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3 \section{Comparing $n$-category definitions} |
3 \section{Comparing $n$-category definitions} |
4 \label{sec:comparing-defs} |
4 \label{sec:comparing-defs} |
5 |
5 |
6 In this appendix we relate the ``topological" category definitions of Section \ref{sec:ncats} |
6 In this appendix we relate the ``topological" category definitions of Section \ref{sec:ncats} |
7 to more traditional definitions, for $n=1$ and 2. |
7 to more traditional definitions, for $n=1$ and 2. |
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8 |
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9 \nn{cases to cover: (a) plain $n$-cats for $n=1,2$; (b) $n$-cat modules for $n=1$, also 2?; |
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10 (c) $A_\infty$ 1-cat; (b) $A_\infty$ 1-cat module?; (e) tensor products?} |
8 |
11 |
9 \subsection{$1$-categories over $\Set$ or $\Vect$} |
12 \subsection{$1$-categories over $\Set$ or $\Vect$} |
10 \label{ssec:1-cats} |
13 \label{ssec:1-cats} |
11 Given a topological $1$-category $\cX$ we construct a $1$-category in the conventional sense, $c(\cX)$. |
14 Given a topological $1$-category $\cX$ we construct a $1$-category in the conventional sense, $c(\cX)$. |
12 This construction is quite straightforward, but we include the details for the sake of completeness, because it illustrates the role of structures (e.g. orientations, spin structures, etc) on the underlying manifolds, and |
15 This construction is quite straightforward, but we include the details for the sake of completeness, because it illustrates the role of structures (e.g. orientations, spin structures, etc) on the underlying manifolds, and |