1765 certainly the simple kind (strictly commute with gluing) arise in nature.} |
1763 certainly the simple kind (strictly commute with gluing) arise in nature.} |
1766 |
1764 |
1767 |
1765 |
1768 |
1766 |
1769 |
1767 |
1770 |
|
1771 |
|
1772 |
|
1773 |
|
1774 \subsection{The $n{+}1$-category of sphere modules} |
1768 \subsection{The $n{+}1$-category of sphere modules} |
1775 \label{ssec:spherecat} |
1769 \label{ssec:spherecat} |
1776 |
1770 |
1777 In this subsection we define an $n{+}1$-category $\cS$ of ``sphere modules" |
1771 In this subsection we define $n{+}1$-categories $\cS$ of ``sphere modules" |
1778 whose objects are $n$-categories. |
1772 whose objects are $n$-categories. |
1779 When $n=2$ |
1773 With future applications in mind, we treat simultaneously the big category |
1780 this is closely related to the familiar $2$-category of algebras, bimodules and intertwiners. |
1774 of all $n$-categories and all sphere modules and also subcategories thereof. |
|
1775 When $n=1$ this is closely related to familiar $2$-categories consisting of |
|
1776 algebras, bimodules and intertwiners (or a subcategory of that). |
|
1777 |
1781 While it is appropriate to call an $S^0$ module a bimodule, |
1778 While it is appropriate to call an $S^0$ module a bimodule, |
1782 this is much less true for higher dimensional spheres, |
1779 this is much less true for higher dimensional spheres, |
1783 so we prefer the term ``sphere module" for the general case. |
1780 so we prefer the term ``sphere module" for the general case. |
1784 |
1781 |
1785 The results of this subsection are not needed for the rest of the paper, |
1782 %The results of this subsection are not needed for the rest of the paper, |
1786 so we will skimp on details in a couple of places. We have included this mostly for the sake of comparing our notion of a topological $n$-category to other definitions. |
1783 %so we will skimp on details in a couple of places. We have included this mostly |
|
1784 %for the sake of comparing our notion of a topological $n$-category to other definitions. |
1787 |
1785 |
1788 For simplicity, we will assume that $n$-categories are enriched over $\c$-vector spaces. |
1786 For simplicity, we will assume that $n$-categories are enriched over $\c$-vector spaces. |
1789 |
1787 |
1790 The $0$- through $n$-dimensional parts of $\cS$ are various sorts of modules, and we describe |
1788 The $0$- through $n$-dimensional parts of $\cS$ are various sorts of modules, and we describe |
1791 these first. |
1789 these first. |
1804 (This, in turn, is very similar to our definition of $n$-category.) |
1802 (This, in turn, is very similar to our definition of $n$-category.) |
1805 Because of this similarity, we only sketch the definitions below. |
1803 Because of this similarity, we only sketch the definitions below. |
1806 |
1804 |
1807 We start with $0$-sphere modules, which also could reasonably be called (categorified) bimodules. |
1805 We start with $0$-sphere modules, which also could reasonably be called (categorified) bimodules. |
1808 (For $n=1$ they are precisely bimodules in the usual, uncategorified sense.) |
1806 (For $n=1$ they are precisely bimodules in the usual, uncategorified sense.) |
|
1807 We prefer the more awkward term ``0-sphere module" to emphasize the analogy |
|
1808 with the higher sphere modules defined below. |
|
1809 |
1809 Define a $0$-marked $k$-ball, $1\le k \le n$, to be a pair $(X, M)$ homeomorphic to the standard |
1810 Define a $0$-marked $k$-ball, $1\le k \le n$, to be a pair $(X, M)$ homeomorphic to the standard |
1810 $(B^k, B^{k-1})$. |
1811 $(B^k, B^{k-1})$. |
1811 See Figure \ref{feb21a}. |
1812 See Figure \ref{feb21a}. |
1812 Another way to say this is that $(X, M)$ is homeomorphic to $B^{k-1}\times([-1,1], \{0\})$. |
1813 Another way to say this is that $(X, M)$ is homeomorphic to $B^{k-1}\times([-1,1], \{0\})$. |
1813 |
1814 |
1814 \begin{figure}[!ht] |
1815 \begin{figure}[t] |
1815 $$\tikz[baseline,line width=2pt]{\draw[blue] (-2,0)--(2,0); \fill[red] (0,0) circle (0.1);} \qquad \qquad \tikz[baseline,line width=2pt]{\draw[blue][fill=blue!30!white] (0,0) circle (2 and 1); \draw[red] (0,1)--(0,-1);}$$ |
1816 $$\tikz[baseline,line width=2pt]{\draw[blue] (-2,0)--(2,0); \fill[red] (0,0) circle (0.1);} \qquad \qquad \tikz[baseline,line width=2pt]{\draw[blue][fill=blue!30!white] (0,0) circle (2 and 1); \draw[red] (0,1)--(0,-1);}$$ |
1816 \caption{0-marked 1-ball and 0-marked 2-ball} |
1817 \caption{0-marked 1-ball and 0-marked 2-ball} |
1817 \label{feb21a} |
1818 \label{feb21a} |
1818 \end{figure} |
1819 \end{figure} |
1819 |
1820 |
1850 The product is pinched over the boundary of $J$. |
1851 The product is pinched over the boundary of $J$. |
1851 The set $\cD$ breaks into ``blocks" according to the restrictions to the pinched points of $X\times J$ |
1852 The set $\cD$ breaks into ``blocks" according to the restrictions to the pinched points of $X\times J$ |
1852 (see Figure \ref{feb21b}). |
1853 (see Figure \ref{feb21b}). |
1853 These restrictions are 0-morphisms $(a, b)$ of $\cA$ and $\cB$. |
1854 These restrictions are 0-morphisms $(a, b)$ of $\cA$ and $\cB$. |
1854 |
1855 |
1855 \begin{figure}[!ht] |
1856 \begin{figure}[t] |
1856 $$ |
1857 $$ |
1857 \begin{tikzpicture}[blue,line width=2pt] |
1858 \begin{tikzpicture}[blue,line width=2pt] |
1858 \draw (0,1) -- (0,-1) node[below] {$X$}; |
1859 \draw (0,1) -- (0,-1) node[below] {$X$}; |
1859 |
1860 |
1860 \draw (2,0) -- (4,0) node[below] {$J$}; |
1861 \draw (2,0) -- (4,0) node[below] {$J$}; |
1873 \label{feb21b} |
1874 \label{feb21b} |
1874 \end{figure} |
1875 \end{figure} |
1875 |
1876 |
1876 More generally, consider an interval with interior marked points, and with the complements |
1877 More generally, consider an interval with interior marked points, and with the complements |
1877 of these points labeled by $n$-categories $\cA_i$ ($0\le i\le l$) and the marked points labeled |
1878 of these points labeled by $n$-categories $\cA_i$ ($0\le i\le l$) and the marked points labeled |
1878 by $\cA_i$-$\cA_{i+1}$ bimodules $\cM_i$. |
1879 by $\cA_i$-$\cA_{i+1}$ 0-sphere modules $\cM_i$. |
1879 (See Figure \ref{feb21c}.) |
1880 (See Figure \ref{feb21c}.) |
1880 To this data we can apply the coend construction as in \S\ref{moddecss} above |
1881 To this data we can apply the coend construction as in \S\ref{moddecss} above |
1881 to obtain an $\cA_0$-$\cA_l$ $0$-sphere module and, forgetfully, an $n{-}1$-category. |
1882 to obtain an $\cA_0$-$\cA_l$ $0$-sphere module and, forgetfully, an $n{-}1$-category. |
1882 This amounts to a definition of taking tensor products of $0$-sphere module over $n$-categories. |
1883 This amounts to a definition of taking tensor products of $0$-sphere modules over $n$-categories. |
1883 |
1884 |
1884 \begin{figure}[!ht] |
1885 \begin{figure}[t] |
1885 $$ |
1886 $$ |
1886 \begin{tikzpicture}[baseline,line width = 2pt] |
1887 \begin{tikzpicture}[baseline,line width = 2pt] |
1887 \draw[blue] (0,0) -- (6,0); |
1888 \draw[blue] (0,0) -- (6,0); |
1888 \foreach \x/\n in {0.5/0,1.5/1,3/2,4.5/3,5.5/4} { |
1889 \foreach \x/\n in {0.5/0,1.5/1,3/2,4.5/3,5.5/4} { |
1889 \path (\x,0) node[below] {\color{green!50!brown}$\cA_{\n}$}; |
1890 \path (\x,0) node[below] {\color{green!50!brown}$\cA_{\n}$}; |
1911 |
1912 |
1912 We could also similarly mark and label a circle, obtaining an $n{-}1$-category |
1913 We could also similarly mark and label a circle, obtaining an $n{-}1$-category |
1913 associated to the marked and labeled circle. |
1914 associated to the marked and labeled circle. |
1914 (See Figure \ref{feb21c}.) |
1915 (See Figure \ref{feb21c}.) |
1915 If the circle is divided into two intervals, we can think of this $n{-}1$-category |
1916 If the circle is divided into two intervals, we can think of this $n{-}1$-category |
1916 as the 2-sided tensor product of the two bimodules associated to the two intervals. |
1917 as the 2-sided tensor product of the two 0-sphere modules associated to the two intervals. |
1917 |
1918 |
1918 \medskip |
1919 \medskip |
1919 |
1920 |
1920 Next we define $n$-category 1-sphere modules. |
1921 Next we define $n$-category 1-sphere modules. |
1921 These are just representations of (modules for) $n{-}1$-categories associated to marked and labeled |
1922 These are just representations of (modules for) $n{-}1$-categories associated to marked and labeled |
1922 circles (1-spheres) which we just introduced. |
1923 circles (1-spheres) which we just introduced. |
1923 |
1924 |
1924 Equivalently, we can define 1-sphere modules in terms of 1-marked $k$-balls, $2\le k\le n$. |
1925 Equivalently, we can define 1-sphere modules in terms of 1-marked $k$-balls, $2\le k\le n$. |
1925 Fix a marked (and labeled) circle $S$. |
1926 Fix a marked (and labeled) circle $S$. |
1926 Let $C(S)$ denote the cone of $S$, a marked 2-ball (Figure \ref{feb21d}). |
1927 Let $C(S)$ denote the cone of $S$, a marked 2-ball (Figure \ref{feb21d}). |
1927 \nn{I need to make up my mind whether marked things are always labeled too. |
1928 %\nn{I need to make up my mind whether marked things are always labeled too. |
1928 For the time being, let's say they are.} |
1929 %For the time being, let's say they are.} |
1929 A 1-marked $k$-ball is anything homeomorphic to $B^j \times C(S)$, $0\le j\le n-2$, |
1930 A 1-marked $k$-ball is anything homeomorphic to $B^j \times C(S)$, $0\le j\le n-2$, |
1930 where $B^j$ is the standard $j$-ball. |
1931 where $B^j$ is the standard $j$-ball. |
1931 A 1-marked $k$-ball can be decomposed in various ways into smaller balls, which are either |
1932 A 1-marked $k$-ball can be decomposed in various ways into smaller balls, which are either |
1932 (a) smaller 1-marked $k$-balls, (b) 0-marked $k$-balls, or (c) plain $k$-balls. |
1933 (a) smaller 1-marked $k$-balls, (b) 0-marked $k$-balls, or (c) plain $k$-balls. |
1933 (See Figure xxxx.) |
1934 (See Figure \nn{need figure}.) |
1934 We now proceed as in the above module definitions. |
1935 We now proceed as in the above module definitions. |
1935 |
1936 |
1936 \begin{figure}[!ht] |
1937 \begin{figure}[!ht] |
1937 $$ |
1938 $$ |
1938 \begin{tikzpicture}[baseline,line width = 2pt] |
1939 \begin{tikzpicture}[baseline,line width = 2pt] |
1975 and a 2-sphere module is a representation of such an $n{-}2$-category. |
1976 and a 2-sphere module is a representation of such an $n{-}2$-category. |
1976 |
1977 |
1977 \medskip |
1978 \medskip |
1978 |
1979 |
1979 We can now define the $n$-or-less-dimensional part of our $n{+}1$-category $\cS$. |
1980 We can now define the $n$-or-less-dimensional part of our $n{+}1$-category $\cS$. |
1980 Choose some collection of $n$-categories, then choose some collections of bimodules between |
1981 Choose some collection of $n$-categories, then choose some collections of 0-sphere modules between |
1981 these $n$-categories, then choose some collection of 1-sphere modules for the various |
1982 these $n$-categories, then choose some collection of 1-sphere modules for the various |
1982 possible marked 1-spheres labeled by the $n$-categories and bimodules, and so on. |
1983 possible marked 1-spheres labeled by the $n$-categories and 0-sphere modules, and so on. |
1983 Let $L_i$ denote the collection of $i{-}1$-sphere modules we have chosen. |
1984 Let $L_i$ denote the collection of $i{-}1$-sphere modules we have chosen. |
1984 (For convenience, we declare a $(-1)$-sphere module to be an $n$-category.) |
1985 (For convenience, we declare a $(-1)$-sphere module to be an $n$-category.) |
1985 There is a wide range of possibilities. |
1986 There is a wide range of possibilities. |
1986 The set $L_0$ could contain infinitely many $n$-categories or just one. |
1987 The set $L_0$ could contain infinitely many $n$-categories or just one. |
1987 For each pair of $n$-categories in $L_0$, $L_1$ could contain no bimodules at all or |
1988 For each pair of $n$-categories in $L_0$, $L_1$ could contain no 0-sphere modules at all or |
1988 it could contain several. |
1989 it could contain several. |
1989 The only requirement is that each $k$-sphere module be a module for a $k$-sphere $n{-}k$-category |
1990 The only requirement is that each $k$-sphere module be a module for a $k$-sphere $n{-}k$-category |
1990 constructed out of labels taken from $L_j$ for $j<k$. |
1991 constructed out of labels taken from $L_j$ for $j<k$. |
1991 |
1992 |
1992 We now define $\cS(X)$, for $X$ a ball of dimension at most $n$, to be the set of all |
1993 We now define $\cS(X)$, for $X$ a ball of dimension at most $n$, to be the set of all |
2019 Define |
2020 Define |
2020 \[ |
2021 \[ |
2021 \cS(X; c; E) \deq \hom_{\cS(E_c)}(\cS(\bd_-X_c), \cS(\bd_+X_c)) . |
2022 \cS(X; c; E) \deq \hom_{\cS(E_c)}(\cS(\bd_-X_c), \cS(\bd_+X_c)) . |
2022 \] |
2023 \] |
2023 |
2024 |
2024 We will show that if the sphere modules are equipped with a `compatible family of |
2025 We will show that if the sphere modules are equipped with a ``compatible family of |
2025 non-degenerate inner products', then there is a coherent family of isomorphisms |
2026 non-degenerate inner products", then there is a coherent family of isomorphisms |
2026 $\cS(X; c; E) \cong \cS(X; c; E')$ for all pairs of choices $E$ and $E'$. |
2027 $\cS(X; c; E) \cong \cS(X; c; E')$ for all pairs of choices $E$ and $E'$. |
2027 This will allow us to define $\cS(X; e)$ independently of the choice of $E$. |
2028 This will allow us to define $\cS(X; c)$ independently of the choice of $E$. |
|
2029 \nn{also need to (simultaneously) show compatibility with action of homeos of boundary} |
2028 |
2030 |
2029 First we must define ``inner product", ``non-degenerate" and ``compatible". |
2031 First we must define ``inner product", ``non-degenerate" and ``compatible". |
2030 Let $Y$ be a decorated $n$-ball, and $\ol{Y}$ it's mirror image. |
2032 Let $Y$ be a decorated $n$-ball, and $\ol{Y}$ it's mirror image. |
2031 (We assume we are working in the unoriented category.) |
2033 (We assume we are working in the unoriented category.) |
2032 Let $Y\cup\ol{Y}$ denote the decorated $n$-sphere obtained by gluing $Y$ and $\ol{Y}$ |
2034 Let $Y\cup\ol{Y}$ denote the decorated $n$-sphere obtained by gluing $Y$ and $\ol{Y}$ |
2156 It follows from the lemma that we can construct an isomorphism |
2158 It follows from the lemma that we can construct an isomorphism |
2157 between $\cS(X; c; E)$ and $\cS(X; c; E')$ for any pair $E$, $E'$. |
2159 between $\cS(X; c; E)$ and $\cS(X; c; E')$ for any pair $E$, $E'$. |
2158 This construction involves on a choice of simple ``moves" (as above) to transform |
2160 This construction involves on a choice of simple ``moves" (as above) to transform |
2159 $E$ to $E'$. |
2161 $E$ to $E'$. |
2160 We must now show that the isomorphism does not depend on this choice. |
2162 We must now show that the isomorphism does not depend on this choice. |
2161 We will show below that it suffice to check two ``movie moves". |
2163 We will show below that it suffice to check three ``movie moves". |
2162 |
2164 |
2163 The first movie move is to push $E$ across an $n$-ball $B$ as above, then push it back. |
2165 The first movie move is to push $E$ across an $n$-ball $B$ as above, then push it back. |
2164 The result is equivalent to doing nothing. |
2166 The result is equivalent to doing nothing. |
2165 As we remarked above, the isomorphisms corresponding to these two pushes are mutually |
2167 As we remarked above, the isomorphisms corresponding to these two pushes are mutually |
2166 inverse, so we have invariance under this movie move. |
2168 inverse, so we have invariance under this movie move. |
2167 |
2169 |
2168 The second movie move replaces to successive pushes in the same direction, |
2170 The second movie move replaces two successive pushes in the same direction, |
2169 across $B_1$ and $B_2$, say, with a single push across $B_1\cup B_2$. |
2171 across $B_1$ and $B_2$, say, with a single push across $B_1\cup B_2$. |
2170 (See Figure \ref{jun23d}.) |
2172 (See Figure \ref{jun23d}.) |
2171 \begin{figure}[t] |
2173 \begin{figure}[t] |
2172 \begin{equation*} |
2174 \begin{equation*} |
2173 \mathfig{.9}{tempkw/jun23d} |
2175 \mathfig{.9}{tempkw/jun23d} |
2175 \caption{A movie move} |
2177 \caption{A movie move} |
2176 \label{jun23d} |
2178 \label{jun23d} |
2177 \end{figure} |
2179 \end{figure} |
2178 Invariance under this movie move follows from the compatibility of the inner |
2180 Invariance under this movie move follows from the compatibility of the inner |
2179 product for $B_1\cup B_2$ with the inner products for $B_1$ and $B_2$. |
2181 product for $B_1\cup B_2$ with the inner products for $B_1$ and $B_2$. |
2180 \nn{should also say something about locality/distant-commutativity} |
2182 |
2181 |
2183 The third movie move could be called ``locality" or ``disjoint commutativity". |
2182 If $n\ge 2$, these two movie move suffice: |
2184 \nn{...} |
|
2185 |
|
2186 If $n\ge 2$, these three movie move suffice: |
2183 |
2187 |
2184 \begin{lem} |
2188 \begin{lem} |
2185 Assume $n\ge 2$ and fix $E$ and $E'$ as above. |
2189 Assume $n\ge 2$ and fix $E$ and $E'$ as above. |
2186 The any two sequences of elementary moves connecting $E$ to $E'$ |
2190 The any two sequences of elementary moves connecting $E$ to $E'$ |
2187 are related by a sequence of the two movie moves defined above. |
2191 are related by a sequence of the three movie moves defined above. |
2188 \end{lem} |
2192 \end{lem} |
2189 |
2193 |
2190 \begin{proof} |
2194 \begin{proof} |
2191 (Sketch) |
2195 (Sketch) |
2192 Consider a two parameter family of diffeomorphisms (one parameter family of isotopies) |
2196 Consider a two parameter family of diffeomorphisms (one parameter family of isotopies) |
2193 of $\bd X$. |
2197 of $\bd X$. |
2194 Up to homotopy, |
2198 Up to homotopy, |
2195 such a family is homotopic to a family which can be decomposed |
2199 such a family is homotopic to a family which can be decomposed |
2196 into small families which are either |
2200 into small families which are either |
2197 (a) supported away from $E$, |
2201 (a) supported away from $E$, |
2198 (b) have boundaries corresponding to the two movie moves above. |
2202 (b) have boundaries corresponding to the three movie moves above. |
2199 Finally, observe that the space of $E$'s is simply connected. |
2203 Finally, observe that the space of $E$'s is simply connected. |
2200 (This fails for $n=1$.) |
2204 (This fails for $n=1$.) |
2201 \end{proof} |
2205 \end{proof} |
2202 |
2206 |
2203 For $n=1$ we have to check an additional ``global" relations corresponding to |
2207 For $n=1$ we have to check an additional ``global" relations corresponding to |