1679 \label{ssec:spherecat} |
1679 \label{ssec:spherecat} |
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 closely related to the familiar $2$-category of algebras, bimodules and intertwiners. |
1685 It is clearly appropriate to call an $S^0$ module a bimodule, |
1685 While it is appropriate to call an $S^0$ module a bimodule, |
1686 but this is much less true for higher dimensional spheres, |
1686 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 |
1688 |
1689 The results of this subsection are not needed for the rest of the paper, |
1689 The results of this subsection are not needed for the rest of the paper, |
1690 so we will skimp on details in a couple of places. |
1690 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. |
1691 |
1691 |
1692 For simplicity, we will assume that $n$-categories are enriched over $\c$-vector spaces. |
1692 For simplicity, we will assume that $n$-categories are enriched over $\c$-vector spaces. |
1693 |
1693 |
1694 The $0$- through $n$-dimensional parts of $\cS$ are various sorts of modules, and we describe |
1694 The $0$- through $n$-dimensional parts of $\cS$ are various sorts of modules, and we describe |
1695 these first. |
1695 these first. |
1696 The $n{+}1$-dimensional part of $\cS$ consists of intertwiners |
1696 The $n{+}1$-dimensional part of $\cS$ consists of intertwiners |
1697 of (garden-variety) $1$-category modules associated to decorated $n$-balls. |
1697 of $1$-category modules associated to decorated $n$-balls. |
1698 We will see below that in order for these $n{+}1$-morphisms to satisfy all of |
1698 We will see below that in order for these $n{+}1$-morphisms to satisfy all of |
1699 the duality requirements of an $n{+}1$-category, we will have to assume |
1699 the axioms of an $n{+}1$-category (in particular, duality requirements), we will have to assume |
1700 that our $n$-categories and modules have non-degenerate inner products. |
1700 that our $n$-categories and modules have non-degenerate inner products. |
1701 (In other words, we need to assume some extra duality on the $n$-categories and modules.) |
1701 (In other words, we need to assume some extra duality on the $n$-categories and modules.) |
1702 |
1702 |
1703 \medskip |
1703 \medskip |
1704 |
1704 |
1708 (This, in turn, is very similar to our definition of $n$-category.) |
1708 (This, in turn, is very similar to our definition of $n$-category.) |
1709 Because of this similarity, we only sketch the definitions below. |
1709 Because of this similarity, we only sketch the definitions below. |
1710 |
1710 |
1711 We start with $0$-sphere modules, which also could reasonably be called (categorified) bimodules. |
1711 We start with $0$-sphere modules, which also could reasonably be called (categorified) bimodules. |
1712 (For $n=1$ they are precisely bimodules in the usual, uncategorified sense.) |
1712 (For $n=1$ they are precisely bimodules in the usual, uncategorified sense.) |
1713 Define a $0$-marked $k$-ball $(X, M)$, $1\le k \le n$, to be a pair homeomorphic to the standard |
1713 Define a $0$-marked $k$-ball, $1\le k \le n$, to be a pair $(X, M)$ homeomorphic to the standard |
1714 $(B^k, B^{k-1})$. |
1714 $(B^k, B^{k-1})$. |
1715 See Figure \ref{feb21a}. |
1715 See Figure \ref{feb21a}. |
1716 Another way to say this is that $(X, M)$ is homeomorphic to $B^{k-1}\times([-1,1], \{0\})$. |
1716 Another way to say this is that $(X, M)$ is homeomorphic to $B^{k-1}\times([-1,1], \{0\})$. |
1717 |
1717 |
1718 \begin{figure}[!ht] |
1718 \begin{figure}[!ht] |
1727 or plain (don't intersect the $0$-marking of the large ball). |
1727 or plain (don't intersect the $0$-marking of the large ball). |
1728 We can also take the boundary of a $0$-marked ball, which is $0$-marked sphere. |
1728 We can also take the boundary of a $0$-marked ball, which is $0$-marked sphere. |
1729 |
1729 |
1730 Fix $n$-categories $\cA$ and $\cB$. |
1730 Fix $n$-categories $\cA$ and $\cB$. |
1731 These will label the two halves of a $0$-marked $k$-ball. |
1731 These will label the two halves of a $0$-marked $k$-ball. |
1732 The $0$-sphere module we define next will depend on $\cA$ and $\cB$ |
1732 |
1733 (it's an $\cA$-$\cB$ bimodule), but we will suppress that from the notation. |
1733 An $n$-category $0$-sphere module $\cM$ over the $n$-categories $\cA$ and $\cB$ is a collection of functors $\cM_k$ from the category |
1734 |
|
1735 An $n$-category $0$-sphere module $\cM$ is a collection of functors $\cM_k$ from the category |
|
1736 of $0$-marked $k$-balls, $1\le k \le n$, |
1734 of $0$-marked $k$-balls, $1\le k \le n$, |
1737 (with the two halves labeled by $\cA$ and $\cB$) to the category of sets. |
1735 (with the two halves labeled by $\cA$ and $\cB$) to the category of sets. |
1738 If $k=n$ these sets should be enriched to the extent $\cA$ and $\cB$ are. |
1736 If $k=n$ these sets should be enriched to the extent $\cA$ and $\cB$ are. |
1739 Given a decomposition of a $0$-marked $k$-ball $X$ into smaller balls $X_i$, we have |
1737 Given a decomposition of a $0$-marked $k$-ball $X$ into smaller balls $X_i$, we have |
1740 morphism sets $\cA_k(X_i)$ (if $X_i$ lies on the $\cA$-labeled side) |
1738 morphism sets $\cA_k(X_i)$ (if $X_i$ lies on the $\cA$-labeled side) |
1741 or $\cB_k(X_i)$ (if $X_i$ lies on the $\cB$-labeled side) |
1739 or $\cB_k(X_i)$ (if $X_i$ lies on the $\cB$-labeled side) |
1742 or $\cM_k(X_i)$ (if $X_i$ intersects the marking and is therefore a smaller 0-marked ball). |
1740 or $\cM_k(X_i)$ (if $X_i$ intersects the marking and is therefore a smaller 0-marked ball). |
1743 Corresponding to this decomposition we have an action and/or composition map |
1741 Corresponding to this decomposition we have a composition (or `gluing') map |
1744 from the product of these various sets into $\cM_k(X)$. |
1742 from the product (fibered over the boundary data) of these various sets into $\cM_k(X)$. |
1745 |
1743 |
1746 \medskip |
1744 \medskip |
1747 |
1745 |
1748 Part of the structure of an $n$-category 0-sphere module $\cM$ is captured by saying it is |
1746 Part of the structure of an $n$-category 0-sphere module $\cM$ is captured by saying it is |
1749 a collection $\cD^{ab}$ of $n{-}1$-categories, indexed by pairs $(a, b)$ of objects (0-morphisms) |
1747 a collection $\cD^{ab}$ of $n{-}1$-categories, indexed by pairs $(a, b)$ of objects (0-morphisms) |
1832 Let $C(S)$ denote the cone of $S$, a marked 2-ball (Figure \ref{feb21d}). |
1830 Let $C(S)$ denote the cone of $S$, a marked 2-ball (Figure \ref{feb21d}). |
1833 \nn{I need to make up my mind whether marked things are always labeled too. |
1831 \nn{I need to make up my mind whether marked things are always labeled too. |
1834 For the time being, let's say they are.} |
1832 For the time being, let's say they are.} |
1835 A 1-marked $k$-ball is anything homeomorphic to $B^j \times C(S)$, $0\le j\le n-2$, |
1833 A 1-marked $k$-ball is anything homeomorphic to $B^j \times C(S)$, $0\le j\le n-2$, |
1836 where $B^j$ is the standard $j$-ball. |
1834 where $B^j$ is the standard $j$-ball. |
1837 1-marked $k$-balls can be decomposed in various ways into smaller balls, which are either |
1835 A 1-marked $k$-balls can be decomposed in various ways into smaller balls, which are either |
1838 smaller 1-marked $k$-balls or the product of an unmarked ball with a marked interval. |
1836 smaller 1-marked $k$-balls or the product of an unmarked ball with a marked interval. \todo{I'm confused by this last sentence. By `the product of an unmarked ball with a marked internal', you mean a 0-marked $k$-ball, right? If so, we should say it that way. Further, there are also just some entirely unmarked balls. -S} |
1839 We now proceed as in the above module definitions. |
1837 We now proceed as in the above module definitions. |
1840 |
1838 |
1841 \begin{figure}[!ht] |
1839 \begin{figure}[!ht] |
1842 $$ |
1840 $$ |
1843 \begin{tikzpicture}[baseline,line width = 2pt] |
1841 \begin{tikzpicture}[baseline,line width = 2pt] |
1867 (e.g.\ 2-ball or 2-sphere) marked by an embedded 1-complex $K$. |
1865 (e.g.\ 2-ball or 2-sphere) marked by an embedded 1-complex $K$. |
1868 The components of $Y\setminus K$ are labeled by $n$-categories, |
1866 The components of $Y\setminus K$ are labeled by $n$-categories, |
1869 the edges of $K$ are labeled by 0-sphere modules, |
1867 the edges of $K$ are labeled by 0-sphere modules, |
1870 and the 0-cells of $K$ are labeled by 1-sphere modules. |
1868 and the 0-cells of $K$ are labeled by 1-sphere modules. |
1871 We can now apply the coend construction and obtain an $n{-}2$-category. |
1869 We can now apply the coend construction and obtain an $n{-}2$-category. |
1872 If $Y$ has boundary then this $n{-}2$-category is a module for the $n{-}1$-manifold |
1870 If $Y$ has boundary then this $n{-}2$-category is a module for the $n{-}1$-category |
1873 associated to the (marked, labeled) boundary of $Y$. |
1871 associated to the (marked, labeled) boundary of $Y$. |
1874 In particular, if $\bd Y$ is a 1-sphere then we get a 1-sphere module as defined above. |
1872 In particular, if $\bd Y$ is a 1-sphere then we get a 1-sphere module as defined above. |
1875 |
1873 |
1876 \medskip |
1874 \medskip |
1877 |
1875 |
1880 and a 2-sphere module is a representation of such an $n{-}2$-category. |
1878 and a 2-sphere module is a representation of such an $n{-}2$-category. |
1881 |
1879 |
1882 \medskip |
1880 \medskip |
1883 |
1881 |
1884 We can now define the $n$-or-less-dimensional part of our $n{+}1$-category $\cS$. |
1882 We can now define the $n$-or-less-dimensional part of our $n{+}1$-category $\cS$. |
1885 Choose some collection of $n$-categories, then choose some collections of bimodules for |
1883 Choose some collection of $n$-categories, then choose some collections of bimodules between |
1886 these $n$-categories, then choose some collection of 1-sphere modules for the various |
1884 these $n$-categories, then choose some collection of 1-sphere modules for the various |
1887 possible marked 1-spheres labeled by the $n$-categories and bimodules, and so on. |
1885 possible marked 1-spheres labeled by the $n$-categories and bimodules, and so on. |
1888 Let $L_i$ denote the collection of $i{-}1$-sphere modules we have chosen. |
1886 Let $L_i$ denote the collection of $i{-}1$-sphere modules we have chosen. |
1889 (For convenience, we declare a $(-1)$-sphere module to be an $n$-category.) |
1887 (For convenience, we declare a $(-1)$-sphere module to be an $n$-category.) |
1890 There is a wide range of possibilities. |
1888 There is a wide range of possibilities. |
1891 $L_0$ could contain infinitely many $n$-categories or just one. |
1889 The set $L_0$ could contain infinitely many $n$-categories or just one. |
1892 For each pair of $n$-categories in $L_0$, $L_1$ could contain no bimodules at all or |
1890 For each pair of $n$-categories in $L_0$, $L_1$ could contain no bimodules at all or |
1893 it could contain several. |
1891 it could contain several. |
1894 The only requirement is that each $k$-sphere module be a module for a $k$-sphere $n{-}k$-category |
1892 The only requirement is that each $k$-sphere module be a module for a $k$-sphere $n{-}k$-category |
1895 constructed out of labels taken from $L_j$ for $j<k$. |
1893 constructed out of labels taken from $L_j$ for $j<k$. |
1896 |
1894 |
1897 We now define $\cS(X)$, for $X$ of dimension at most $n$, to be the set of all |
1895 We now define $\cS(X)$, for $X$ a ball of dimension at most $n$, to be the set of all |
1898 cell-complexes $K$ embedded in $X$, with the codimension-$j$ parts of $(X, K)$ labeled |
1896 cell-complexes $K$ embedded in $X$, with the codimension-$j$ parts of $(X, K)$ labeled |
1899 by elements of $L_j$. |
1897 by elements of $L_j$. |
1900 As described above, we can think of each decorated $k$-ball as defining a $k{-}1$-sphere module |
1898 As described above, we can think of each decorated $k$-ball as defining a $k{-}1$-sphere module |
1901 for the $n{-}k{+}1$-category associated to its decorated boundary. |
1899 for the $n{-}k{+}1$-category associated to its decorated boundary. |
1902 Thus the $k$-morphisms of $\cS$ (for $k\le n$) can be thought |
1900 Thus the $k$-morphisms of $\cS$ (for $k\le n$) can be thought |
1903 of as $n$-category $k{-}1$-sphere modules |
1901 of as $n$-category $k{-}1$-sphere modules |
1904 (generalizations of bimodules). |
1902 (generalizations of bimodules). |
1905 On the other hand, we can equally well think of the $k$-morphisms as decorations on $k$-balls, |
1903 On the other hand, we can equally well think of the $k$-morphisms as decorations on $k$-balls, |
1906 and from this (official) point of view it is clear that they satisfy all of the axioms of an |
1904 and from this point of view it is clear that they satisfy all of the axioms of an |
1907 $n{+}1$-category. |
1905 $n{+}1$-category. |
1908 (All of the axioms for the less-than-$n{+}1$-dimensional part of an $n{+}1$-category, that is.) |
1906 (All of the axioms for the less-than-$n{+}1$-dimensional part of an $n{+}1$-category, that is.) |
1909 |
1907 |
1910 \medskip |
1908 \medskip |
1911 |
1909 |
1912 Next we define the $n{+}1$-morphisms of $\cS$. |
1910 Next we define the $n{+}1$-morphisms of $\cS$. |
1913 The construction of the 0- through $n$-morphisms was easy and tautological, but the |
1911 The construction of the 0- through $n$-morphisms was easy and tautological, but the |
1914 $n{+}1$-morphisms will require a bit of combinatorial topology effort, as well as addition |
1912 $n{+}1$-morphisms will require some effort and combinatorial topology, as well as additional |
1915 duality assumptions on the lower morphisms. |
1913 duality assumptions on the lower morphisms. These are required because we define the spaces of $n{+}1$-morphisms by making arbitrary choices of incoming and outgoing boundaries for each $n$-ball. The additional duality assumptions are needed to prove independence of our definition form these choices. |
1916 |
1914 |
1917 Let $X$ be an $n{+}1$-ball, and let $c$ be a decoration of its boundary |
1915 Let $X$ be an $n{+}1$-ball, and let $c$ be a decoration of its boundary |
1918 by a cell complex labeled by 0- through $n$-morphisms, as above. |
1916 by a cell complex labeled by 0- through $n$-morphisms, as above. |
1919 Choose an $n{-}1$-sphere $E\sub \bd X$ which divides |
1917 Choose an $n{-}1$-sphere $E\sub \bd X$ which divides |
1920 $\bd X$ into ``incoming" and ``outgoing" boundary $\bd_-X$ and $\bd_+X$. |
1918 $\bd X$ into ``incoming" and ``outgoing" boundary $\bd_-X$ and $\bd_+X$. |
1924 Define |
1922 Define |
1925 \[ |
1923 \[ |
1926 \cS(X; c; E) \deq \hom_{\cS(E_c)}(\cS(\bd_-X_c), \cS(\bd_+X_c)) . |
1924 \cS(X; c; E) \deq \hom_{\cS(E_c)}(\cS(\bd_-X_c), \cS(\bd_+X_c)) . |
1927 \] |
1925 \] |
1928 |
1926 |
1929 We will show that if the sphere modules are equipped with a compatible family of |
1927 We will show that if the sphere modules are equipped with a `compatible family of |
1930 non-degenerate inner products, then there is a coherent family of isomorphisms |
1928 non-degenerate inner products', then there is a coherent family of isomorphisms |
1931 $\cS(X; c; E) \cong \cS(X; c; E')$ for all pairs of choices $E$ and $E'$. |
1929 $\cS(X; c; E) \cong \cS(X; c; E')$ for all pairs of choices $E$ and $E'$. |
1932 This will allow us to define $\cS(X; e)$ independently of the choice of $E$. |
1930 This will allow us to define $\cS(X; e)$ independently of the choice of $E$. |
1933 |
1931 |
1934 First we must define ``inner product", ``non-degenerate" and ``compatible". |
1932 First we must define ``inner product", ``non-degenerate" and ``compatible". |
1935 Let $Y$ be a decorated $n$-ball, and $\ol{Y}$ it's mirror image. |
1933 Let $Y$ be a decorated $n$-ball, and $\ol{Y}$ it's mirror image. |
1960 (One can think of these inner products as giving some duality in dimension $n{+}1$; |
1958 (One can think of these inner products as giving some duality in dimension $n{+}1$; |
1961 heretofore we have only assumed duality in dimensions 0 through $n$.) |
1959 heretofore we have only assumed duality in dimensions 0 through $n$.) |
1962 |
1960 |
1963 Next we define compatibility. |
1961 Next we define compatibility. |
1964 Let $Y = Y_1\cup Y_2$ with $D = Y_1\cap Y_2$. |
1962 Let $Y = Y_1\cup Y_2$ with $D = Y_1\cap Y_2$. |
1965 Let $X_1$ and $X_2$ be the two components of $Y\times I$ (pinched) cut along |
1963 Let $X_1$ and $X_2$ be the two components of $Y\times I$ cut along |
1966 $D\times I$. |
1964 $D\times I$, in both cases using the pinched product. |
1967 (Here we are overloading notation and letting $D$ denote both a decorated and an undecorated |
1965 (Here we are overloading notation and letting $D$ denote both a decorated and an undecorated |
1968 manifold.) |
1966 manifold.) |
1969 We have $\bd X_i = Y_i \cup \ol{Y}_i \cup (D\times I)$ |
1967 We have $\bd X_i = Y_i \cup \ol{Y}_i \cup (D\times I)$ |
1970 (see Figure \ref{jun23a}). |
1968 (see Figure \ref{jun23a}). |
1971 \begin{figure}[t] |
1969 \begin{figure}[t] |