1273 \end{example} |
1273 \end{example} |
1274 |
1274 |
1275 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. |
1275 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. |
1276 |
1276 |
1277 The next example is only intended to be illustrative, as we don't specify |
1277 The next example is only intended to be illustrative, as we don't specify |
1278 which definition of a ``traditional $n$-category" we intend. |
1278 which definition of a ``traditional $n$-category with strong duality" we intend. |
1279 Further, most of these definitions don't even have an agreed-upon notion of |
1279 %Further, most of these definitions don't even have an agreed-upon notion of |
1280 ``strong duality", which we assume here. |
1280 %``strong duality", which we assume here. |
1281 \begin{example}[Traditional $n$-categories] |
1281 \begin{example}[Traditional $n$-categories] |
1282 \rm |
1282 \rm |
1283 \label{ex:traditional-n-categories} |
1283 \label{ex:traditional-n-categories} |
1284 Given a ``traditional $n$-category with strong duality" $C$ |
1284 Given a ``traditional $n$-category with strong duality" $C$ |
1285 define $\cC(X)$, for $X$ a $k$-ball with $k < n$, |
1285 define $\cC(X)$, for $X$ a $k$-ball with $k < n$, |
1366 We think of this as providing a ``free resolution" |
1366 We think of this as providing a ``free resolution" |
1367 of the ordinary $n$-category. |
1367 of the ordinary $n$-category. |
1368 %\nn{say something about cofibrant replacements?} |
1368 %\nn{say something about cofibrant replacements?} |
1369 In fact, there is also a trivial, but mostly uninteresting, way to do this: |
1369 In fact, there is also a trivial, but mostly uninteresting, way to do this: |
1370 we can think of each vector space associated to an $n$-ball as a chain complex concentrated in degree $0$, |
1370 we can think of each vector space associated to an $n$-ball as a chain complex concentrated in degree $0$, |
1371 and take $\CD{B}$ to act trivially. |
1371 and let $\CH{B}$ act trivially. |
1372 |
1372 |
1373 Beware that the ``free resolution" of the ordinary $n$-category $\pi_{\leq n}(T)$ |
1373 Beware that the ``free resolution" of the ordinary $n$-category $\pi_{\leq n}(T)$ |
1374 is not the $A_\infty$ $n$-category $\pi^\infty_{\leq n}(T)$. |
1374 is not the $A_\infty$ $n$-category $\pi^\infty_{\leq n}(T)$. |
1375 It's easy to see that with $n=0$, the corresponding system of fields is just |
1375 It's easy to see that with $n=0$, the corresponding system of fields is just |
1376 linear combinations of connected components of $T$, and the local relations are trivial. |
1376 linear combinations of connected components of $T$, and the local relations are trivial. |