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672 Taking singular chains converts such a space type $A_\infty$ $n$-category into a chain complex |
672 Taking singular chains converts such a space type $A_\infty$ $n$-category into a chain complex |
673 type $A_\infty$ $n$-category. |
673 type $A_\infty$ $n$-category. |
674 |
674 |
675 \medskip |
675 \medskip |
676 |
676 |
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677 We define a $j$ times monoidal $n$-category to be an $(n{+}j)$-category $\cC$ where |
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678 $\cC(X)$ is a trivial 1-element set if $X$ is a $k$-ball with $k<j$. |
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679 See Example \ref{ex:bord-cat}. |
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680 |
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681 \medskip |
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682 |
677 The alert reader will have already noticed that our definition of a (ordinary) $n$-category |
683 The alert reader will have already noticed that our definition of a (ordinary) $n$-category |
678 is extremely similar to our definition of a system of fields. |
684 is extremely similar to our definition of a system of fields. |
679 There are two differences. |
685 There are two differences. |
680 First, for the $n$-category definition we restrict our attention to balls |
686 First, for the $n$-category definition we restrict our attention to balls |
681 (and their boundaries), while for fields we consider all manifolds. |
687 (and their boundaries), while for fields we consider all manifolds. |
803 to be the set of all $C$-labeled embedded cell complexes of $X\times F$. |
809 to be the set of all $C$-labeled embedded cell complexes of $X\times F$. |
804 Define $\cC(X; c)$, for $X$ an $n$-ball, |
810 Define $\cC(X; c)$, for $X$ an $n$-ball, |
805 to be the dual Hilbert space $A(X\times F; c)$. |
811 to be the dual Hilbert space $A(X\times F; c)$. |
806 (See \S\ref{sec:constructing-a-tqft}.) |
812 (See \S\ref{sec:constructing-a-tqft}.) |
807 \end{example} |
813 \end{example} |
808 |
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809 \noop{ |
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810 \nn{shouldn't this go elsewhere? we haven't yet discussed constructing a system of fields from |
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811 an n-cat} |
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812 Recall we described a system of fields and local relations based on a ``traditional $n$-category" |
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813 $C$ in Example \ref{ex:traditional-n-categories(fields)} above. |
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814 \nn{KW: We already refer to \S \ref{sec:fields} above} |
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815 Constructing a system of fields from $\cC$ recovers that example. |
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816 \todo{Except that it doesn't: pasting diagrams v.s. string diagrams.} |
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817 \nn{KW: but the above example is all about string diagrams. the only difference is at the top level, |
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818 where the quotient is built in. |
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819 but (string diagrams)/(relations) is isomorphic to |
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820 (pasting diagrams composed of smaller string diagrams)/(relations)} |
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821 } |
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822 |
814 |
823 |
815 |
824 \begin{example}[The bordism $n$-category of $d$-manifolds, ordinary version] |
816 \begin{example}[The bordism $n$-category of $d$-manifolds, ordinary version] |
825 \label{ex:bord-cat} |
817 \label{ex:bord-cat} |
826 \rm |
818 \rm |