1753 |
1753 |
1754 |
1754 |
1755 \subsection{The \texorpdfstring{$n{+}1$}{n+1}-category of sphere modules} |
1755 \subsection{The \texorpdfstring{$n{+}1$}{n+1}-category of sphere modules} |
1756 \label{ssec:spherecat} |
1756 \label{ssec:spherecat} |
1757 |
1757 |
1758 In this subsection we define $n{+}1$-categories $\cS$ of ``sphere modules" |
1758 In this subsection we define $n{+}1$-categories $\cS$ of ``sphere modules". |
1759 whose objects are $n$-categories. |
1759 The objects are $n$-categories, the $k$-morphisms are $k{-}1$-sphere modules for $1\le k \le n$, |
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1760 and the $n{+}1$-morphisms are intertwinors. |
1760 With future applications in mind, we treat simultaneously the big category |
1761 With future applications in mind, we treat simultaneously the big category |
1761 of all $n$-categories and all sphere modules and also subcategories thereof. |
1762 of all $n$-categories and all sphere modules and also subcategories thereof. |
1762 When $n=1$ this is closely related to familiar $2$-categories consisting of |
1763 When $n=1$ this is closely related to familiar $2$-categories consisting of |
1763 algebras, bimodules and intertwiners (or a subcategory of that). |
1764 algebras, bimodules and intertwiners (or a subcategory of that). |
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1765 The sphere module $n{+}1$-category is a natural generalization of the |
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1766 algebra-bimodule-intertwinor 2-category to higher dimensions. |
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1767 |
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1768 Another possible name for this $n{+}1$-category is $n{+}1$-category of defects. |
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1769 The $n$-categories are thought of as representing field theories, and the |
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1770 $0$-sphere modules are codimension 1 defects between adjacent theories. |
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1771 In general, $m$-sphere modules are codimension $m{+}1$ defects; |
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1772 the link of such a defect is an $m$-sphere decorated with defects of smaller codimension. |
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1773 |
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1774 \medskip |
1764 |
1775 |
1765 While it is appropriate to call an $S^0$ module a bimodule, |
1776 While it is appropriate to call an $S^0$ module a bimodule, |
1766 this is much less true for higher dimensional spheres, |
1777 this is much less true for higher dimensional spheres, |
1767 so we prefer the term ``sphere module" for the general case. |
1778 so we prefer the term ``sphere module" for the general case. |
1768 |
|
1769 %The results of this subsection are not needed for the rest of the paper, |
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1770 %so we will skimp on details in a couple of places. We have included this mostly |
|
1771 %for the sake of comparing our notion of a disk-like $n$-category to other definitions. |
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1772 |
1779 |
1773 For simplicity, we will assume that $n$-categories are enriched over $\c$-vector spaces. |
1780 For simplicity, we will assume that $n$-categories are enriched over $\c$-vector spaces. |
1774 |
1781 |
1775 The $0$- through $n$-dimensional parts of $\cS$ are various sorts of modules, and we describe |
1782 The $0$- through $n$-dimensional parts of $\cS$ are various sorts of modules, and we describe |
1776 these first. |
1783 these first. |
1781 that our $n$-categories and modules have non-degenerate inner products. |
1788 that our $n$-categories and modules have non-degenerate inner products. |
1782 (In other words, we need to assume some extra duality on the $n$-categories and modules.) |
1789 (In other words, we need to assume some extra duality on the $n$-categories and modules.) |
1783 |
1790 |
1784 \medskip |
1791 \medskip |
1785 |
1792 |
1786 Our first task is to define an $n$-category $m$-sphere module, for $0\le m \le n-1$. |
1793 Our first task is to define an $n$-category $m$-sphere modules, for $0\le m \le n-1$. |
1787 These will be defined in terms of certain classes of marked balls, very similarly |
1794 These will be defined in terms of certain classes of marked balls, very similarly |
1788 to the definition of $n$-category modules above. |
1795 to the definition of $n$-category modules above. |
1789 (This, in turn, is very similar to our definition of $n$-category.) |
1796 (This, in turn, is very similar to our definition of $n$-category.) |
1790 Because of this similarity, we only sketch the definitions below. |
1797 Because of this similarity, we only sketch the definitions below. |
1791 |
1798 |
1812 We can also take the boundary of a $0$-marked ball, which is $0$-marked sphere. |
1819 We can also take the boundary of a $0$-marked ball, which is $0$-marked sphere. |
1813 |
1820 |
1814 Fix $n$-categories $\cA$ and $\cB$. |
1821 Fix $n$-categories $\cA$ and $\cB$. |
1815 These will label the two halves of a $0$-marked $k$-ball. |
1822 These will label the two halves of a $0$-marked $k$-ball. |
1816 |
1823 |
1817 An $n$-category $0$-sphere module $\cM$ over the $n$-categories $\cA$ and $\cB$ is a collection of functors $\cM_k$ from the category |
1824 An $n$-category $0$-sphere module $\cM$ over the $n$-categories $\cA$ and $\cB$ is |
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1825 a collection of functors $\cM_k$ from the category |
1818 of $0$-marked $k$-balls, $1\le k \le n$, |
1826 of $0$-marked $k$-balls, $1\le k \le n$, |
1819 (with the two halves labeled by $\cA$ and $\cB$) to the category of sets. |
1827 (with the two halves labeled by $\cA$ and $\cB$) to the category of sets. |
1820 If $k=n$ these sets should be enriched to the extent $\cA$ and $\cB$ are. |
1828 If $k=n$ these sets should be enriched to the extent $\cA$ and $\cB$ are. |
1821 Given a decomposition of a $0$-marked $k$-ball $X$ into smaller balls $X_i$, we have |
1829 Given a decomposition of a $0$-marked $k$-ball $X$ into smaller balls $X_i$, we have |
1822 morphism sets $\cA_k(X_i)$ (if $X_i$ lies on the $\cA$-labeled side) |
1830 morphism sets $\cA_k(X_i)$ (if $X_i$ lies on the $\cA$-labeled side) |
2022 \medskip |
2030 \medskip |
2023 |
2031 |
2024 Next we define the $n{+}1$-morphisms of $\cS$. |
2032 Next we define the $n{+}1$-morphisms of $\cS$. |
2025 The construction of the 0- through $n$-morphisms was easy and tautological, but the |
2033 The construction of the 0- through $n$-morphisms was easy and tautological, but the |
2026 $n{+}1$-morphisms will require some effort and combinatorial topology, as well as additional |
2034 $n{+}1$-morphisms will require some effort and combinatorial topology, as well as additional |
2027 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. |
2035 duality assumptions on the lower morphisms. |
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2036 These are required because we define the spaces of $n{+}1$-morphisms by |
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2037 making arbitrary choices of incoming and outgoing boundaries for each $n$-ball. |
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2038 The additional duality assumptions are needed to prove independence of our definition form these choices. |
2028 |
2039 |
2029 Let $X$ be an $n{+}1$-ball, and let $c$ be a decoration of its boundary |
2040 Let $X$ be an $n{+}1$-ball, and let $c$ be a decoration of its boundary |
2030 by a cell complex labeled by 0- through $n$-morphisms, as above. |
2041 by a cell complex labeled by 0- through $n$-morphisms, as above. |
2031 Choose an $n{-}1$-sphere $E\sub \bd X$ which divides |
2042 Choose an $n{-}1$-sphere $E\sub \bd X$ which divides |
2032 $\bd X$ into ``incoming" and ``outgoing" boundary $\bd_-X$ and $\bd_+X$. |
2043 $\bd X$ into ``incoming" and ``outgoing" boundary $\bd_-X$ and $\bd_+X$. |