PS

Kan extensionによるlimit

Functor:

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

について、

命題

そのdual:

  •  (\eta _ n : Tn \rightarrow a)_ n : \text{colimiting cocone of } T
    •  \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

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