equal
deleted
inserted
replaced
1843 \path (\qa:1) node {\color{green!50!brown} $\cA_\n$}; |
1843 \path (\qa:1) node {\color{green!50!brown} $\cA_\n$}; |
1844 \path (\qm+20:2.5) node(M\n) {\color{green!50!brown} $\cM_\n$}; |
1844 \path (\qm+20:2.5) node(M\n) {\color{green!50!brown} $\cM_\n$}; |
1845 \draw[line width=1pt, green!50!brown, ->] (M\n.\qm+135) to[out=\qm+135,in=\qm+90] (\qm+5:1.3); |
1845 \draw[line width=1pt, green!50!brown, ->] (M\n.\qm+135) to[out=\qm+135,in=\qm+90] (\qm+5:1.3); |
1846 } |
1846 } |
1847 \end{tikzpicture} |
1847 \end{tikzpicture} |
1848 \caption{Cone on a marked circle} |
1848 \caption{Cone on a marked circle, the prototypical 1-marked ball} |
1849 \label{feb21d} |
1849 \label{feb21d} |
1850 \end{figure} |
1850 \end{figure} |
1851 |
1851 |
1852 A $n$-category 1-sphere module is, among other things, an $n{-}2$-category $\cD$ with |
1852 A $n$-category 1-sphere module is, among other things, an $n{-}2$-category $\cD$ with |
1853 \[ |
1853 \[ |
2235 |
2235 |
2236 We define product $n{+}1$-morphisms to be identity maps of modules. |
2236 We define product $n{+}1$-morphisms to be identity maps of modules. |
2237 |
2237 |
2238 To define (binary) composition of $n{+}1$-morphisms, choose the obvious common equator |
2238 To define (binary) composition of $n{+}1$-morphisms, choose the obvious common equator |
2239 then compose the module maps. |
2239 then compose the module maps. |
2240 |
2240 Associativity of this composition rules follows from repeated application of the adjoint identity between |
2241 |
2241 the maps of Figures \ref{jun23b} and \ref{jun23c}. |
2242 \nn{still to do: associativity} |
2242 |
|
2243 |
|
2244 %\nn{still to do: associativity} |
2243 |
2245 |
2244 \medskip |
2246 \medskip |
2245 |
2247 |
2246 %\nn{Stuff that remains to be done (either below or in an appendix or in a separate section or in |
2248 %\nn{Stuff that remains to be done (either below or in an appendix or in a separate section or in |
2247 %a separate paper): discuss Morita equivalence; functors} |
2249 %a separate paper): discuss Morita equivalence; functors} |