46 \end{example} |
46 \end{example} |
47 |
47 |
48 \begin{example} |
48 \begin{example} |
49 \label{ex:traditional-n-categories(fields)} |
49 \label{ex:traditional-n-categories(fields)} |
50 Fix an $n$-category $C$, and let $\cC(X)$ be |
50 Fix an $n$-category $C$, and let $\cC(X)$ be |
51 the set of sub-cell-complexes of $X$ with codimension-$j$ cells labeled by |
51 the set of embedded cell complexes in $X$ with codimension-$j$ cells labeled by |
52 $j$-morphisms of $C$. |
52 $j$-morphisms of $C$. |
53 One can think of such sub-cell-complexes as dual to pasting diagrams for $C$. |
53 One can think of such embedded cell complexes as dual to pasting diagrams for $C$. |
54 This is described in more detail in \S \ref{sec:example:traditional-n-categories(fields)}. |
54 This is described in more detail in \S \ref{sec:example:traditional-n-categories(fields)}. |
55 \end{example} |
55 \end{example} |
56 |
56 |
57 Now for the rest of the definition of system of fields. |
57 Now for the rest of the definition of system of fields. |
58 (Readers desiring a more precise definition should refer to Subsection \ref{ss:n-cat-def} |
58 (Readers desiring a more precise definition should refer to Subsection \ref{ss:n-cat-def} |
197 |
197 |
198 |
198 |
199 \subsection{Systems of fields from $n$-categories} |
199 \subsection{Systems of fields from $n$-categories} |
200 \label{sec:example:traditional-n-categories(fields)} |
200 \label{sec:example:traditional-n-categories(fields)} |
201 We now describe in more detail Example \ref{ex:traditional-n-categories(fields)}, |
201 We now describe in more detail Example \ref{ex:traditional-n-categories(fields)}, |
202 systems of fields coming from sub-cell-complexes labeled |
202 systems of fields coming from embedded cell complexes labeled |
203 by $n$-category morphisms. |
203 by $n$-category morphisms. |
204 |
204 |
205 Given an $n$-category $C$ with the right sort of duality |
205 Given an $n$-category $C$ with the right sort of duality |
206 (e.g. a pivotal 2-category, *-1-category), |
206 (e.g. a pivotal 2-category, *-1-category), |
207 we can construct a system of fields as follows. |
207 we can construct a system of fields as follows. |
306 A {\it local relation} is a collection subspaces $U(B; c) \sub \lf(B; c)$, |
306 A {\it local relation} is a collection subspaces $U(B; c) \sub \lf(B; c)$, |
307 for all $n$-manifolds $B$ which are |
307 for all $n$-manifolds $B$ which are |
308 homeomorphic to the standard $n$-ball and all $c \in \cC(\bd B)$, |
308 homeomorphic to the standard $n$-ball and all $c \in \cC(\bd B)$, |
309 satisfying the following properties. |
309 satisfying the following properties. |
310 \begin{enumerate} |
310 \begin{enumerate} |
311 \item functoriality: |
311 \item Functoriality: |
312 $f(U(B; c)) = U(B', f(c))$ for all homeomorphisms $f: B \to B'$ |
312 $f(U(B; c)) = U(B', f(c))$ for all homeomorphisms $f: B \to B'$ |
313 \item local relations imply extended isotopy: |
313 \item Local relations imply extended isotopy: |
314 if $x, y \in \cC(B; c)$ and $x$ is extended isotopic |
314 if $x, y \in \cC(B; c)$ and $x$ is extended isotopic |
315 to $y$, then $x-y \in U(B; c)$. |
315 to $y$, then $x-y \in U(B; c)$. |
316 \item ideal with respect to gluing: |
316 \item Ideal with respect to gluing: |
317 if $B = B' \cup B''$, $x\in U(B')$, and $c\in \cC(B'')$, then $x\bullet r \in U(B)$ |
317 if $B = B' \cup B''$, $x\in U(B')$, and $c\in \cC(B'')$, then $x\bullet r \in U(B)$ |
318 \end{enumerate} |
318 \end{enumerate} |
319 \end{defn} |
319 \end{defn} |
320 See \cite{kw:tqft} for further details. |
320 See \cite{kw:tqft} for further details. |
321 |
321 |