1665 |
1665 |
1666 If the disk-like $n$-category $\cC$ is enriched over some other category (e.g.\ vector spaces), |
1666 If the disk-like $n$-category $\cC$ is enriched over some other category (e.g.\ vector spaces), |
1667 then for each marked $n$-ball $M=(B,N)$ and $c\in \cC(\bd B \setminus N)$, the set $\cM(M; c)$ should be an object in that category. |
1667 then for each marked $n$-ball $M=(B,N)$ and $c\in \cC(\bd B \setminus N)$, the set $\cM(M; c)$ should be an object in that category. |
1668 |
1668 |
1669 \begin{lem}[Boundary from domain and range] |
1669 \begin{lem}[Boundary from domain and range] |
|
1670 \label{lem:module-boundary} |
1670 {Let $H = M_1 \cup_E M_2$, where $H$ is a marked $k{-}1$-hemisphere ($1\le k\le n$), |
1671 {Let $H = M_1 \cup_E M_2$, where $H$ is a marked $k{-}1$-hemisphere ($1\le k\le n$), |
1671 $M_i$ is a marked $k{-}1$-ball, and $E = M_1\cap M_2$ is a marked $k{-}2$-hemisphere. |
1672 $M_i$ is a marked $k{-}1$-ball, and $E = M_1\cap M_2$ is a marked $k{-}2$-hemisphere. |
1672 Let $\cM(M_1) \times_{\cM(E)} \cM(M_2)$ denote the fibered product of the |
1673 Let $\cM(M_1) \times_{\cM(E)} \cM(M_2)$ denote the fibered product of the |
1673 two maps $\bd: \cM(M_i)\to \cl\cM(E)$. |
1674 two maps $\bd: \cM(M_i)\to \cl\cM(E)$. |
1674 Then we have an injective map |
1675 Then we have an injective map |
1675 \[ |
1676 \[ |
1676 \gl_E : \cM(M_1) \times_{\cl\cM(E)} \cM(M_2) \hookrightarrow \cl\cM(H) |
1677 \gl_E : \cM(M_1) \times_{\cl\cM(E)} \cM(M_2) \hookrightarrow \cl\cM(H) |
1677 \] |
1678 \] |
1678 which is natural with respect to the actions of homeomorphisms.} |
1679 which is natural with respect to the actions of homeomorphisms.} |
1679 \end{lem} |
1680 \end{lem} |
1680 Again, this is in exact analogy with Lemma \ref{lem:domain-and-range}. |
1681 Again, this is in exact analogy with Lemma \ref{lem:domain-and-range}, and illustrated in Figure \ref{fig:module-boundary}. |
|
1682 \begin{figure}[t] |
|
1683 \tikzset{marked/.style={line width=5pt}} |
|
1684 |
|
1685 \begin{equation*} |
|
1686 \begin{tikzpicture}[baseline=0] |
|
1687 \coordinate (a) at (0,1); |
|
1688 \coordinate (b) at (4,1); |
|
1689 \draw[marked] (a) arc (180:0:2); |
|
1690 \draw (b) -- (a); |
|
1691 \node at (2,2) {$M_1$}; |
|
1692 |
|
1693 \draw (0,0) node[fill, circle] {} -- (4,0) node[fill,circle] {}; |
|
1694 \node at (-0.6,0) {$E$}; |
|
1695 |
|
1696 \draw[marked] (0,-1) arc(-180:0:2); |
|
1697 \draw (4,-1) -- (0,-1); |
|
1698 \node at (2,-2) {$M_2$}; |
|
1699 \end{tikzpicture} |
|
1700 \qquad \qquad \qquad |
|
1701 \begin{tikzpicture}[baseline=0] |
|
1702 \draw[marked] (0,0) node {$H$} circle (2); |
|
1703 \end{tikzpicture} |
|
1704 \end{equation*}\caption{The marked hemispheres and marked balls from Lemma \ref{lem:module-boundary}.} |
|
1705 \label{fig:module-boundary} |
|
1706 \end{figure} |
1681 |
1707 |
1682 Let $\cl\cM(H)\trans E$ denote the image of $\gl_E$. |
1708 Let $\cl\cM(H)\trans E$ denote the image of $\gl_E$. |
1683 We will refer to elements of $\cl\cM(H)\trans E$ as ``splittable along $E$" or ``transverse to $E$". |
1709 We will refer to elements of $\cl\cM(H)\trans E$ as ``splittable along $E$" or ``transverse to $E$". |
1684 |
1710 |
1685 \noop{ %%%%%%% |
1711 \noop{ %%%%%%% |