PS

Weighted limits via conical limits

圏論 命題

Conical (co)limits

Weighted (co)limit に対してフツーの (co)limit のことを conical (co)limit という。

に注意すると、特に weight が constant functor:

  • \Delta \lbrace \ast \rbrace : \mathcal{K} \to \mathcal{Set}

の場合の weighted (co)limit が conical (co)limit である:

  • \operatorname{lim} _ K GK \cong \operatorname{lim} ^ { \lbrace \ast \rbrace} _ K GK
  • \operatorname{colim} _ K GK \cong \operatorname{colim} ^ {\lbrace \ast \rbrace} _ K G K
    • G : \mathcal{K} \to \mathcal{B}

Category of elements

  • \operatorname{el} F := ( \Delta \lbrace \ast \rbrace \downarrow F )
    • F : \mathcal{K} \to \mathcal{Set}

Any set-valued functor is a colimit of representables

Co-Yoneda lemma in colimits - PS の系より:

  • F \cong \operatorname{colim} _ {(K,x) {\in} \operatorname{el}F} \lambda _ A \mathcal{K}(K,A)
    • F : \mathcal{K} \to \mathcal{Set}

Continuity in weights

Weighted limits of representations - PS の系より:

  • \operatorname{lim} _ A ^ {(\operatorname{colim} _ K ^ {FK} GK)A} TA \cong \operatorname{lim} _ K ^ {FK} \operatorname{lim} _ A ^ {(GK)A} TA
    • F : \mathcal{K} ^ {\operatorname{op}} \to (\mathcal{V})
    • G : \mathcal{K} \to \lbrack \mathcal{A}, (\mathcal{V}) \rbrack

命題

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

証明

\begin{aligned} & \operatorname{lim} _ A ^ {FA} G A \\ \cong & \lbrace \text{any set-valued functor is a colimit of representables} \rbrace \\ & \operatorname{lim} _ A ^ {\operatorname{colim} _ {(K,x) {\in} \operatorname{el}F} \mathcal{K}(K,A) } GA \\ \cong & \lbrace \text{continuity in weights} \rbrace \\ & \operatorname{lim} _ {(K,x) {\in} \operatorname{el}F} \operatorname{lim} ^ { \mathcal{K}(K,A) } _ A GA \\ \cong & \lbrace \text{Yoneda lemma via weighted limits} \rbrace \\ & \operatorname{lim} _ {(K,x) {\in} \operatorname{el}F} GK \\ \end{aligned}

Dual

  • \operatorname{colm} _ K ^ {FK} G K \cong \operatorname{colim} _ {(K,x) {\in} \operatorname{el}F} GK
    • F : \mathcal{K} \to \mathcal{Set}
    • G : \mathcal{K} ^ {\operatorname{op}} \to \mathcal{B}

参考文献

*1:Self-enriched category - PS