1704 $(B^k, B^{k-1})$. |
1704 $(B^k, B^{k-1})$. |
1705 See Figure \ref{feb21a}. |
1705 See Figure \ref{feb21a}. |
1706 Another way to say this is that $(X, M)$ is homeomorphic to $B^{k-1}\times([-1,1], \{0\})$. |
1706 Another way to say this is that $(X, M)$ is homeomorphic to $B^{k-1}\times([-1,1], \{0\})$. |
1707 |
1707 |
1708 \begin{figure}[!ht] |
1708 \begin{figure}[!ht] |
1709 \begin{equation*} |
1709 $$\tikz[baseline,line width=2pt]{\draw[blue] (-2,0)--(2,0); \fill[red] (0,0) circle (0.1);} \qquad \qquad \tikz[baseline,line width=2pt]{\draw[blue] (0,0) circle (2 and 1); \draw[red] (0,1)--(0,-1);}$$ |
1710 \mathfig{.85}{tempkw/feb21a} |
|
1711 \end{equation*} |
|
1712 \caption{0-marked 1-ball and 0-marked 2-ball} |
1710 \caption{0-marked 1-ball and 0-marked 2-ball} |
1713 \label{feb21a} |
1711 \label{feb21a} |
1714 \end{figure} |
1712 \end{figure} |
1715 |
1713 |
1716 The $0$-marked balls can be cut into smaller balls in various ways. |
1714 The $0$-marked balls can be cut into smaller balls in various ways. |
1749 The set $\cD$ breaks into ``blocks" according to the restrictions to the pinched points of $X\times J$ |
1747 The set $\cD$ breaks into ``blocks" according to the restrictions to the pinched points of $X\times J$ |
1750 (see Figure \ref{feb21b}). |
1748 (see Figure \ref{feb21b}). |
1751 These restrictions are 0-morphisms $(a, b)$ of $\cA$ and $\cB$. |
1749 These restrictions are 0-morphisms $(a, b)$ of $\cA$ and $\cB$. |
1752 |
1750 |
1753 \begin{figure}[!ht] |
1751 \begin{figure}[!ht] |
1754 \begin{equation*} |
1752 $$ |
1755 \mathfig{1}{tempkw/feb21b} |
1753 \begin{tikzpicture}[blue,line width=2pt] |
1756 \end{equation*} |
1754 \draw (0,1) -- (0,-1) node[below] {$X$}; |
|
1755 |
|
1756 \draw (2,0) -- (4,0) node[below] {$J$}; |
|
1757 \fill[red] (3,0) circle (0.1); |
|
1758 |
|
1759 \draw (6,0) node(a) {} arc (135:90:4) node(top) {} arc (90:45:4) node(b) {} arc (-45:-90:4) node(bottom) {} arc(-90:-135:4); |
|
1760 \draw[red] (top.center) -- (bottom.center); |
|
1761 \fill (a) circle (0.1) node[left] {\color{green!50!brown} $a$}; |
|
1762 \fill (b) circle (0.1) node[right] {\color{green!50!brown} $b$}; |
|
1763 |
|
1764 \path (bottom) node[below]{$X \times J$}; |
|
1765 |
|
1766 \end{tikzpicture} |
|
1767 $$ |
1757 \caption{The pinched product $X\times J$} |
1768 \caption{The pinched product $X\times J$} |
1758 \label{feb21b} |
1769 \label{feb21b} |
1759 \end{figure} |
1770 \end{figure} |
1760 |
1771 |
1761 More generally, consider an interval with interior marked points, and with the complements |
1772 More generally, consider an interval with interior marked points, and with the complements |
1765 To this data we can apply the coend construction as in Subsection \ref{moddecss} above |
1776 To this data we can apply the coend construction as in Subsection \ref{moddecss} above |
1766 to obtain an $\cA_0$-$\cA_l$ $0$-sphere module and, forgetfully, an $n{-}1$-category. |
1777 to obtain an $\cA_0$-$\cA_l$ $0$-sphere module and, forgetfully, an $n{-}1$-category. |
1767 This amounts to a definition of taking tensor products of $0$-sphere module over $n$-categories. |
1778 This amounts to a definition of taking tensor products of $0$-sphere module over $n$-categories. |
1768 |
1779 |
1769 \begin{figure}[!ht] |
1780 \begin{figure}[!ht] |
1770 \begin{equation*} |
1781 $$ |
1771 \mathfig{1}{tempkw/feb21c} |
1782 \begin{tikzpicture}[baseline,line width = 2pt] |
1772 \end{equation*} |
1783 \draw[blue] (0,0) -- (6,0); |
|
1784 \foreach \x/\n in {0.5/0,1.5/1,3/2,4.5/3,5.5/4} { |
|
1785 \path (\x,0) node[below] {\color{green!50!brown}$\cA_{\n}$}; |
|
1786 } |
|
1787 \foreach \x/\n in {1/0,2/1,4/2,5/3} { |
|
1788 \fill[red] (\x,0) circle (0.1) node[above] {\color{green!50!brown}$\cM_{\n}$}; |
|
1789 } |
|
1790 \end{tikzpicture} |
|
1791 \qquad |
|
1792 \qquad |
|
1793 \begin{tikzpicture}[baseline,line width = 2pt] |
|
1794 \draw[blue] (0,0) circle (2); |
|
1795 \foreach \q/\n in {-45/0,90/1,180/2} { |
|
1796 \path (\q:2.4) node {\color{green!50!brown}$\cA_{\n}$}; |
|
1797 } |
|
1798 \foreach \q/\n in {60/0,120/1,-120/2} { |
|
1799 \fill[red] (\q:2) circle (0.1); |
|
1800 \path (\q:2.4) node {\color{green!50!brown}$\cM_{\n}$}; |
|
1801 } |
|
1802 \end{tikzpicture} |
|
1803 $$ |
1773 \caption{Marked and labeled 1-manifolds} |
1804 \caption{Marked and labeled 1-manifolds} |
1774 \label{feb21c} |
1805 \label{feb21c} |
1775 \end{figure} |
1806 \end{figure} |
1776 |
1807 |
1777 We could also similarly mark and label a circle, obtaining an $n{-}1$-category |
1808 We could also similarly mark and label a circle, obtaining an $n{-}1$-category |
1796 1-marked $k$-balls can be decomposed in various ways into smaller balls, which are either |
1827 1-marked $k$-balls can be decomposed in various ways into smaller balls, which are either |
1797 smaller 1-marked $k$-balls or the product of an unmarked ball with a marked interval. |
1828 smaller 1-marked $k$-balls or the product of an unmarked ball with a marked interval. |
1798 We now proceed as in the above module definitions. |
1829 We now proceed as in the above module definitions. |
1799 |
1830 |
1800 \begin{figure}[!ht] |
1831 \begin{figure}[!ht] |
1801 \begin{equation*} |
1832 $$ |
1802 \mathfig{.4}{tempkw/feb21d} |
1833 \begin{tikzpicture}[baseline,line width = 2pt] |
1803 \end{equation*} |
1834 \draw[blue] (0,0) circle (2); |
|
1835 \fill[red] (0,0) circle (0.1); |
|
1836 \foreach \qm/\qa/\n in {70/-30/0, 120/95/1, -120/180/2} { |
|
1837 \draw[red] (0,0) -- (\qm:2); |
|
1838 \path (\qa:1) node {\color{green!50!brown} $\cA_\n$}; |
|
1839 \path (\qm+20:2.5) node(M\n) {\color{green!50!brown} $\cM_\n$}; |
|
1840 \draw[line width=1pt, green!50!brown, ->] (M\n.\qm+135) to[out=\qm+135,in=\qm+90] (\qm+5:1.3); |
|
1841 } |
|
1842 \end{tikzpicture} |
|
1843 $$ |
1804 \caption{Cone on a marked circle} |
1844 \caption{Cone on a marked circle} |
1805 \label{feb21d} |
1845 \label{feb21d} |
1806 \end{figure} |
1846 \end{figure} |
1807 |
1847 |
1808 A $n$-category 1-sphere module is, among other things, an $n{-}2$-category $\cD$ with |
1848 A $n$-category 1-sphere module is, among other things, an $n{-}2$-category $\cD$ with |