1169 \begin{example}[Blob complexes of balls (with a fiber)] |
1169 \begin{example}[Blob complexes of balls (with a fiber)] |
1170 \rm |
1170 \rm |
1171 \label{ex:blob-complexes-of-balls} |
1171 \label{ex:blob-complexes-of-balls} |
1172 Fix an $n{-}k$-dimensional manifold $F$ and an $n$-dimensional system of fields $\cE$. |
1172 Fix an $n{-}k$-dimensional manifold $F$ and an $n$-dimensional system of fields $\cE$. |
1173 We will define an $A_\infty$ disk-like $k$-category $\cC$. |
1173 We will define an $A_\infty$ disk-like $k$-category $\cC$. |
1174 When $X$ is a $m$-ball, with $m<k$, define $\cC(X) = \cE(X\times F)$. |
1174 When $X$ is an $m$-ball, with $m<k$, define $\cC(X) = \cE(X\times F)$. |
1175 When $X$ is an $k$-ball, |
1175 When $X$ is a $k$-ball, |
1176 define $\cC(X; c) = \bc^\cE_*(X\times F; c)$ |
1176 define $\cC(X; c) = \bc^\cE_*(X\times F; c)$ |
1177 where $\bc^\cE_*$ denotes the blob complex based on $\cE$. |
1177 where $\bc^\cE_*$ denotes the blob complex based on $\cE$. |
1178 \end{example} |
1178 \end{example} |
1179 |
1179 |
1180 This example will be used in Theorem \ref{thm:product} below, which allows us to compute the blob complex of a product. |
1180 This example will be used in Theorem \ref{thm:product} below, which allows us to compute the blob complex of a product. |
1281 |
1281 |
1282 In the case of ordinary disk-like $n$-categories, this construction factors into a construction of a |
1282 In the case of ordinary disk-like $n$-categories, this construction factors into a construction of a |
1283 system of fields and local relations, followed by the usual TQFT definition of a |
1283 system of fields and local relations, followed by the usual TQFT definition of a |
1284 vector space invariant of manifolds given as Definition \ref{defn:TQFT-invariant}. |
1284 vector space invariant of manifolds given as Definition \ref{defn:TQFT-invariant}. |
1285 For an $A_\infty$ disk-like $n$-category, $\cl{\cC}$ is defined using a homotopy colimit instead. |
1285 For an $A_\infty$ disk-like $n$-category, $\cl{\cC}$ is defined using a homotopy colimit instead. |
1286 Recall that we can take a ordinary disk-like $n$-category $\cC$ and pass to the ``free resolution", |
1286 Recall that we can take an ordinary disk-like $n$-category $\cC$ and pass to the ``free resolution", |
1287 an $A_\infty$ disk-like $n$-category $\bc_*(\cC)$, by computing the blob complex of balls |
1287 an $A_\infty$ disk-like $n$-category $\bc_*(\cC)$, by computing the blob complex of balls |
1288 (recall Example \ref{ex:blob-complexes-of-balls} above). |
1288 (recall Example \ref{ex:blob-complexes-of-balls} above). |
1289 We will show in Corollary \ref{cor:new-old} below that the homotopy colimit invariant |
1289 We will show in Corollary \ref{cor:new-old} below that the homotopy colimit invariant |
1290 for a manifold $M$ associated to this $A_\infty$ disk-like $n$-category is actually the |
1290 for a manifold $M$ associated to this $A_\infty$ disk-like $n$-category is actually the |
1291 same as the original blob complex for $M$ with coefficients in $\cC$. |
1291 same as the original blob complex for $M$ with coefficients in $\cC$. |