882 $\cC(D\times W, \cN)$, where in this case $\cN_i$ labels the submanifold |
882 $\cC(D\times W, \cN)$, where in this case $\cN_i$ labels the submanifold |
883 $D\times Y_i \sub \bd(D\times W)$. |
883 $D\times Y_i \sub \bd(D\times W)$. |
884 |
884 |
885 It is not hard to see that the assignment $D \mapsto \cT(W, \cN)(D) \deq \cC(D\times W, \cN)$ |
885 It is not hard to see that the assignment $D \mapsto \cT(W, \cN)(D) \deq \cC(D\times W, \cN)$ |
886 has the structure of an $n{-}k$-category. |
886 has the structure of an $n{-}k$-category. |
887 We will use a simple special case of this construction in the next subsection to define tensor products |
887 |
|
888 \medskip |
|
889 |
|
890 |
|
891 %\subsection{Tensor products} |
|
892 |
|
893 We will use a simple special case of the above |
|
894 construction to define tensor products |
888 of modules. |
895 of modules. |
889 |
896 Let $\cM_1$ and $\cM_2$ be modules for an $n$-category $\cC$. |
890 \subsection{Tensor products} |
|
891 |
|
892 Next we consider tensor products. |
|
893 |
|
894 \nn{what about self tensor products /coends ?} |
|
895 |
|
896 \nn{maybe ``tensor product" is not the best name?} |
|
897 |
|
898 \nn{start with (less general) tensor products; maybe change this later} |
|
899 |
|
900 |
|
901 Let $\cM$ and $\cM'$ be modules for an $n$-category $\cC$. |
|
902 (If $k=1$ and manifolds are oriented, then one should be |
897 (If $k=1$ and manifolds are oriented, then one should be |
903 a left module and the other a right module.) |
898 a left module and the other a right module.) |
904 We will define an $n{-}1$-category $\cM\ot_\cC\cM'$, which depends (functorially) |
899 Choose a 1-ball $J$, and label the two boundary points of $J$ by $\cM_1$ and $\cM_2$. |
905 on a choice of 1-ball (interval) $J$. |
900 Define the tensor product of $\cM_1$ and $\cM_2$ to be the |
906 |
901 $n{-}1$-category $\cT(J, \cM_1, \cM_2)$, |
907 Let $p$ and $p'$ be the boundary points of $J$. |
902 \[ |
908 Given a $k$-ball $X$, let $(X\times J, \cM, \cM')$ denote $X\times J$ with |
903 \cM_1\otimes \cM_2 \deq \cT(J, \cM_1, \cM_2) . |
909 $X\times\{p\}$ labeled by $\cM$ and $X\times\{p'\}$ labeled by $\cM'$, as in Subsection \ref{moddecss}. |
904 \] |
910 Let |
905 This of course depends (functorially) |
911 \[ |
906 on the choice of 1-ball $J$. |
912 \cT(X) \deq \cC(X\times J, \cM, \cM') , |
907 |
913 \] |
908 We will define a more general self tensor product (categorified coend) below. |
914 where the right hand side is the colimit construction defined in Subsection \ref{moddecss}. |
909 |
915 It is not hard to see that $\cT$ becomes an $n{-}1$-category. |
910 |
916 \nn{maybe follows from stuff (not yet written) in previous subsection?} |
911 |
|
912 |
|
913 %\nn{what about self tensor products /coends ?} |
|
914 |
|
915 \nn{maybe ``tensor product" is not the best name?} |
|
916 |
|
917 %\nn{start with (less general) tensor products; maybe change this later} |
917 |
918 |
918 |
919 |
919 |
920 |
920 \subsection{The $n{+}1$-category of sphere modules} |
921 \subsection{The $n{+}1$-category of sphere modules} |
|
922 |
|
923 |
921 |
924 |
922 Outline: |
925 Outline: |
923 \begin{itemize} |
926 \begin{itemize} |
924 \item |
927 \item |
925 \end{itemize} |
928 \end{itemize} |
926 |
929 |
927 |
930 |
|
931 \nn{need to assume a little extra structure to define the top ($n+1$) part (?)} |
928 |
932 |
929 \medskip |
933 \medskip |
930 \hrule |
934 \hrule |
931 \medskip |
935 \medskip |
932 |
936 |