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replaced
2115 First we explain the remark about derived hom above. |
2115 First we explain the remark about derived hom above. |
2116 Let $L$ be a marked 1-ball and let $\cl{\cX}(L)$ denote the local homotopy colimit construction |
2116 Let $L$ be a marked 1-ball and let $\cl{\cX}(L)$ denote the local homotopy colimit construction |
2117 associated to $L$ by $\cX$ and $\cC$. |
2117 associated to $L$ by $\cX$ and $\cC$. |
2118 (See \S \ref{ss:ncat_fields} and \S \ref{moddecss}.) |
2118 (See \S \ref{ss:ncat_fields} and \S \ref{moddecss}.) |
2119 Define $\cl{\cY}(L)$ similarly. |
2119 Define $\cl{\cY}(L)$ similarly. |
2120 For $K$ an unmarked 1-ball let $\cl{\cC(K)}$ denote the local homotopy colimit |
2120 For $K$ an unmarked 1-ball let $\cl{\cC}(K)$ denote the local homotopy colimit |
2121 construction associated to $K$ by $\cC$. |
2121 construction associated to $K$ by $\cC$. |
2122 Then we have an injective gluing map |
2122 Then we have an injective gluing map |
2123 \[ |
2123 \[ |
2124 \gl: \cl{\cX}(L) \ot \cl{\cC}(K) \to \cl{\cX}(L\cup K) |
2124 \gl: \cl{\cX}(L) \ot \cl{\cC}(K) \to \cl{\cX}(L\cup K) |
2125 \] |
2125 \] |
2223 |
2223 |
2224 The $0$-marked balls can be cut into smaller balls in various ways. |
2224 The $0$-marked balls can be cut into smaller balls in various ways. |
2225 We only consider those decompositions in which the smaller balls are either |
2225 We only consider those decompositions in which the smaller balls are either |
2226 $0$-marked (i.e. intersect the $0$-marking of the large ball in a disc) |
2226 $0$-marked (i.e. intersect the $0$-marking of the large ball in a disc) |
2227 or plain (don't intersect the $0$-marking of the large ball). |
2227 or plain (don't intersect the $0$-marking of the large ball). |
2228 We can also take the boundary of a $0$-marked ball, which is $0$-marked sphere. |
2228 We can also take the boundary of a $0$-marked ball, which is a $0$-marked sphere. |
2229 |
2229 |
2230 Fix $n$-categories $\cA$ and $\cB$. |
2230 Fix $n$-categories $\cA$ and $\cB$. |
2231 These will label the two halves of a $0$-marked $k$-ball. |
2231 These will label the two halves of a $0$-marked $k$-ball. |
2232 |
2232 |
2233 An $n$-category $0$-sphere module $\cM$ over the $n$-categories $\cA$ and $\cB$ is |
2233 An $n$-category $0$-sphere module $\cM$ over the $n$-categories $\cA$ and $\cB$ is |
2616 \end{tikzpicture} |
2616 \end{tikzpicture} |
2617 \end{equation*} |
2617 \end{equation*} |
2618 \caption{Moving $B$ from bottom to top} |
2618 \caption{Moving $B$ from bottom to top} |
2619 \label{jun23c} |
2619 \label{jun23c} |
2620 \end{figure} |
2620 \end{figure} |
2621 Let $D' = B\cap C$. |
|
2622 It is not hard too show that the above two maps are mutually inverse. |
2621 It is not hard too show that the above two maps are mutually inverse. |
2623 |
2622 |
2624 \begin{lem} \label{equator-lemma} |
2623 \begin{lem} \label{equator-lemma} |
2625 Any two choices of $E$ and $E'$ are related by a series of modifications as above. |
2624 Any two choices of $E$ and $E'$ are related by a series of modifications as above. |
2626 \end{lem} |
2625 \end{lem} |