207 we can construct a system of fields as follows. |
207 we can construct a system of fields as follows. |
208 Roughly speaking, $\cC(X)$ will the set of all embedded cell complexes in $X$ |
208 Roughly speaking, $\cC(X)$ will the set of all embedded cell complexes in $X$ |
209 with codimension $i$ cells labeled by $i$-morphisms of $C$. |
209 with codimension $i$ cells labeled by $i$-morphisms of $C$. |
210 We'll spell this out for $n=1,2$ and then describe the general case. |
210 We'll spell this out for $n=1,2$ and then describe the general case. |
211 |
211 |
|
212 This way of decorating an $n$-manifold with an $n$-category is sometimes referred to |
|
213 as a ``string diagram". |
|
214 It can be thought of as (geometrically) dual to a pasting diagram. |
|
215 One of the advantages of string diagrams over pasting diagrams is that one has more |
|
216 flexibility in slicing them up in various ways. |
|
217 In addition, string diagrams are traditional in quantum topology. |
|
218 The diagrams predate by many years the terms ``string diagram" and ``quantum topology". |
|
219 \nn{?? cite penrose, kauffman, jones(?)} |
|
220 |
212 If $X$ has boundary, we require that the cell decompositions are in general |
221 If $X$ has boundary, we require that the cell decompositions are in general |
213 position with respect to the boundary --- the boundary intersects each cell |
222 position with respect to the boundary --- the boundary intersects each cell |
214 transversely, so cells meeting the boundary are mere half-cells. |
223 transversely, so cells meeting the boundary are mere half-cells. |
215 Put another way, the cell decompositions we consider are dual to standard cell |
224 Put another way, the cell decompositions we consider are dual to standard cell |
216 decompositions of $X$. |
225 decompositions of $X$. |