1680 |
1680 |
1681 In this subsection we define an $n{+}1$-category $\cS$ of ``sphere modules" |
1681 In this subsection we define an $n{+}1$-category $\cS$ of ``sphere modules" |
1682 whose objects are $n$-categories. |
1682 whose objects are $n$-categories. |
1683 When $n=2$ |
1683 When $n=2$ |
1684 this is a version of the familiar algebras-bimodules-intertwiners $2$-category. |
1684 this is a version of the familiar algebras-bimodules-intertwiners $2$-category. |
1685 While it is clearly appropriate to call an $S^0$ module a bimodule, |
1685 It is clearly appropriate to call an $S^0$ module a bimodule, |
1686 but this is much less true for higher dimensional spheres, |
1686 but this is much less true for higher dimensional spheres, |
1687 so we prefer the term ``sphere module" for the general case. |
1687 so we prefer the term ``sphere module" for the general case. |
|
1688 |
|
1689 For simplicity, we will assume that $n$-categories are enriched over $\c$-vector spaces. |
1688 |
1690 |
1689 The $0$- through $n$-dimensional parts of $\cC$ are various sorts of modules, and we describe |
1691 The $0$- through $n$-dimensional parts of $\cC$ are various sorts of modules, and we describe |
1690 these first. |
1692 these first. |
1691 The $n{+}1$-dimensional part of $\cS$ consists of intertwiners |
1693 The $n{+}1$-dimensional part of $\cS$ consists of intertwiners |
1692 of (garden-variety) $1$-category modules associated to decorated $n$-balls. |
1694 of (garden-variety) $1$-category modules associated to decorated $n$-balls. |
1709 $(B^k, B^{k-1})$. |
1711 $(B^k, B^{k-1})$. |
1710 See Figure \ref{feb21a}. |
1712 See Figure \ref{feb21a}. |
1711 Another way to say this is that $(X, M)$ is homeomorphic to $B^{k-1}\times([-1,1], \{0\})$. |
1713 Another way to say this is that $(X, M)$ is homeomorphic to $B^{k-1}\times([-1,1], \{0\})$. |
1712 |
1714 |
1713 \begin{figure}[!ht] |
1715 \begin{figure}[!ht] |
1714 $$\tikz[baseline,line width=2pt]{\draw[blue] (-2,0)--(2,0); \fill[red] (0,0) circle (0.1);} \qquad \qquad \tikz[baseline,line width=2pt]{\draw[blue] (0,0) circle (2 and 1); \draw[red] (0,1)--(0,-1);}$$ |
1716 $$\tikz[baseline,line width=2pt]{\draw[blue] (-2,0)--(2,0); \fill[red] (0,0) circle (0.1);} \qquad \qquad \tikz[baseline,line width=2pt]{\draw[blue][fill=blue!30!white] (0,0) circle (2 and 1); \draw[red] (0,1)--(0,-1);}$$ |
1715 \caption{0-marked 1-ball and 0-marked 2-ball} |
1717 \caption{0-marked 1-ball and 0-marked 2-ball} |
1716 \label{feb21a} |
1718 \label{feb21a} |
1717 \end{figure} |
1719 \end{figure} |
1718 |
1720 |
1719 The $0$-marked balls can be cut into smaller balls in various ways. |
1721 The $0$-marked balls can be cut into smaller balls in various ways. |
1734 Given a decomposition of a $0$-marked $k$-ball $X$ into smaller balls $X_i$, we have |
1736 Given a decomposition of a $0$-marked $k$-ball $X$ into smaller balls $X_i$, we have |
1735 morphism sets $\cA_k(X_i)$ (if $X_i$ lies on the $\cA$-labeled side) |
1737 morphism sets $\cA_k(X_i)$ (if $X_i$ lies on the $\cA$-labeled side) |
1736 or $\cB_k(X_i)$ (if $X_i$ lies on the $\cB$-labeled side) |
1738 or $\cB_k(X_i)$ (if $X_i$ lies on the $\cB$-labeled side) |
1737 or $\cM_k(X_i)$ (if $X_i$ intersects the marking and is therefore a smaller 0-marked ball). |
1739 or $\cM_k(X_i)$ (if $X_i$ intersects the marking and is therefore a smaller 0-marked ball). |
1738 Corresponding to this decomposition we have an action and/or composition map |
1740 Corresponding to this decomposition we have an action and/or composition map |
1739 from the product of these various sets into $\cM(X)$. |
1741 from the product of these various sets into $\cM_k(X)$. |
1740 |
1742 |
1741 \medskip |
1743 \medskip |
1742 |
1744 |
1743 Part of the structure of an $n$-category 0-sphere module $\cM$ is captured by saying it is |
1745 Part of the structure of an $n$-category 0-sphere module $\cM$ is captured by saying it is |
1744 a collection $\cD^{ab}$ of $n{-}1$-categories, indexed by pairs $(a, b)$ of objects (0-morphisms) |
1746 a collection $\cD^{ab}$ of $n{-}1$-categories, indexed by pairs $(a, b)$ of objects (0-morphisms) |
1759 \draw (0,1) -- (0,-1) node[below] {$X$}; |
1761 \draw (0,1) -- (0,-1) node[below] {$X$}; |
1760 |
1762 |
1761 \draw (2,0) -- (4,0) node[below] {$J$}; |
1763 \draw (2,0) -- (4,0) node[below] {$J$}; |
1762 \fill[red] (3,0) circle (0.1); |
1764 \fill[red] (3,0) circle (0.1); |
1763 |
1765 |
1764 \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); |
1766 \draw[fill=blue!30!white] (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); |
1765 \draw[red] (top.center) -- (bottom.center); |
1767 \draw[red] (top.center) -- (bottom.center); |
1766 \fill (a) circle (0.1) node[left] {\color{green!50!brown} $a$}; |
1768 \fill (a) circle (0.1) node[left] {\color{green!50!brown} $a$}; |
1767 \fill (b) circle (0.1) node[right] {\color{green!50!brown} $b$}; |
1769 \fill (b) circle (0.1) node[right] {\color{green!50!brown} $b$}; |
1768 |
1770 |
1769 \path (bottom) node[below]{$X \times J$}; |
1771 \path (bottom) node[below]{$X \times J$}; |
1834 We now proceed as in the above module definitions. |
1836 We now proceed as in the above module definitions. |
1835 |
1837 |
1836 \begin{figure}[!ht] |
1838 \begin{figure}[!ht] |
1837 $$ |
1839 $$ |
1838 \begin{tikzpicture}[baseline,line width = 2pt] |
1840 \begin{tikzpicture}[baseline,line width = 2pt] |
1839 \draw[blue] (0,0) circle (2); |
1841 \draw[blue][fill=blue!15!white] (0,0) circle (2); |
1840 \fill[red] (0,0) circle (0.1); |
1842 \fill[red] (0,0) circle (0.1); |
1841 \foreach \qm/\qa/\n in {70/-30/0, 120/95/1, -120/180/2} { |
1843 \foreach \qm/\qa/\n in {70/-30/0, 120/95/1, -120/180/2} { |
1842 \draw[red] (0,0) -- (\qm:2); |
1844 \draw[red] (0,0) -- (\qm:2); |
1843 \path (\qa:1) node {\color{green!50!brown} $\cA_\n$}; |
1845 \path (\qa:1) node {\color{green!50!brown} $\cA_\n$}; |
1844 \path (\qm+20:2.5) node(M\n) {\color{green!50!brown} $\cM_\n$}; |
1846 \path (\qm+20:2.5) node(M\n) {\color{green!50!brown} $\cM_\n$}; |
1874 For example, there is an $n{-}2$-category associated to a marked, labeled 2-sphere, |
1876 For example, there is an $n{-}2$-category associated to a marked, labeled 2-sphere, |
1875 and a 2-sphere module is a representation of such an $n{-}2$-category. |
1877 and a 2-sphere module is a representation of such an $n{-}2$-category. |
1876 |
1878 |
1877 \medskip |
1879 \medskip |
1878 |
1880 |
1879 We can now define the $n$- or less dimensional part of our $n{+}1$-category $\cS$. |
1881 We can now define the $n$-or-less-dimensional part of our $n{+}1$-category $\cS$. |
1880 Choose some collection of $n$-categories, then choose some collections of bimodules for |
1882 Choose some collection of $n$-categories, then choose some collections of bimodules for |
1881 these $n$-categories, then choose some collection of 1-sphere modules for the various |
1883 these $n$-categories, then choose some collection of 1-sphere modules for the various |
1882 possible marked 1-spheres labeled by the $n$-categories and bimodules, and so on. |
1884 possible marked 1-spheres labeled by the $n$-categories and bimodules, and so on. |
1883 Let $L_i$ denote the collection of $i{-}1$-sphere modules we have chosen. |
1885 Let $L_i$ denote the collection of $i{-}1$-sphere modules we have chosen. |
1884 (For convenience, we declare a $(-1)$-sphere module to be an $n$-category.) |
1886 (For convenience, we declare a $(-1)$-sphere module to be an $n$-category.) |
1895 As described above, we can think of each decorated $k$-ball as defining a $k{-}1$-sphere module |
1897 As described above, we can think of each decorated $k$-ball as defining a $k{-}1$-sphere module |
1896 for the $n{-}k{+}1$-category associated to its decorated boundary. |
1898 for the $n{-}k{+}1$-category associated to its decorated boundary. |
1897 Thus the $k$-morphisms of $\cS$ (for $k\le n$) can be thought |
1899 Thus the $k$-morphisms of $\cS$ (for $k\le n$) can be thought |
1898 of as $n$-category $k{-}1$-sphere modules |
1900 of as $n$-category $k{-}1$-sphere modules |
1899 (generalizations of bimodules). |
1901 (generalizations of bimodules). |
1900 On the other hand, we can equally think of the $k$-morphisms as decorations on $k$-balls, |
1902 On the other hand, we can equally well think of the $k$-morphisms as decorations on $k$-balls, |
1901 and from this (official) point of view it is clear that they satisfy all of the axioms of an |
1903 and from this (official) point of view it is clear that they satisfy all of the axioms of an |
1902 $n{+}1$-category. |
1904 $n{+}1$-category. |
1903 (All of the axioms for the less-than-$n{+}1$-dimensional part of an $n{+}1$-category, that is.) |
1905 (All of the axioms for the less-than-$n{+}1$-dimensional part of an $n{+}1$-category, that is.) |
1904 |
1906 |
1905 \medskip |
1907 \medskip |
1906 |
1908 |
1907 Next we define the $n{+}1$-morphisms of $\cS$. |
1909 Next we define the $n{+}1$-morphisms of $\cS$. |
1908 |
1910 The construction of the 0- through $n$-morphisms was easy and tautological, but the |
1909 |
1911 $n{+}1$-morphisms will require a bit of combinatorial topology effort, as well as addition |
1910 |
1912 duality assumptions on the lower morphisms. |
1911 |
1913 |
1912 |
1914 Let $X$ be an $n{+}1$-ball, and let $c$ be a decoration of its boundary |
1913 |
1915 by a cell complex labeled by 0- through $n$-morphisms, as above. |
|
1916 Choose an $n{-}1$-sphere $E\sub \bd X$ which divides |
|
1917 $\bd X$ into ``incoming" and ``outgoing" boundary $\bd_-X$ and $\bd_+X$. |
|
1918 Let $E_c$ denote $E$ decorated by the restriction of $c$ to $E$. |
|
1919 Recall from above the associated 1-category $\cS(E_c)$. |
|
1920 We can also have $\cS(E_c)$ modules $\cS(\bd_-X_c)$ and $\cS(\bd_+X_c)$. |
|
1921 Define |
|
1922 \[ |
|
1923 \cS(X; c; E) \deq \hom_{\cS(E_c)}(\cS(\bd_-X_c), \cS(\bd_+X_c)) . |
|
1924 \] |
|
1925 |
|
1926 We will show that if the sphere modules are equipped with a compatible family of |
|
1927 non-degenerate inner products, then there is a coherent family of isomorphisms |
|
1928 $\cS(X; c; E) \cong \cS(X; c; E')$ for all pairs of choices $E$ and $E'$. |
|
1929 This will allow us to define $\cS(X; e)$ independently of the choice of $E$. |
|
1930 |
|
1931 Let $Y$ be a decorated $n$-ball, and $\ol{Y}$ it's mirror image. |
|
1932 (We assume we are working in the unoriented category.) |
|
1933 Let $Y\cup\ol{Y}$ denote the decorated $n$-sphere obtained by gluing $Y$ and $\ol{Y}$ |
|
1934 along their common boundary. |
|
1935 An {\it inner product} on $\cS(Y)$ is a dual vector |
|
1936 \[ |
|
1937 z_Y : \cS(Y\cup\ol{Y}) \to \c. |
|
1938 \] |
|
1939 We will also use the notation |
|
1940 \[ |
|
1941 \langle a, b\rangle \deq z_Y(a\bullet \ol{b}) \in \c . |
|
1942 \] |
|
1943 An inner product is {\it non-degenerate} if |
1914 |
1944 |
1915 \nn{...} |
1945 \nn{...} |
1916 |
1946 |
1917 \medskip |
1947 \medskip |
1918 \hrule |
1948 \hrule |