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1 %!TEX root = ../../blob1.tex |
1 %!TEX root = ../../blob1.tex |
2 |
2 |
3 \section{Comparing $n$-category definitions} |
3 \section{Comparing \texorpdfstring{$n$}{n}-category definitions} |
4 \label{sec:comparing-defs} |
4 \label{sec:comparing-defs} |
5 |
5 |
6 In \S\ref{sec:example:traditional-n-categories(fields)} we showed how to construct |
6 In \S\ref{sec:example:traditional-n-categories(fields)} we showed how to construct |
7 a topological $n$-category from a traditional $n$-category; the morphisms of the |
7 a topological $n$-category from a traditional $n$-category; the morphisms of the |
8 topological $n$-category are string diagrams labeled by the traditional $n$-category. |
8 topological $n$-category are string diagrams labeled by the traditional $n$-category. |
20 it falls well short of proving the definitions are equivalent. |
20 it falls well short of proving the definitions are equivalent. |
21 |
21 |
22 %\nn{cases to cover: (a) plain $n$-cats for $n=1,2$; (b) $n$-cat modules for $n=1$, also 2?; |
22 %\nn{cases to cover: (a) plain $n$-cats for $n=1,2$; (b) $n$-cat modules for $n=1$, also 2?; |
23 %(c) $A_\infty$ 1-cat; (b) $A_\infty$ 1-cat module?; (e) tensor products?} |
23 %(c) $A_\infty$ 1-cat; (b) $A_\infty$ 1-cat module?; (e) tensor products?} |
24 |
24 |
25 \subsection{$1$-categories over $\Set$ or $\Vect$} |
25 \subsection{1-categories over \texorpdfstring{$\Set$ or $\Vect$}{Set or Vect}} |
26 \label{ssec:1-cats} |
26 \label{ssec:1-cats} |
27 Given a topological $1$-category $\cX$ we construct a $1$-category in the conventional sense, $c(\cX)$. |
27 Given a topological $1$-category $\cX$ we construct a $1$-category in the conventional sense, $c(\cX)$. |
28 This construction is quite straightforward, but we include the details for the sake of completeness, |
28 This construction is quite straightforward, but we include the details for the sake of completeness, |
29 because it illustrates the role of structures (e.g. orientations, spin structures, etc) |
29 because it illustrates the role of structures (e.g. orientations, spin structures, etc) |
30 on the underlying manifolds, and |
30 on the underlying manifolds, and |
558 \end{figure} |
558 \end{figure} |
559 |
559 |
560 %\nn{need to find a list of axioms for pivotal 2-cats to check} |
560 %\nn{need to find a list of axioms for pivotal 2-cats to check} |
561 |
561 |
562 |
562 |
563 \subsection{$A_\infty$ $1$-categories} |
563 \subsection{\texorpdfstring{$A_\infty$}{A-infinity} 1-categories} |
564 \label{sec:comparing-A-infty} |
564 \label{sec:comparing-A-infty} |
565 In this section, we make contact between the usual definition of an $A_\infty$ category |
565 In this section, we make contact between the usual definition of an $A_\infty$ category |
566 and our definition of a topological $A_\infty$ $1$-category, from \S \ref{ss:n-cat-def}. |
566 and our definition of a topological $A_\infty$ $1$-category, from \S \ref{ss:n-cat-def}. |
567 |
567 |
568 \medskip |
568 \medskip |