30 Before finishing the definition of fields, we give two motivating examples of systems of fields. |
30 Before finishing the definition of fields, we give two motivating examples of systems of fields. |
31 |
31 |
32 \begin{example} |
32 \begin{example} |
33 \label{ex:maps-to-a-space(fields)} |
33 \label{ex:maps-to-a-space(fields)} |
34 Fix a target space $T$, and let $\cC(X)$ be the set of continuous maps |
34 Fix a target space $T$, and let $\cC(X)$ be the set of continuous maps |
35 from X to $B$. |
35 from $X$ to $T$. |
36 \end{example} |
36 \end{example} |
37 |
37 |
38 \begin{example} |
38 \begin{example} |
39 \label{ex:traditional-n-categories(fields)} |
39 \label{ex:traditional-n-categories(fields)} |
40 Fix an $n$-category $C$, and let $\cC(X)$ be |
40 Fix an $n$-category $C$, and let $\cC(X)$ be |
182 above tensor products. |
182 above tensor products. |
183 |
183 |
184 |
184 |
185 \subsection{Systems of fields from $n$-categories} |
185 \subsection{Systems of fields from $n$-categories} |
186 \label{sec:example:traditional-n-categories(fields)} |
186 \label{sec:example:traditional-n-categories(fields)} |
187 We now describe in more detail systems of fields coming from sub-cell-complexes labeled |
187 We now describe in more detail Example \ref{ex:traditional-n-categories(fields)}, systems of fields coming from sub-cell-complexes labeled |
188 by $n$-category morphisms. |
188 by $n$-category morphisms. |
189 |
189 |
190 Given an $n$-category $C$ with the right sort of duality |
190 Given an $n$-category $C$ with the right sort of duality |
191 (e.g. pivotal 2-category, 1-category with duals, star 1-category, disklike $n$-category), |
191 (e.g. a pivotal 2-category, 1-category with duals, star 1-category, disklike $n$-category), |
192 we can construct a system of fields as follows. |
192 we can construct a system of fields as follows. |
193 Roughly speaking, $\cC(X)$ will the set of all embedded cell complexes in $X$ |
193 Roughly speaking, $\cC(X)$ will the set of all embedded cell complexes in $X$ |
194 with codimension $i$ cells labeled by $i$-morphisms of $C$. |
194 with codimension $i$ cells labeled by $i$-morphisms of $C$. |
195 We'll spell this out for $n=1,2$ and then describe the general case. |
195 We'll spell this out for $n=1,2$ and then describe the general case. |
196 |
196 |
197 If $X$ has boundary, we require that the cell decompositions are in general |
197 If $X$ has boundary, we require that the cell decompositions are in general |
198 position with respect to the boundary --- the boundary intersects each cell |
198 position with respect to the boundary --- the boundary intersects each cell |
199 transversely, so cells meeting the boundary are mere half-cells. |
199 transversely, so cells meeting the boundary are mere half-cells. Put another way, the cell decompositions we consider are dual to standard cell |
200 |
|
201 Put another way, the cell decompositions we consider are dual to standard cell |
|
202 decompositions of $X$. |
200 decompositions of $X$. |
203 |
201 |
204 We will always assume that our $n$-categories have linear $n$-morphisms. |
202 We will always assume that our $n$-categories have linear $n$-morphisms. |
205 |
203 |
206 For $n=1$, a field on a 0-manifold $P$ is a labeling of each point of $P$ with |
204 For $n=1$, a field on a 0-manifold $P$ is a labeling of each point of $P$ with |
207 an object (0-morphism) of the 1-category $C$. |
205 an object (0-morphism) of the 1-category $C$. |
208 A field on a 1-manifold $S$ consists of |
206 A field on a 1-manifold $S$ consists of |
209 \begin{itemize} |
207 \begin{itemize} |
210 \item A cell decomposition of $S$ (equivalently, a finite collection |
208 \item a cell decomposition of $S$ (equivalently, a finite collection |
211 of points in the interior of $S$); |
209 of points in the interior of $S$); |
212 \item a labeling of each 1-cell (and each half 1-cell adjacent to $\bd S$) |
210 \item a labeling of each 1-cell (and each half 1-cell adjacent to $\bd S$) |
213 by an object (0-morphism) of $C$; |
211 by an object (0-morphism) of $C$; |
214 \item a transverse orientation of each 0-cell, thought of as a choice of |
212 \item a transverse orientation of each 0-cell, thought of as a choice of |
215 ``domain" and ``range" for the two adjacent 1-cells; and |
213 ``domain" and ``range" for the two adjacent 1-cells; and |
231 A field on a 0-manifold $P$ is a labeling of each point of $P$ with |
229 A field on a 0-manifold $P$ is a labeling of each point of $P$ with |
232 an object of the 2-category $C$. |
230 an object of the 2-category $C$. |
233 A field of a 1-manifold is defined as in the $n=1$ case, using the 0- and 1-morphisms of $C$. |
231 A field of a 1-manifold is defined as in the $n=1$ case, using the 0- and 1-morphisms of $C$. |
234 A field on a 2-manifold $Y$ consists of |
232 A field on a 2-manifold $Y$ consists of |
235 \begin{itemize} |
233 \begin{itemize} |
236 \item A cell decomposition of $Y$ (equivalently, a graph embedded in $Y$ such |
234 \item a cell decomposition of $Y$ (equivalently, a graph embedded in $Y$ such |
237 that each component of the complement is homeomorphic to a disk); |
235 that each component of the complement is homeomorphic to a disk); |
238 \item a labeling of each 2-cell (and each partial 2-cell adjacent to $\bd Y$) |
236 \item a labeling of each 2-cell (and each partial 2-cell adjacent to $\bd Y$) |
239 by a 0-morphism of $C$; |
237 by a 0-morphism of $C$; |
240 \item a transverse orientation of each 1-cell, thought of as a choice of |
238 \item a transverse orientation of each 1-cell, thought of as a choice of |
241 ``domain" and ``range" for the two adjacent 2-cells; |
239 ``domain" and ``range" for the two adjacent 2-cells; |