PS

Enriched isomorphism まとめ

圏論定義 Enriched

記法

$\mathcal{V} := (\mathcal{V} _ 0, \otimes, I)$
$\operatorname{\mathcal{V}-nat} _ A(TA,SA) := \operatorname{\mathcal{V}-nat}(T, S)$ *1
$\operatorname{\mathcal{V}-nat} _ A(K,T(A,A) ) := \operatorname{\mathcal{V}-wedge}(K, T)$ *2

Ends

- $T : \mathcal{A} ^ {\operatorname{op}} \otimes \mathcal{A} \to (\mathcal{V})$

Fubini

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

Yoneda bijections

- $F : \mathcal{A} \to (\mathcal{V})$
- $F : \mathcal{B} ^ {\operatorname{op}} \otimes \mathcal{A} \to (\mathcal{V})$
- $K : \mathcal{B} \to \mathcal{A}$

(Co)representations

- $F : \mathcal{A} \to (\mathcal{V})$
- $F : \mathcal{A} ^ {\operatorname{op}} \to (\mathcal{V})$

Yoneda

- $F : \mathcal{A} \to (\mathcal{V})$

Limits

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

Colimits

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

Limits in self-enriched categories

- $F, G : \mathcal{K} \to (\mathcal{V})$

Yoneda via (co)limits

- $G : \mathcal{K} \to \mathcal{B}$

Fubini via (co)limits

- $F : \mathcal{K} \to (\mathcal{V})$
- $H : \mathcal{A} \to (\mathcal{V})$
- $P : \mathcal{K} \otimes \mathcal{A} \to \mathcal{B}$
- $F : \mathcal{K} ^ {\operatorname{op}} \to (\mathcal{V})$
- $H : \mathcal{A} ^ {\operatorname{op}} \to (\mathcal{V})$
- $P : \mathcal{K} \otimes \mathcal{A} \to \mathcal{B}$

Powers

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

Copowers

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

Continuity in weights

- $F : \mathcal{K} ^ {\operatorname{op}} \to (\mathcal{V})$
- $G : \mathcal{K} \to \lbrack \mathcal{A}, (\mathcal{V}) \rbrack$
- $T : \mathcal{A} \to \mathcal{B}$
- $F : \mathcal{K} ^ {\operatorname{op}} \to (\mathcal{V})$
- $G : \mathcal{K} \to \lbrack \mathcal{A} ^ {\operatorname{op}}, (\mathcal{V}) \rbrack$
- $T : \mathcal{A} \to \mathcal{B}$

Ends in enriched categories

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

Coends in enriched categories

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

(Co)limits via (co)powers

- $F : \mathcal{K} \to (\mathcal{V})$
- $G : \mathcal{K} \to \mathcal{B}$
- $F : \mathcal{K} ^ {\operatorname{op}} \to (\mathcal{V})$
- $G : \mathcal{K} \to \mathcal{B}$

Yoneda via (co)powers

- $G : \mathcal{K} \to \mathcal{B}$

参考文献

*1:Enriched Yoneda lemma (weak form) - PS

*2:Enriched end - PS