1637 the above antirefinements of the fixed interval $J$, but with the rightmost subinterval $I_m$ always |
1637 the above antirefinements of the fixed interval $J$, but with the rightmost subinterval $I_m$ always |
1638 omitted. |
1638 omitted. |
1639 More specifically, $D\to D'$ is an antirefinement if $D'$ is obtained from $D$ by |
1639 More specifically, $D\to D'$ is an antirefinement if $D'$ is obtained from $D$ by |
1640 gluing subintervals together and/or omitting some of the rightmost subintervals. |
1640 gluing subintervals together and/or omitting some of the rightmost subintervals. |
1641 (See Figure \ref{fig:lmar}.) |
1641 (See Figure \ref{fig:lmar}.) |
1642 \begin{figure}[t]$$ |
1642 \begin{figure}[t] \centering |
1643 \definecolor{arcolor}{rgb}{.75,.4,.1} |
1643 \definecolor{arcolor}{rgb}{.75,.4,.1} |
1644 \begin{tikzpicture}[line width=1pt] |
1644 \begin{tikzpicture}[line width=1pt] |
1645 \fill (0,0) circle (.1); |
1645 \fill (0,0) circle (.1); |
1646 \draw (0,0) -- (2,0); |
1646 \draw (0,0) -- (2,0); |
1647 \draw (1,0.1) -- (1,-0.1); |
1647 \draw (1,0.1) -- (1,-0.1); |
1677 \foreach \x in {1.0, 1.5} { |
1677 \foreach \x in {1.0, 1.5} { |
1678 \draw (\x,1.1) -- (\x,0.9); |
1678 \draw (\x,1.1) -- (\x,0.9); |
1679 } |
1679 } |
1680 |
1680 |
1681 \end{tikzpicture} |
1681 \end{tikzpicture} |
1682 $$ |
|
1683 \caption{Antirefinements of left-marked intervals}\label{fig:lmar}\end{figure} |
1682 \caption{Antirefinements of left-marked intervals}\label{fig:lmar}\end{figure} |
1684 |
1683 |
1685 Now we define the chain complex $\hom_\cC(\cX_\cC \to \cY_\cC)$. |
1684 Now we define the chain complex $\hom_\cC(\cX_\cC \to \cY_\cC)$. |
1686 The underlying vector space is |
1685 The underlying vector space is |
1687 \[ |
1686 \[ |
1733 \] |
1732 \] |
1734 for each left-marked interval $K$. |
1733 for each left-marked interval $K$. |
1735 These are required to commute with gluing; |
1734 These are required to commute with gluing; |
1736 for each subdivision $K = I_1\cup\cdots\cup I_q$ the following diagram commutes: |
1735 for each subdivision $K = I_1\cup\cdots\cup I_q$ the following diagram commutes: |
1737 \[ \xymatrix{ |
1736 \[ \xymatrix{ |
1738 \cX(I_1)\ot\cC(I_2)\ot\cdots\ot\cC(I_q) \ar[r]^{h_{I_0}\ot \id} |
1737 \cX(I_1)\ot\cC(I_2)\ot\cdots\ot\cC(I_q) \ar[r]^{h_{I_1}\ot \id} |
1739 \ar[d]_{\gl} & \cY(I_1)\ot\cC(I_2)\ot\cdots\ot\cC(I_q) |
1738 \ar[d]_{\gl} & \cY(I_1)\ot\cC(I_2)\ot\cdots\ot\cC(I_q) |
1740 \ar[d]^{\gl} \\ |
1739 \ar[d]^{\gl} \\ |
1741 \cX(K) \ar[r]^{h_{K}} & \cY(K) |
1740 \cX(K) \ar[r]^{h_{K}} & \cY(K) |
1742 } \] |
1741 } \] |
1743 Given such an $h$ we can construct a morphism $g$, with $\bd g = 0$, as follows. |
1742 Given such an $h$ we can construct a morphism $g$, with $\bd g = 0$, as follows. |
1873 The product is pinched over the boundary of $J$. |
1872 The product is pinched over the boundary of $J$. |
1874 The set $\cD$ breaks into ``blocks" according to the restrictions to the pinched points of $X\times J$ |
1873 The set $\cD$ breaks into ``blocks" according to the restrictions to the pinched points of $X\times J$ |
1875 (see Figure \ref{feb21b}). |
1874 (see Figure \ref{feb21b}). |
1876 These restrictions are 0-morphisms $(a, b)$ of $\cA$ and $\cB$. |
1875 These restrictions are 0-morphisms $(a, b)$ of $\cA$ and $\cB$. |
1877 |
1876 |
1878 \begin{figure}[t] |
1877 \begin{figure}[t] \centering |
1879 $$ |
|
1880 \begin{tikzpicture}[blue,line width=2pt] |
1878 \begin{tikzpicture}[blue,line width=2pt] |
1881 \draw (0,1) -- (0,-1) node[below] {$X$}; |
1879 \draw (0,1) -- (0,-1) node[below] {$X$}; |
1882 |
1880 |
1883 \draw (2,0) -- (4,0) node[below] {$J$}; |
1881 \draw (2,0) -- (4,0) node[below] {$J$}; |
1884 \fill[red] (3,0) circle (0.1); |
1882 \fill[red] (3,0) circle (0.1); |
1889 \fill (b) circle (0.1) node[right] {\color{green!50!brown} $b$}; |
1887 \fill (b) circle (0.1) node[right] {\color{green!50!brown} $b$}; |
1890 |
1888 |
1891 \path (bottom) node[below]{$X \times J$}; |
1889 \path (bottom) node[below]{$X \times J$}; |
1892 |
1890 |
1893 \end{tikzpicture} |
1891 \end{tikzpicture} |
1894 $$ |
|
1895 \caption{The pinched product $X\times J$} |
1892 \caption{The pinched product $X\times J$} |
1896 \label{feb21b} |
1893 \label{feb21b} |
1897 \end{figure} |
1894 \end{figure} |
1898 |
1895 |
1899 More generally, consider an interval with interior marked points, and with the complements |
1896 More generally, consider an interval with interior marked points, and with the complements |
1902 (See Figure \ref{feb21c}.) |
1899 (See Figure \ref{feb21c}.) |
1903 To this data we can apply the coend construction as in \S\ref{moddecss} above |
1900 To this data we can apply the coend construction as in \S\ref{moddecss} above |
1904 to obtain an $\cA_0$-$\cA_l$ $0$-sphere module and, forgetfully, an $n{-}1$-category. |
1901 to obtain an $\cA_0$-$\cA_l$ $0$-sphere module and, forgetfully, an $n{-}1$-category. |
1905 This amounts to a definition of taking tensor products of $0$-sphere modules over $n$-categories. |
1902 This amounts to a definition of taking tensor products of $0$-sphere modules over $n$-categories. |
1906 |
1903 |
1907 \begin{figure}[t] |
1904 \begin{figure}[t] \centering |
1908 $$ |
|
1909 \begin{tikzpicture}[baseline,line width = 2pt] |
1905 \begin{tikzpicture}[baseline,line width = 2pt] |
1910 \draw[blue] (0,0) -- (6,0); |
1906 \draw[blue] (0,0) -- (6,0); |
1911 \foreach \x/\n in {0.5/0,1.5/1,3/2,4.5/3,5.5/4} { |
1907 \foreach \x/\n in {0.5/0,1.5/1,3/2,4.5/3,5.5/4} { |
1912 \path (\x,0) node[below] {\color{green!50!brown}$\cA_{\n}$}; |
1908 \path (\x,0) node[below] {\color{green!50!brown}$\cA_{\n}$}; |
1913 } |
1909 } |
1925 \foreach \q/\n in {60/0,120/1,-120/2} { |
1921 \foreach \q/\n in {60/0,120/1,-120/2} { |
1926 \fill[red] (\q:2) circle (0.1); |
1922 \fill[red] (\q:2) circle (0.1); |
1927 \path (\q:2.4) node {\color{green!50!brown}$\cM_{\n}$}; |
1923 \path (\q:2.4) node {\color{green!50!brown}$\cM_{\n}$}; |
1928 } |
1924 } |
1929 \end{tikzpicture} |
1925 \end{tikzpicture} |
1930 $$ |
|
1931 \caption{Marked and labeled 1-manifolds} |
1926 \caption{Marked and labeled 1-manifolds} |
1932 \label{feb21c} |
1927 \label{feb21c} |
1933 \end{figure} |
1928 \end{figure} |
1934 |
1929 |
1935 We could also similarly mark and label a circle, obtaining an $n{-}1$-category |
1930 We could also similarly mark and label a circle, obtaining an $n{-}1$-category |
1954 A 1-marked $k$-ball can be decomposed in various ways into smaller balls, which are either |
1949 A 1-marked $k$-ball can be decomposed in various ways into smaller balls, which are either |
1955 (a) smaller 1-marked $k$-balls, (b) 0-marked $k$-balls, or (c) plain $k$-balls. |
1950 (a) smaller 1-marked $k$-balls, (b) 0-marked $k$-balls, or (c) plain $k$-balls. |
1956 (See Figure \nn{need figure}.) |
1951 (See Figure \nn{need figure}.) |
1957 We now proceed as in the above module definitions. |
1952 We now proceed as in the above module definitions. |
1958 |
1953 |
1959 \begin{figure}[!ht] |
1954 \begin{figure}[t] \centering |
1960 $$ |
|
1961 \begin{tikzpicture}[baseline,line width = 2pt] |
1955 \begin{tikzpicture}[baseline,line width = 2pt] |
1962 \draw[blue][fill=blue!15!white] (0,0) circle (2); |
1956 \draw[blue][fill=blue!15!white] (0,0) circle (2); |
1963 \fill[red] (0,0) circle (0.1); |
1957 \fill[red] (0,0) circle (0.1); |
1964 \foreach \qm/\qa/\n in {70/-30/0, 120/95/1, -120/180/2} { |
1958 \foreach \qm/\qa/\n in {70/-30/0, 120/95/1, -120/180/2} { |
1965 \draw[red] (0,0) -- (\qm:2); |
1959 \draw[red] (0,0) -- (\qm:2); |
1966 \path (\qa:1) node {\color{green!50!brown} $\cA_\n$}; |
1960 \path (\qa:1) node {\color{green!50!brown} $\cA_\n$}; |
1967 \path (\qm+20:2.5) node(M\n) {\color{green!50!brown} $\cM_\n$}; |
1961 \path (\qm+20:2.5) node(M\n) {\color{green!50!brown} $\cM_\n$}; |
1968 \draw[line width=1pt, green!50!brown, ->] (M\n.\qm+135) to[out=\qm+135,in=\qm+90] (\qm+5:1.3); |
1962 \draw[line width=1pt, green!50!brown, ->] (M\n.\qm+135) to[out=\qm+135,in=\qm+90] (\qm+5:1.3); |
1969 } |
1963 } |
1970 \end{tikzpicture} |
1964 \end{tikzpicture} |
1971 $$ |
|
1972 \caption{Cone on a marked circle} |
1965 \caption{Cone on a marked circle} |
1973 \label{feb21d} |
1966 \label{feb21d} |
1974 \end{figure} |
1967 \end{figure} |
1975 |
1968 |
1976 A $n$-category 1-sphere module is, among other things, an $n{-}2$-category $\cD$ with |
1969 A $n$-category 1-sphere module is, among other things, an $n{-}2$-category $\cD$ with |