1782 %\nn{need to finish explaining why we have a system of fields; |
1782 %\nn{need to finish explaining why we have a system of fields; |
1783 %define $k$-cat $\cC(\cdot\times W)$} |
1783 %define $k$-cat $\cC(\cdot\times W)$} |
1784 |
1784 |
1785 \subsection{Modules} |
1785 \subsection{Modules} |
1786 \label{sec:modules} |
1786 \label{sec:modules} |
|
1787 |
|
1788 \tikzset{marked/.style={line width=3pt,red}} |
|
1789 |
1787 Next we define ordinary and $A_\infty$ $n$-category modules. |
1790 Next we define ordinary and $A_\infty$ $n$-category modules. |
1788 The definition will be very similar to that of $n$-categories, |
1791 The definition will be very similar to that of $n$-categories, |
1789 but with $k$-balls replaced by {\it marked $k$-balls,} defined below. |
1792 but with $k$-balls replaced by {\it marked $k$-balls,} defined below. |
1790 |
1793 |
1791 Our motivating example comes from an $(m{-}n{+}1)$-dimensional manifold $W$ with boundary |
1794 Our motivating example comes from an $(m{-}n{+}1)$-dimensional manifold $W$ with boundary |
1864 \] |
1867 \] |
1865 which is natural with respect to the actions of homeomorphisms.} |
1868 which is natural with respect to the actions of homeomorphisms.} |
1866 \end{lem} |
1869 \end{lem} |
1867 Again, this is in exact analogy with Lemma \ref{lem:domain-and-range}, and illustrated in Figure \ref{fig:module-boundary}. |
1870 Again, this is in exact analogy with Lemma \ref{lem:domain-and-range}, and illustrated in Figure \ref{fig:module-boundary}. |
1868 \begin{figure}[t] |
1871 \begin{figure}[t] |
1869 \tikzset{marked/.style={line width=3pt}} |
|
1870 |
|
1871 \begin{equation*} |
1872 \begin{equation*} |
1872 \begin{tikzpicture}[baseline=0] |
1873 \begin{tikzpicture}[baseline=0] |
1873 \coordinate (a) at (0,1); |
1874 \coordinate (a) at (0,1); |
1874 \coordinate (b) at (4,1); |
1875 \coordinate (b) at (4,1); |
1875 \draw[marked] (a) arc (180:0:2); |
1876 \draw[marked] (a) arc (180:0:2); |
1876 \draw (b) -- (a); |
1877 \draw (b) -- (a); |
1877 \node at (2,2) {$M_1$}; |
1878 \node at (2,2) {$M_1$}; |
1878 |
1879 |
1879 \draw (0,0) node[fill, circle] {} -- (4,0) node[fill,circle] {}; |
1880 \draw (0,0) node[fill, circle, red] {} -- (4,0) node[fill,circle,red] {}; |
1880 \node at (-0.6,0) {$E$}; |
1881 \node at (-0.6,0) {$E$}; |
1881 |
1882 |
1882 \draw[marked] (0,-1) arc(-180:0:2); |
1883 \draw[marked] (0,-1) arc(-180:0:2); |
1883 \draw (4,-1) -- (0,-1); |
1884 \draw (4,-1) -- (0,-1); |
1884 \node at (2,-2) {$M_2$}; |
1885 \node at (2,-2) {$M_2$}; |
1885 \end{tikzpicture} |
1886 \end{tikzpicture} |
1886 \qquad \qquad \qquad |
1887 \qquad \qquad \qquad |
1887 \begin{tikzpicture}[baseline=0] |
1888 \begin{tikzpicture}[baseline=0] |
1888 \draw[marked] (0,0) node {$H$} circle (2); |
1889 \draw[marked] (0,0) node[black] {$H$} circle (2); |
1889 \end{tikzpicture} |
1890 \end{tikzpicture} |
1890 \end{equation*}\caption{The marked hemispheres and marked balls from Lemma \ref{lem:module-boundary}.} |
1891 \end{equation*}\caption{The marked hemispheres and marked balls from Lemma \ref{lem:module-boundary}.} |
1891 \label{fig:module-boundary} |
1892 \label{fig:module-boundary} |
1892 \end{figure} |
1893 \end{figure} |
1893 |
1894 |
2023 \begin{figure}[ht] |
2024 \begin{figure}[ht] |
2024 \begin{equation*} |
2025 \begin{equation*} |
2025 \begin{tikzpicture} |
2026 \begin{tikzpicture} |
2026 \draw (0,2) -- (2,2.5); |
2027 \draw (0,2) -- (2,2.5); |
2027 \draw (0,2) -- (2,1.5); |
2028 \draw (0,2) -- (2,1.5); |
2028 \draw[line width=2pt] (2,1.5) -- (2,2.5); |
2029 \draw[marked] (2,1.5) -- (2,2.5); |
2029 \draw (0,0) -- (2,0) node[circle,fill,inner sep=2pt] {}; |
2030 \draw (0,0) -- (2,0) node[red,circle,fill,inner sep=2pt] {}; |
2030 \draw[->] (1,1.5) -- (1,0.25); |
2031 \draw[->] (1,1.5) -- (1,0.25); |
|
2032 % fibres |
|
2033 \path[clip] (0,2) -- (2,2.5) -- (2,1.5) -- cycle; |
|
2034 \foreach \x in {0, 0.25, ..., 1.75} { |
|
2035 \draw[green!50!brown] (\x,1) -- (\x,3); |
|
2036 } |
2031 \end{tikzpicture} |
2037 \end{tikzpicture} |
2032 \qquad \qquad \qquad |
2038 \qquad \qquad \qquad |
2033 \begin{tikzpicture} |
2039 \begin{tikzpicture} |
2034 \draw (2,2.5) -- (0,2.5) -- (0,1.5) -- (2,1.5); |
2040 \draw (2,2.5) -- (0,2.5) -- (0,1.5) -- (2,1.5); |
2035 \draw[line width=2pt] (2,1.5) -- (2,2.5); |
2041 \draw[marked] (2,1.5) -- (2,2.5); |
2036 \draw (0,0) -- (2,0) node[circle,fill,inner sep=2pt] {}; |
2042 \draw (0,0) -- (2,0) node[red,circle,fill,inner sep=2pt] {}; |
2037 \draw[->] (1,1.2) -- (1,0.25); |
2043 \draw[->] (1,1.2) -- (1,0.25); |
|
2044 % fibres |
|
2045 \path[clip] (2,2.5) -- (0,2.5) -- (0,1.5) -- (2,1.5); |
|
2046 \foreach \x in {0, 0.25, ..., 1.75} { |
|
2047 \draw[green!50!brown] (\x,1) -- (\x,3); |
|
2048 } |
2038 \end{tikzpicture} |
2049 \end{tikzpicture} |
2039 \end{equation*} |
2050 \end{equation*} |
2040 \caption{Two examples of marked pinched products.} |
2051 \caption{Two examples of marked pinched products.} |
2041 \label{fig:marked-pinched-products} |
2052 \label{fig:marked-pinched-products} |
2042 \end{figure} |
2053 \end{figure} |
2047 \begin{figure}[ht] |
2058 \begin{figure}[ht] |
2048 \begin{equation*} |
2059 \begin{equation*} |
2049 \begin{tikzpicture} |
2060 \begin{tikzpicture} |
2050 \draw (0,2) -- (2,2.5); |
2061 \draw (0,2) -- (2,2.5); |
2051 \draw (0,2) -- (2,1.5); |
2062 \draw (0,2) -- (2,1.5); |
2052 \draw[dashed] (1.333,2.333) -- (1.333,1.666); |
2063 \draw[marked] (2,1.5) -- (2,2.5); |
2053 \draw[line width=2pt] (2,1.5) -- (2,2.5); |
2064 \draw (0,0) -- (2,0) node[red,circle,fill,inner sep=2pt] {}; |
2054 \draw (0,0) -- (2,0) node[circle,fill,inner sep=2pt] {}; |
|
2055 \draw[->] (1,1.5) -- (1,0.25); |
2065 \draw[->] (1,1.5) -- (1,0.25); |
|
2066 % fibres |
|
2067 \path[clip] (0,2) -- (2,2.5) -- (2,1.5) -- cycle; |
|
2068 \draw[dashed] (1.4,2.5) -- (1.4,1.5); |
|
2069 \foreach \x in {0, 0.25, ..., 1.75} { |
|
2070 \draw[green!50!brown] (\x,1) -- (\x,3); |
|
2071 } |
2056 \end{tikzpicture} |
2072 \end{tikzpicture} |
2057 \qquad \qquad \qquad |
2073 \qquad \qquad \qquad |
2058 \begin{tikzpicture} |
2074 \begin{tikzpicture} |
2059 \draw (0,2) -- (2,2.5); |
2075 \draw (0,2) -- (2,2.5); |
2060 \draw (0,2) -- (2,1.5); |
2076 \draw (0,2) -- (2,1.5); |
2061 \draw[dashed] (0.666,2.166) -- (2,1.833); |
2077 \draw[dashed] (0.666,2.166) -- (2,1.833); |
2062 \draw[line width=2pt] (2,1.5) -- (2,2.5); |
2078 \draw[marked] (2,1.5) -- (2,2.5); |
2063 \draw (0,0) -- (2,0) node[circle,fill,inner sep=2pt] {}; |
2079 \draw (0,0) -- (2,0) node[red,circle,fill,inner sep=2pt] {}; |
2064 \draw[->] (1,1.5) -- (1,0.25); |
2080 \draw[->] (1,1.5) -- (1,0.25); |
|
2081 % fibres |
|
2082 \path[clip] (0,2) -- (2,2.5) -- (2,1.5) -- cycle; |
|
2083 \foreach \x in {0, 0.25, ..., 1.75} { |
|
2084 \draw[green!50!brown] (\x,1) -- (\x,3); |
|
2085 } |
2065 \end{tikzpicture} |
2086 \end{tikzpicture} |
2066 \end{equation*} |
2087 \end{equation*} |
2067 \caption{Two examples of decompositions of marked pinched products.} |
2088 \caption{Two examples of decompositions of marked pinched products.} |
2068 \label{fig:decomposing-marked-pinched-products} |
2089 \label{fig:decomposing-marked-pinched-products} |
2069 \end{figure} |
2090 \end{figure} |