PS

Enriched hom functors preserve weighted limits

命題: Limits in  (\mathcal{V})

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

証明



\begin{aligned}
      & \lbrack B, \lbrack \mathcal{K}, (\mathcal{V}) \rbrack (F, G) \rbrack = \lbrack B, \textstyle\int _ K \lbrack FK,GK \rbrack \rbrack \\
      & \lbrace \text{internal hom functors preserve ends} \rbrace \\
\cong & \displaystyle\int _ K \lbrack B, \lbrack FK, GK \rbrack \rbrack \\
      & \lbrace \text{flip} \rbrace \\
\cong & \textstyle\int _ K \lbrack FK, \lbrack B, GK \rbrack \rbrack = \lbrack \mathcal{K}, (\mathcal{V})\rbrack (F, \lambda _ K \lbrack B, GK \rbrack ) \\
\end{aligned}

実装

f:id:mbps:20150912121919p:plain

補題: A reduction of limits in  (\mathcal{V})

f:id:mbps:20150912132344p:plain

命題: Hom functors preserve limits (Limits via limits in  (\mathcal{V}))

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

証明

 \begin{aligned}
& \mathcal{B}(B, \textstyle\lim _ K ^ {FK} GK) \\
\cong & \lbrace \text{limit} \rbrace \\
&  \lbrack \mathcal{K},(\mathcal{V})\rbrack(F, \lambda _ K \mathcal{B}(B, GK) ) \\
\cong & \lbrace \text{limits in} (\mathcal{V}) \rbrace \\
& \textstyle\lim _ K ^ {FK} \mathcal{B}(B, GK)
\end{aligned}

および、補題により確かに Preservation of weighted limits - PS の形になる。

参考文献