Enriched hom functors preserve weighted limits

命題: Limits in $(\mathcal{V})$

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

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 $(\mathcal{V})$ )

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

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

Basic Concepts of Enriched Category Theory

命題: Limits in

証明

実装

補題: A reduction of limits in

命題: Hom functors preserve limits (Limits via limits in )

証明

参考文献

命題: Limits in $(\mathcal{V})$

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

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