254 In the presence of strong duality, these $k$ distinct compositions are subsumed into |
254 In the presence of strong duality, these $k$ distinct compositions are subsumed into |
255 one general type of composition which can be in any direction. |
255 one general type of composition which can be in any direction. |
256 |
256 |
257 \begin{axiom}[Composition] |
257 \begin{axiom}[Composition] |
258 \label{axiom:composition} |
258 \label{axiom:composition} |
259 Let $B = B_1 \cup_Y B_2$, where $B$, $B_1$ and $B_2$ are $k$-balls ($0\le k\le n$) |
259 Let $B = B_1 \cup_Y B_2$, where $B$, $B_1$ and $B_2$ are $k$-balls ($1\le k\le n$) |
260 and $Y = B_1\cap B_2$ is a $k{-}1$-ball (Figure \ref{blah5}). |
260 and $Y = B_1\cap B_2$ is a $k{-}1$-ball (Figure \ref{blah5}). |
261 Let $E = \bd Y$, which is a $k{-}2$-sphere. |
261 Let $E = \bd Y$, which is a $k{-}2$-sphere. |
262 Note that each of $B$, $B_1$ and $B_2$ has its boundary split into two $k{-}1$-balls by $E$. |
262 Note that each of $B$, $B_1$ and $B_2$ has its boundary split into two $k{-}1$-balls by $E$. |
263 We have restriction (domain or range) maps $\cC(B_i)\trans E \to \cC(Y)$. |
263 We have restriction (domain or range) maps $\cC(B_i)\trans E \to \cC(Y)$. |
264 Let $\cC(B_1)\trans E \times_{\cC(Y)} \cC(B_2)\trans E$ denote the fibered product of these two maps. |
264 Let $\cC(B_1)\trans E \times_{\cC(Y)} \cC(B_2)\trans E$ denote the fibered product of these two maps. |
1569 $\cl\cC(N_0) \to \cl\cC(\bd M_1)$. |
1569 $\cl\cC(N_0) \to \cl\cC(\bd M_1)$. |
1570 The second condition is that the image of $\psi_{\cC;W}(x)$ in $\cl\cC(\bd M_1)$ is splittable |
1570 The second condition is that the image of $\psi_{\cC;W}(x)$ in $\cl\cC(\bd M_1)$ is splittable |
1571 along $\bd Y_1$ and $\bd Y'_1$, and that the restrictions to $\cl\cC(Y_1)$ and $\cl\cC(Y'_1)$ agree |
1571 along $\bd Y_1$ and $\bd Y'_1$, and that the restrictions to $\cl\cC(Y_1)$ and $\cl\cC(Y'_1)$ agree |
1572 (with respect to the identification of $Y_1$ and $Y'_1$ provided by the gluing map). |
1572 (with respect to the identification of $Y_1$ and $Y'_1$ provided by the gluing map). |
1573 The $i$-th condition is defined similarly. |
1573 The $i$-th condition is defined similarly. |
1574 Note that these conditions depend on the boundaries of elements of $\prod_a \cC(X_a)$. |
1574 Note that these conditions depend only on the boundaries of elements of $\prod_a \cC(X_a)$. |
1575 |
1575 |
1576 We define $\psi_{\cC;W}(x)$ to be the subset of $\prod_a \cC(X_a)$ which satisfies the |
1576 We define $\psi_{\cC;W}(x)$ to be the subset of $\prod_a \cC(X_a)$ which satisfies the |
1577 above conditions for all $i$ and also all |
1577 above conditions for all $i$ and also all |
1578 ball decompositions compatible with $x$. |
1578 ball decompositions compatible with $x$. |
1579 (If $x$ is a nice, non-pathological cell decomposition, then it is easy to see that gluing |
1579 (If $x$ is a nice, non-pathological cell decomposition, then it is easy to see that gluing |
1727 Eilenberg-Zilber type subdivision argument. |
1727 Eilenberg-Zilber type subdivision argument. |
1728 |
1728 |
1729 \medskip |
1729 \medskip |
1730 |
1730 |
1731 $\cl{\cC}(W)$ is functorial with respect to homeomorphisms of $k$-manifolds. |
1731 $\cl{\cC}(W)$ is functorial with respect to homeomorphisms of $k$-manifolds. |
1732 Restricting the $k$-spheres, we have now proved Lemma \ref{lem:spheres}. |
1732 Restricting to $k$-spheres, we have now proved Lemma \ref{lem:spheres}. |
1733 |
1733 |
1734 \begin{lem} |
1734 \begin{lem} |
1735 \label{lem:colim-injective} |
1735 \label{lem:colim-injective} |
1736 Let $W$ be a manifold of dimension less than $n$. Then for each |
1736 Let $W$ be a manifold of dimension $j<n$. Then for each |
1737 decomposition $x$ of $W$ the natural map $\psi_{\cC;W}(x)\to \cl{\cC}(W)$ is injective. |
1737 decomposition $x$ of $W$ the natural map $\psi_{\cC;W}(x)\to \cl{\cC}(W)$ is injective. |
1738 \end{lem} |
1738 \end{lem} |
1739 \begin{proof} |
1739 \begin{proof} |
1740 $\cl{\cC}(W)$ is a colimit of a diagram of sets, and each of the arrows in the diagram is |
1740 $\cl{\cC}(W)$ is a colimit of a diagram of sets, and each of the arrows in the diagram is |
1741 injective. |
1741 injective. |
1794 The definition will be very similar to that of $n$-categories, |
1794 The definition will be very similar to that of $n$-categories, |
1795 but with $k$-balls replaced by {\it marked $k$-balls,} defined below. |
1795 but with $k$-balls replaced by {\it marked $k$-balls,} defined below. |
1796 |
1796 |
1797 Our motivating example comes from an $(m{-}n{+}1)$-dimensional manifold $W$ with boundary |
1797 Our motivating example comes from an $(m{-}n{+}1)$-dimensional manifold $W$ with boundary |
1798 in the context of an $m{+}1$-dimensional TQFT. |
1798 in the context of an $m{+}1$-dimensional TQFT. |
1799 Such a $W$ gives rise to a module for the $n$-category associated to $\bd W$. |
1799 Such a $W$ gives rise to a module for the $n$-category associated to $\bd W$ (see Example \ref{ex:ncats-from-tqfts}). |
1800 This will be explained in more detail as we present the axioms. |
1800 This will be explained in more detail as we present the axioms. |
1801 |
1801 |
1802 Throughout, we fix an $n$-category $\cC$. |
1802 Throughout, we fix an $n$-category $\cC$. |
1803 For all but one axiom, it doesn't matter whether $\cC$ is an ordinary $n$-category or an $A_\infty$ $n$-category. |
1803 For all but one axiom, it doesn't matter whether $\cC$ is an ordinary $n$-category or an $A_\infty$ $n$-category. |
1804 We state the final axiom, regarding actions of homeomorphisms, differently in the two cases. |
1804 We state the final axiom, regarding actions of homeomorphisms, differently in the two cases. |
1816 \end{module-axiom} |
1816 \end{module-axiom} |
1817 |
1817 |
1818 (As with $n$-categories, we will usually omit the subscript $k$.) |
1818 (As with $n$-categories, we will usually omit the subscript $k$.) |
1819 |
1819 |
1820 For example, let $\cD$ be the TQFT which assigns to a $k$-manifold $N$ the set |
1820 For example, let $\cD$ be the TQFT which assigns to a $k$-manifold $N$ the set |
1821 of maps from $N$ to $T$ (for $k\le m$), modulo homotopy (and possibly linearized) if $k=m$. |
1821 of maps from $N$ to $T$ (for $k\le m$), modulo homotopy (and possibly linearized) if $k=m$ |
|
1822 (see Example \ref{ex:maps-with-fiber}). |
1822 Let $W$ be an $(m{-}n{+}1)$-dimensional manifold with boundary. |
1823 Let $W$ be an $(m{-}n{+}1)$-dimensional manifold with boundary. |
1823 Let $\cC$ be the $n$-category with $\cC(X) \deq \cD(X\times \bd W)$. |
1824 Let $\cC$ be the $n$-category with $\cC(X) \deq \cD(X\times \bd W)$. |
1824 Let $\cM(B, N) \deq \cD((B\times \bd W)\cup (N\times W))$ |
1825 Let $\cM(B, N) \deq \cD((B\times \bd W)\cup (N\times W))$. |
1825 (see Example \ref{ex:maps-with-fiber}). |
|
1826 (The union is along $N\times \bd W$.) |
1826 (The union is along $N\times \bd W$.) |
|
1827 See Figure \ref{blah15}. |
1827 %(If $\cD$ were a general TQFT, we would define $\cM(B, N)$ to be |
1828 %(If $\cD$ were a general TQFT, we would define $\cM(B, N)$ to be |
1828 %the subset of $\cD((B\times \bd W)\cup (N\times W))$ which is splittable along $N\times \bd W$.) |
1829 %the subset of $\cD((B\times \bd W)\cup (N\times W))$ which is splittable along $N\times \bd W$.) |
1829 |
1830 |
1830 \begin{figure}[t] |
1831 \begin{figure}[t] |
1831 $$\mathfig{.55}{ncat/boundary-collar}$$ |
1832 $$\mathfig{.55}{ncat/boundary-collar}$$ |
1837 We call it a hemisphere instead of a ball because it plays a role analogous |
1838 We call it a hemisphere instead of a ball because it plays a role analogous |
1838 to the $k{-}1$-spheres in the $n$-category definition.) |
1839 to the $k{-}1$-spheres in the $n$-category definition.) |
1839 |
1840 |
1840 \begin{lem} |
1841 \begin{lem} |
1841 \label{lem:hemispheres} |
1842 \label{lem:hemispheres} |
1842 {For each $0 \le k \le n-1$, we have a functor $\cl\cM_k$ from |
1843 {For each $1 \le k \le n$, we have a functor $\cl\cM_{k-1}$ from |
1843 the category of marked $k$-hemispheres and |
1844 the category of marked $k$-hemispheres and |
1844 homeomorphisms to the category of sets and bijections.} |
1845 homeomorphisms to the category of sets and bijections.} |
1845 \end{lem} |
1846 \end{lem} |
1846 The proof is exactly analogous to that of Lemma \ref{lem:spheres}, and we omit the details. |
1847 The proof is exactly analogous to that of Lemma \ref{lem:spheres}, and we omit the details. |
1847 We use the same type of colimit construction. |
1848 We use the same type of colimit construction. |
1868 \[ |
1869 \[ |
1869 \gl_E : \cM(M_1) \times_{\cl\cM(E)} \cM(M_2) \hookrightarrow \cl\cM(H) |
1870 \gl_E : \cM(M_1) \times_{\cl\cM(E)} \cM(M_2) \hookrightarrow \cl\cM(H) |
1870 \] |
1871 \] |
1871 which is natural with respect to the actions of homeomorphisms.} |
1872 which is natural with respect to the actions of homeomorphisms.} |
1872 \end{lem} |
1873 \end{lem} |
1873 Again, this is in exact analogy with Lemma \ref{lem:domain-and-range}, and illustrated in Figure \ref{fig:module-boundary}. |
1874 This is in exact analogy with Lemma \ref{lem:domain-and-range}, and illustrated in Figure \ref{fig:module-boundary}. |
1874 \begin{figure}[t] |
1875 \begin{figure}[t] |
1875 \begin{equation*} |
1876 \begin{equation*} |
1876 \begin{tikzpicture}[baseline=0] |
1877 \begin{tikzpicture}[baseline=0] |
1877 \coordinate (a) at (0,1); |
1878 \coordinate (a) at (0,1); |
1878 \coordinate (b) at (4,1); |
1879 \coordinate (b) at (4,1); |
1940 \end{figure} |
1941 \end{figure} |
1941 |
1942 |
1942 First, we can compose two module morphisms to get another module morphism. |
1943 First, we can compose two module morphisms to get another module morphism. |
1943 |
1944 |
1944 \begin{module-axiom}[Module composition] |
1945 \begin{module-axiom}[Module composition] |
1945 {Let $M = M_1 \cup_Y M_2$, where $M$, $M_1$ and $M_2$ are marked $k$-balls (with $0\le k\le n$) |
1946 {Let $M = M_1 \cup_Y M_2$, where $M$, $M_1$ and $M_2$ are marked $k$-balls (with $2\le k\le n$) |
1946 and $Y = M_1\cap M_2$ is a marked $k{-}1$-ball. |
1947 and $Y = M_1\cap M_2$ is a marked $k{-}1$-ball. |
1947 Let $E = \bd Y$, which is a marked $k{-}2$-hemisphere. |
1948 Let $E = \bd Y$, which is a marked $k{-}2$-hemisphere. |
1948 Note that each of $M$, $M_1$ and $M_2$ has its boundary split into two marked $k{-}1$-balls by $E$. |
1949 Note that each of $M$, $M_1$ and $M_2$ has its boundary split into two marked $k{-}1$-balls by $E$. |
1949 We have restriction (domain or range) maps $\cM(M_i)\trans E \to \cM(Y)$. |
1950 We have restriction (domain or range) maps $\cM(M_i)\trans E \to \cM(Y)$. |
1950 Let $\cM(M_1) \trans E \times_{\cM(Y)} \cM(M_2) \trans E$ denote the fibered product of these two maps. |
1951 Let $\cM(M_1) \trans E \times_{\cM(Y)} \cM(M_2) \trans E$ denote the fibered product of these two maps. |
1961 Second, we can compose an $n$-category morphism with a module morphism to get another |
1962 Second, we can compose an $n$-category morphism with a module morphism to get another |
1962 module morphism. |
1963 module morphism. |
1963 We'll call this the action map to distinguish it from the other kind of composition. |
1964 We'll call this the action map to distinguish it from the other kind of composition. |
1964 |
1965 |
1965 \begin{module-axiom}[$n$-category action] |
1966 \begin{module-axiom}[$n$-category action] |
1966 {Let $M = X \cup_Y M'$, where $M$ and $M'$ are marked $k$-balls ($0\le k\le n$), |
1967 {Let $M = X \cup_Y M'$, where $M$ and $M'$ are marked $k$-balls ($1\le k\le n$), |
1967 $X$ is a plain $k$-ball, |
1968 $X$ is a plain $k$-ball, |
1968 and $Y = X\cap M'$ is a $k{-}1$-ball. |
1969 and $Y = X\cap M'$ is a $k{-}1$-ball. |
1969 Let $E = \bd Y$, which is a $k{-}2$-sphere. |
1970 Let $E = \bd Y$, which is a $k{-}2$-sphere. |
1970 We have restriction maps $\cM(M') \trans E \to \cC(Y)$ and $\cC(X) \trans E\to \cC(Y)$. |
1971 We have restriction maps $\cM(M') \trans E \to \cC(Y)$ and $\cC(X) \trans E\to \cC(Y)$. |
1971 Let $\cC(X)\trans E \times_{\cC(Y)} \cM(M') \trans E$ denote the fibered product of these two maps. |
1972 Let $\cC(X)\trans E \times_{\cC(Y)} \cM(M') \trans E$ denote the fibered product of these two maps. |
2151 Note that there are two cases: |
2152 Note that there are two cases: |
2152 the collar could intersect the marking of the marked ball $M$, in which case |
2153 the collar could intersect the marking of the marked ball $M$, in which case |
2153 we use a product on a morphism of $\cM$; or the collar could be disjoint from the marking, |
2154 we use a product on a morphism of $\cM$; or the collar could be disjoint from the marking, |
2154 in which case we use a product on a morphism of $\cC$. |
2155 in which case we use a product on a morphism of $\cC$. |
2155 |
2156 |
2156 In our example, elements $a$ of $\cM(M)$ maps to $T$, and $\pi^*(a)$ is the pullback of |
2157 In our example, elements $a$ of $\cM(M)$ are maps to $T$, and $\pi^*(a)$ is the pullback of |
2157 $a$ along a map associated to $\pi$. |
2158 $a$ along the map associated to $\pi$. |
2158 |
2159 |
2159 \medskip |
2160 \medskip |
2160 |
2161 |
2161 %There are two alternatives for the next axiom, according to whether we are defining |
2162 %There are two alternatives for the next axiom, according to whether we are defining |
2162 %modules for ordinary $n$-categories or $A_\infty$ $n$-categories. |
2163 %modules for ordinary $n$-categories or $A_\infty$ $n$-categories. |