14 should be equivalent to the fields-and-local-relations definition. |
14 should be equivalent to the fields-and-local-relations definition. |
15 |
15 |
16 A system of fields is very closely related to an $n$-category. |
16 A system of fields is very closely related to an $n$-category. |
17 In one direction, Example \ref{ex:traditional-n-categories(fields)} |
17 In one direction, Example \ref{ex:traditional-n-categories(fields)} |
18 shows how to construct a system of fields from a (traditional) $n$-category. |
18 shows how to construct a system of fields from a (traditional) $n$-category. |
19 We do this in detail for $n=1,2$ (Subsection \ref{sec:example:traditional-n-categories(fields)}) |
19 We do this in detail for $n=1,2$ (\S\ref{sec:example:traditional-n-categories(fields)}) |
20 and more informally for general $n$. |
20 and more informally for general $n$. |
21 In the other direction, |
21 In the other direction, |
22 our preferred definition of an $n$-category in Section \ref{sec:ncats} is essentially |
22 our preferred definition of an $n$-category in \S\ref{sec:ncats} is essentially |
23 just a system of fields restricted to balls of dimensions 0 through $n$; |
23 just a system of fields restricted to balls of dimensions 0 through $n$; |
24 one could call this the ``local" part of a system of fields. |
24 one could call this the ``local" part of a system of fields. |
25 |
25 |
26 Since this section is intended primarily to motivate |
26 Since this section is intended primarily to motivate |
27 the blob complex construction of Section \ref{sec:blob-definition}, |
27 the blob complex construction of \S\ref{sec:blob-definition}, |
28 we suppress some technical details. |
28 we suppress some technical details. |
29 In Section \ref{sec:ncats} the analogous details are treated more carefully. |
29 In \S\ref{sec:ncats} the analogous details are treated more carefully. |
30 |
30 |
31 \medskip |
31 \medskip |
32 |
32 |
33 We only consider compact manifolds, so if $Y \sub X$ is a closed codimension 0 |
33 We only consider compact manifolds, so if $Y \sub X$ is a closed codimension 0 |
34 submanifold of $X$, then $X \setmin Y$ implicitly means the closure |
34 submanifold of $X$, then $X \setmin Y$ implicitly means the closure |
69 One can think of such embedded cell complexes as dual to pasting diagrams for $C$. |
69 One can think of such embedded cell complexes as dual to pasting diagrams for $C$. |
70 This is described in more detail in \S \ref{sec:example:traditional-n-categories(fields)}. |
70 This is described in more detail in \S \ref{sec:example:traditional-n-categories(fields)}. |
71 \end{example} |
71 \end{example} |
72 |
72 |
73 Now for the rest of the definition of system of fields. |
73 Now for the rest of the definition of system of fields. |
74 (Readers desiring a more precise definition should refer to Subsection \ref{ss:n-cat-def} |
74 (Readers desiring a more precise definition should refer to \S\ref{ss:n-cat-def} |
75 and replace $k$-balls with $k$-manifolds.) |
75 and replace $k$-balls with $k$-manifolds.) |
76 \begin{enumerate} |
76 \begin{enumerate} |
77 \item There are boundary restriction maps $\cC_k(X) \to \cC_{k-1}(\bd X)$, |
77 \item There are boundary restriction maps $\cC_k(X) \to \cC_{k-1}(\bd X)$, |
78 and these maps comprise a natural |
78 and these maps comprise a natural |
79 transformation between the functors $\cC_k$ and $\cC_{k-1}\circ\bd$. |
79 transformation between the functors $\cC_k$ and $\cC_{k-1}\circ\bd$. |