1141 (and their boundaries), while for fields we consider all manifolds. |
1141 (and their boundaries), while for fields we consider all manifolds. |
1142 Second, in the category definition we directly impose isotopy |
1142 Second, in the category definition we directly impose isotopy |
1143 invariance in dimension $n$, while in the fields definition we |
1143 invariance in dimension $n$, while in the fields definition we |
1144 instead remember a subspace of local relations which contain differences of isotopic fields. |
1144 instead remember a subspace of local relations which contain differences of isotopic fields. |
1145 (Recall that the compensation for this complication is that we can demand that the gluing map for fields is injective.) |
1145 (Recall that the compensation for this complication is that we can demand that the gluing map for fields is injective.) |
1146 Thus a \nn{lemma-ize} system of fields and local relations $(\cF,U)$ determines an $n$-category $\cC_ {\cF,U}$ simply by restricting our attention to |
1146 Thus |
|
1147 \begin{lem} |
|
1148 \label{lem:ncat-from-fields} |
|
1149 A system of fields and local relations $(\cF,U)$ determines an $n$-category $\cC_ {\cF,U}$ simply by restricting our attention to |
1147 balls and, at level $n$, quotienting out by the local relations: |
1150 balls and, at level $n$, quotienting out by the local relations: |
1148 \begin{align*} |
1151 \begin{align*} |
1149 \cC_{\cF,U}(B^k) & = \begin{cases}\cF(B) & \text{when $k<n$,} \\ \cF(B) / U(B) & \text{when $k=n$.}\end{cases} |
1152 \cC_{\cF,U}(B^k) & = \begin{cases}\cF(B) & \text{when $k<n$,} \\ \cF(B) / U(B) & \text{when $k=n$.}\end{cases} |
1150 \end{align*} |
1153 \end{align*} |
|
1154 \end{lem} |
1151 This $n$-category can be thought of as the local part of the fields. |
1155 This $n$-category can be thought of as the local part of the fields. |
1152 Conversely, given a disk-like $n$-category we can construct a system of fields via |
1156 Conversely, given a disk-like $n$-category we can construct a system of fields via |
1153 a colimit construction; see \S \ref{ss:ncat_fields} below. |
1157 a colimit construction; see \S \ref{ss:ncat_fields} below. |
1154 |
1158 |
1155 \medskip |
1159 \medskip |
1248 If $X$ is a $k$-ball with $k<j$, let $\cF(W)(X) \deq \cF(W\times X)$. |
1252 If $X$ is a $k$-ball with $k<j$, let $\cF(W)(X) \deq \cF(W\times X)$. |
1249 If $X$ is a $j$-ball and $c\in \cl{\cF(W)}(\bd X)$, |
1253 If $X$ is a $j$-ball and $c\in \cl{\cF(W)}(\bd X)$, |
1250 let $\cF(W)(X; c) \deq A_\cF(W\times X; c)$. |
1254 let $\cF(W)(X; c) \deq A_\cF(W\times X; c)$. |
1251 \end{example} |
1255 \end{example} |
1252 |
1256 |
|
1257 This last example generalizes Lemma \ref{lem:ncat-from-fields} above which produced an $n$-category from an $n$-dimensional system of fields and local relations. Taking $W$ to be the point recovers that statement. |
|
1258 |
1253 The next example is only intended to be illustrative, as we don't specify |
1259 The next example is only intended to be illustrative, as we don't specify |
1254 which definition of a ``traditional $n$-category" we intend. |
1260 which definition of a ``traditional $n$-category" we intend. |
1255 Further, most of these definitions don't even have an agreed-upon notion of |
1261 Further, most of these definitions don't even have an agreed-upon notion of |
1256 ``strong duality", which we assume here. |
1262 ``strong duality", which we assume here. |
1257 \begin{example}[Traditional $n$-categories] |
1263 \begin{example}[Traditional $n$-categories] |
1320 we get an $A_\infty$ $n$-category enriched over spaces. |
1326 we get an $A_\infty$ $n$-category enriched over spaces. |
1321 \end{example} |
1327 \end{example} |
1322 |
1328 |
1323 See also Theorem \ref{thm:map-recon} below, recovering $C_*(\Maps(M \to T))$ up to |
1329 See also Theorem \ref{thm:map-recon} below, recovering $C_*(\Maps(M \to T))$ up to |
1324 homotopy as the blob complex of $M$ with coefficients in $\pi^\infty_{\le n}(T)$. |
1330 homotopy as the blob complex of $M$ with coefficients in $\pi^\infty_{\le n}(T)$. |
|
1331 |
|
1332 Instead of using the TQFT invariant $\cA$ as in Example \ref{ex:ncats-from-tqfts} above, we can turn an $n$-dimensional system of fields and local relations into an $A_\infty$ $n$-category using the blob complex. With a codimension $k$ fiber, we obtain an $A_\infty$ $k$-category: |
1325 |
1333 |
1326 \begin{example}[Blob complexes of balls (with a fiber)] |
1334 \begin{example}[Blob complexes of balls (with a fiber)] |
1327 \rm |
1335 \rm |
1328 \label{ex:blob-complexes-of-balls} |
1336 \label{ex:blob-complexes-of-balls} |
1329 Fix an $n{-}k$-dimensional manifold $F$ and an $n$-dimensional system of fields $\cE$. |
1337 Fix an $n{-}k$-dimensional manifold $F$ and an $n$-dimensional system of fields $\cE$. |