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deleted
inserted
replaced
95 Morphisms are modeled on balls, so their boundaries are modeled on spheres. |
95 Morphisms are modeled on balls, so their boundaries are modeled on spheres. |
96 In other words, we need to extend the functors $\cC_{k-1}$ from balls to spheres, for |
96 In other words, we need to extend the functors $\cC_{k-1}$ from balls to spheres, for |
97 $1\le k \le n$. |
97 $1\le k \le n$. |
98 At first it might seem that we need another axiom for this, but in fact once we have |
98 At first it might seem that we need another axiom for this, but in fact once we have |
99 all the axioms in this subsection for $0$ through $k-1$ we can use a colimit |
99 all the axioms in this subsection for $0$ through $k-1$ we can use a colimit |
100 construction, as described in Subsection \ref{ss:ncat-coend} below, to extend $\cC_{k-1}$ |
100 construction, as described in \S\ref{ss:ncat-coend} below, to extend $\cC_{k-1}$ |
101 to spheres (and any other manifolds): |
101 to spheres (and any other manifolds): |
102 |
102 |
103 \begin{lem} |
103 \begin{lem} |
104 \label{lem:spheres} |
104 \label{lem:spheres} |
105 For each $1 \le k \le n$, we have a functor $\cl{\cC}_{k-1}$ from |
105 For each $1 \le k \le n$, we have a functor $\cl{\cC}_{k-1}$ from |
744 More generally, start with an $n{+}m$-category $C$ and a closed $m$-manifold $F$. |
744 More generally, start with an $n{+}m$-category $C$ and a closed $m$-manifold $F$. |
745 Define $\cC(X)$, for $\dim(X) < n$, |
745 Define $\cC(X)$, for $\dim(X) < n$, |
746 to be the set of all $C$-labeled embedded cell complexes of $X\times F$. |
746 to be the set of all $C$-labeled embedded cell complexes of $X\times F$. |
747 Define $\cC(X; c)$, for $X$ an $n$-ball, |
747 Define $\cC(X; c)$, for $X$ an $n$-ball, |
748 to be the dual Hilbert space $A(X\times F; c)$. |
748 to be the dual Hilbert space $A(X\times F; c)$. |
749 (See Subsection \ref{sec:constructing-a-tqft}.) |
749 (See \S\ref{sec:constructing-a-tqft}.) |
750 \end{example} |
750 \end{example} |
751 |
751 |
752 \noop{ |
752 \noop{ |
753 \nn{shouldn't this go elsewhere? we haven't yet discussed constructing a system of fields from |
753 \nn{shouldn't this go elsewhere? we haven't yet discussed constructing a system of fields from |
754 an n-cat} |
754 an n-cat} |
1506 |
1506 |
1507 \subsection{Morphisms of $A_\infty$ $1$-category modules} |
1507 \subsection{Morphisms of $A_\infty$ $1$-category modules} |
1508 \label{ss:module-morphisms} |
1508 \label{ss:module-morphisms} |
1509 |
1509 |
1510 In order to state and prove our version of the higher dimensional Deligne conjecture |
1510 In order to state and prove our version of the higher dimensional Deligne conjecture |
1511 (Section \ref{sec:deligne}), |
1511 (\S\ref{sec:deligne}), |
1512 we need to define morphisms of $A_\infty$ $1$-category modules and establish |
1512 we need to define morphisms of $A_\infty$ $1$-category modules and establish |
1513 some of their elementary properties. |
1513 some of their elementary properties. |
1514 |
1514 |
1515 To motivate the definitions which follow, consider algebras $A$ and $B$, |
1515 To motivate the definitions which follow, consider algebras $A$ and $B$, |
1516 right modules $X_B$ and $Z_A$ and a bimodule $\leftidx{_B}{Y}{_A}$, and the familiar adjunction |
1516 right modules $X_B$ and $Z_A$ and a bimodule $\leftidx{_B}{Y}{_A}$, and the familiar adjunction |
1875 |
1875 |
1876 More generally, consider an interval with interior marked points, and with the complements |
1876 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 |
1877 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 $\cM_i$. |
1878 by $\cA_i$-$\cA_{i+1}$ bimodules $\cM_i$. |
1879 (See Figure \ref{feb21c}.) |
1879 (See Figure \ref{feb21c}.) |
1880 To this data we can apply the coend construction as in Subsection \ref{moddecss} above |
1880 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. |
1881 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. |
1882 This amounts to a definition of taking tensor products of $0$-sphere module over $n$-categories. |
1883 |
1883 |
1884 \begin{figure}[!ht] |
1884 \begin{figure}[!ht] |
1885 $$ |
1885 $$ |