PS

Weighted limit による enriched Yoneda lemma

Enriched Yoneda lemma with hom functors @deprecated

  •  \mathcal{V}-functor:  P : \mathcal{C} \otimes \mathcal{A} \to (\mathcal{V})

について

  •  P(C,K) \cong \lbrack \mathcal{A}, (\mathcal{V}) \rbrack ( \mathcal{A}(K, \unicode{0x2013}), P(C, \unicode{0x2013}))
    •  \mathcal{V}-natural in  C,K

Enriched Yoneda lemma via weighted limits

 \mathcal{V}-functor:

  •  G : \mathcal{K} \to \mathcal{B}

について

  •  \textstyle\lim _ A ^ {\mathcal{K}(K, A)} GA \cong G K
    •  \mathcal{V}-natural in  K,G

証明

Enriched Yoneda lemma - PS で特に  F := \lambda _ A \mathcal{B}(B,GA) とすると

  •  \mathcal{B}(B,GK) \cong \textstyle\int _ A \lbrack \mathcal{K}(K,A), \mathcal{B}(B,GA) \rbrack

証明 @deprecated

Enriched Yoneda lemma with hom functors で特に

  •  P( (B, G), K) := \mathcal{B}(B, E(G,K) )

とすると

  •  \mathcal{B}(B, GK) \cong \lbrack \mathcal{K}, (\mathcal{V}) \rbrack ( \mathcal{K}(K,\unicode{0x2013}),\mathcal{B}(B,G\unicode{0x2013}))
    •  \mathcal{V}-natural in  B,K,G

となるが、representation の mediating bijection は naturality を preserve するので。

命題: Weighted limits in self-enriched categories

(Enriched hom functors preserve weighted limits - PS も参照)

 \mathcal{V} functor:

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

について

  •  \textstyle\lim _ K ^{FK} GK \cong \lbrack \mathcal{K},(\mathcal{V}) \rbrack ( F, G )
    •  \mathcal{V}-natural in  F,G

証明

右辺から左辺:



\begin{aligned}
    \lbrack B,  \lbrack \mathcal{K}, ( \mathcal{V} ) \rbrack (F, G) \rbrack
    &= \lbrack B, \displaystyle\int _ K \lbrack FK,GK \rbrack \rbrack \\
    &\cong \displaystyle\int _ K \lbrack B, \lbrack FK, GK \rbrack \rbrack \\
    &\cong \displaystyle\int _ K \lbrack FK, \lbrack B, GK \rbrack \rbrack \\
    &= \lbrack \mathcal{K}, (\mathcal{V})\rbrack (F, \lambda _ K \lbrack B, GK \rbrack ) \\
\end{aligned}

左辺から右辺:



\begin{aligned}
    \textstyle\lim _ K ^{FK} GK
    &\cong \lbrack I, \textstyle\lim _ K ^{FK} GK \rbrack \\
    &\cong \displaystyle\int_K \lbrack FK, \lbrack I, GK \rbrack \rbrack \\
    &\cong \displaystyle\int_K \lbrack FK, GK \rbrack \\
    &= \lbrack \mathcal{K},(\mathcal{V}) \rbrack ( F, G )
\end{aligned}

Enriched Yoneda lemma via weighted limits とより enriched Yoneda lemma を導ける:

  •  \lbrack \mathcal{K}, (\mathcal{V}) \rbrack( \lambda _ A \mathcal{K}(K,A), G )  \cong \textstyle\lim _ A ^{\mathcal{K}(K,A)} GA \cong GK

Co-Yoneda lemma @deprecated

Enriched Yoneda lemma via weighted limits で特に

  •  G := H ^ {\operatorname{op}} : \mathcal{K} ^ {\operatorname{op}} \to \mathcal{B} ^ {\operatorname{op}}

とすると

  •  \mathcal{K}(\unicode{0x2013},K) \star H \cong HK

参考文献