275 Put another way, the cell decompositions we consider are dual to standard cell

   275 Put another way, the cell decompositions we consider are dual to standard cell

   276 decompositions of $X$.

   276 decompositions of $X$.

   277 

   277 

   278 We will always assume that our $n$-categories have linear $n$-morphisms.

   278 We will always assume that our $n$-categories have linear $n$-morphisms.

   279 

   279 

   284 For $n=1$, a field on a 0-manifold $P$ is a labeling of each point of $P$ with

   280 For $n=1$, a field on a 0-manifold $P$ is a labeling of each point of $P$ with

   285 an object (0-morphism) of the 1-category $C$.

   281 an object (0-morphism) of the 1-category $C$.

   286 A field on a 1-manifold $S$ consists of

   282 A field on a 1-manifold $S$ consists of

   287 \begin{itemize}

   283 \begin{itemize}

   288     \item a cell decomposition of $S$ (equivalently, a finite collection

   284     \item a cell decomposition of $S$ (equivalently, a finite collection

   354     \item a labeling of each $j$-cell by a $(k-j)$-dimensional morphism of $C$, with

   350     \item a labeling of each $j$-cell by a $(k-j)$-dimensional morphism of $C$, with

   355 domain and range determined by the labelings of the link of $j$-cell.

   351 domain and range determined by the labelings of the link of $j$-cell.

   356 \end{itemize}

   352 \end{itemize}

   357 

   353 

   358 

   354 

   359 

   362 

   360 \subsection{Local relations}

   363 \subsection{Local relations}

   361 \label{sec:local-relations}

   364 \label{sec:local-relations}

   362 

   365 

   363 For convenience we assume that fields are enriched over Vect.

   366 For convenience we assume that fields are enriched over Vect.