PS

Kan extensionによるlimit

圏論命題 Kan-extension

Functor:

$T : \mathcal{M} \rightarrow \mathcal{A}$
$K _ 1 : \mathcal{M} \rightarrow 1$

について、

命題

- $\iff (\epsilon _ n : (\Delta a \circ K _ 1)n \rightarrow Tn)_ n : \text{right Kan extension of } T \text{ along } K _ 1$ *1

そのdual:

- $\iff (\eta _ n : Tn \rightarrow (\Delta a \circ K _ 1)n )_ n : \text{left Kan extension of } T \text{ along } K _ 1$

記法

$\lim T \cong (\text{Ran}_{K _ 1} T)(\ast)$
$\text{colim} T \cong (\text{Lan}_{K _ 1} T)(\ast)$

参考文献

Anderson.pdf

*1: $\Delta a : 1 \rightarrow \mathcal{A}$ はconstant functor