} \to (\mathcal{V})
    •  G : \mathcal{K} \to \lbrack \mathcal{A} ^ {\operatorname{op}}, (\mathcal{V}) \rbrack
    •  T : \mathcal{A} \to \mathcal{B}

Ends in enriched categories

  •  \textstyle\int _ A G(A,A) := \operatorname{corep} _ B \textstyle\int _ A \mathcal{B}(B,G(A,A) )
  •  \mathcal{B}(B, \textstyle\int _ A G(A,A)) \cong \textstyle\int _ A \mathcal{B}(B, G(A,A) )
    •  G : \mathcal{A} ^ {\operatorname{op}} \otimes \mathcal{A} \to \mathcal{B}

Coends in enriched categories

  •  \textstyle\int ^ A G(A,A) := \operatorname{rep} _ B \textstyle\int _ A \mathcal{B}(G(A,A),B)
  •  \mathcal{B}(\textstyle\int ^ A G(A,A), B) \cong \textstyle\int _ A \mathcal{B}(G(A,A),B )
    •  G : \mathcal{A} ^ {\operatorname{op}} \otimes \mathcal{A}   \to \mathcal{B}

(Co)limits via (co)powers

  •  \operatorname{lim} _ K ^ {FK} GK \cong \textstyle\int _ K  \lbrack FK, GK \rbrack
    •  F : \mathcal{K} \to (\mathcal{V})
    •  G : \mathcal{K} \to \mathcal{B}
  •  \operatorname{colim} _ K ^ {FK} GK \cong \textstyle\int ^ K FK \otimes GK
    •  F : \mathcal{K} ^ {\operatorname{op}} \to (\mathcal{V})
    •  G : \mathcal{K} \to \mathcal{B}

Yoneda via (co)powers

  •  \textstyle\int _ A \lbrack \mathcal{K}(K,A), GA \rbrack  \cong GK \cong \textstyle\int ^ A \mathcal{K}(A,K) \otimes GA
    •  G : \mathcal{K} \to \mathcal{B}

参考文献