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785 of marked 1-balls, call them left-marked and right-marked, |
785 of marked 1-balls, call them left-marked and right-marked, |
786 and hence there are two types of modules, call them right modules and left modules. |
786 and hence there are two types of modules, call them right modules and left modules. |
787 In all other cases ($k>1$ or unoriented or $\text{Pin}_\pm$), |
787 In all other cases ($k>1$ or unoriented or $\text{Pin}_\pm$), |
788 there is no left/right module distinction. |
788 there is no left/right module distinction. |
789 |
789 |
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790 \medskip |
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791 |
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792 Examples of modules: |
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793 \begin{itemize} |
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794 \item |
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795 \end{itemize} |
790 |
796 |
791 \subsection{Modules as boundary labels} |
797 \subsection{Modules as boundary labels} |
792 \label{moddecss} |
798 \label{moddecss} |
793 |
799 |
794 Let $\cC$ be an [$A_\infty$] $n$-category, let $W$ be a $k$-manifold ($k\le n$), |
800 Let $\cC$ be an [$A_\infty$] $n$-category, let $W$ be a $k$-manifold ($k\le n$), |
892 vs traditional defs, $n = 1,2$, $A_\infty$ or not, cats, modules, tensor products |
898 vs traditional defs, $n = 1,2$, $A_\infty$ or not, cats, modules, tensor products |
893 \item traditional $A_\infty$ 1-cat def implies our def |
899 \item traditional $A_\infty$ 1-cat def implies our def |
894 \item ... and vice-versa (already done in appendix) |
900 \item ... and vice-versa (already done in appendix) |
895 \item say something about unoriented vs oriented vs spin vs pin for $n=1$ (and $n=2$?) |
901 \item say something about unoriented vs oriented vs spin vs pin for $n=1$ (and $n=2$?) |
896 \item spell out what difference (if any) Top vs PL vs Smooth makes |
902 \item spell out what difference (if any) Top vs PL vs Smooth makes |
897 \item explain relation between old-fashioned blob homology and new-fangled blob homology |
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898 (follows as special case of product formula (product with a point)). |
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899 \item define $n{+}1$-cat of $n$-cats (a.k.a.\ $n{+}1$-category of generalized bimodules |
903 \item define $n{+}1$-cat of $n$-cats (a.k.a.\ $n{+}1$-category of generalized bimodules |
900 a.k.a.\ $n{+}1$-category of sphere modules); discuss Morita equivalence |
904 a.k.a.\ $n{+}1$-category of sphere modules); discuss Morita equivalence |
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905 \item morphisms of modules; show that it's adjoint to tensor product |
901 \end{itemize} |
906 \end{itemize} |
902 |
907 |
903 |
908 |