PS

Weighted limits of representations

圏論 命題 Enriched Representation

記法

Representation について

  • \operatorname{rep} _ A FA := \operatorname{rep} F := \text{representation of } F
    • \mathcal{A}(\operatorname{rep} _ A FA, A) \cong FA
    • F : \mathcal{A} \to (\mathcal{V})
  • \operatorname{corep} _ A FA := \operatorname{corep} F := \text{corepresentation of } F
    • \mathcal{A}(A,\operatorname{corep} _ A FA) \cong FA
    • F : \mathcal{A} ^ {\operatorname{op}} \to (\mathcal{V})

と書くことにする。定義により

  • \operatorname{lim} _ K ^ {FK} GK := \operatorname{corep} _ B \lbrack \mathcal{K}, (\mathcal{V}) \rbrack (F, \lambda _ K \mathcal{B}(B,GK) )
    • F : \mathcal{K} \to (\mathcal{V})
    • G : \mathcal{K} \to \mathcal{B}
  • \operatorname{colim} _ K ^ {FK} GK := \operatorname{rep} _ B \lbrack \mathcal{K}, (\mathcal{V}) \rbrack (F, \lambda _ K \mathcal{B}(GK, B) )
    • F : \mathcal{K} ^ {\operatorname{op}} \to (\mathcal{V})
    • G : \mathcal{K} \to \mathcal{B}

命題

  • \mathcal{V}-functor: Q : \mathcal{C} ^ {\operatorname{op}} \otimes \mathcal{B} \to (\mathcal{V})

について

  • \exists \operatorname{lim} _ K ^ {FK} GK
    • F : \mathcal{K} \to (\mathcal{V})
    • G : \mathcal{K} \to \mathcal{B}
  • \forall K, \exists \operatorname{corep} _ C Q(C,GK)
  • \forall C, \lambda _ B Q(C, B) \text{ preserves the} \operatorname{lim} _ K ^ {FK} GK

ならば

  • \lambda _ B \operatorname{corep} _ C Q(C,B) preserves the \operatorname{lim} _ K ^ {FK} GK

記法

  • \operatorname{corep} _ C Q(C, \operatorname{lim} _ K ^ {FK} GK ) \cong \operatorname{lim} _ K ^ {FK} \operatorname{corep} _ C Q(C,GK)

証明

まず、Enriched representability - PS の命題より右辺は well-formed。

\begin{aligned} & \mathcal{C}(C, \operatorname{corep} _ C Q(C, \operatorname{lim} _ K ^ {FK} GK ) ) \\ \cong & \lbrace \text{definition of representations} \rbrace \\ & Q(C, \operatorname{lim} _ K ^ {FK} GK) \\ \cong & \lbrace \text{preservation of limits} \rbrace \\ & \operatorname{lim} _ K ^ {FK} Q(C, GK) \\ \cong & \lbrace \text{limits in} (\mathcal{V}) \rbrace \\ & \lbrack \mathcal{K}, (\mathcal{V}) \rbrack (F, \lambda _ K Q(C, GK) ) \\ \cong & \lbrace \text{definition of representations} \rbrace \\ & \lbrack \mathcal{K}, (\mathcal{V}) \rbrack (F, \lambda _ K \mathcal{C}(C, \operatorname{corep} _ C Q(C,GK) ) ) \\ \cong & \lbrace \text{definition of limits} \rbrace \\ & \mathcal{C}(C,\operatorname{lim} _ K ^ {FK} \operatorname{corep} _ C Q(C,GK) ) \end{aligned}

これを計算すると確かに Preservation of weighted limits - PS の形になる。

系: Continuity in limit weights

特に

  • Q(B,H) := \lbrack \mathcal{A}, (\mathcal{V}) \rbrack (H, \lambda _ A \mathcal{B}(B, TA) )
    • T : \mathcal{A} \to \mathcal{B}

とすると

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

参考文献