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1224 \label{ex:e-n-alg} |
1224 \label{ex:e-n-alg} |
1225 Let $A$ be an $\cE\cB_n$-algebra. |
1225 Let $A$ be an $\cE\cB_n$-algebra. |
1226 Note that this implies a $\Diff(B^n)$ action on $A$, |
1226 Note that this implies a $\Diff(B^n)$ action on $A$, |
1227 since $\cE\cB_n$ contains a copy of $\Diff(B^n)$. |
1227 since $\cE\cB_n$ contains a copy of $\Diff(B^n)$. |
1228 We will define a strict $A_\infty$ $n$-category $\cC^A$. |
1228 We will define a strict $A_\infty$ $n$-category $\cC^A$. |
|
1229 (We enrich in topological spaces, though this could easily be adapted to, say, chain complexes.) |
1229 If $X$ is a ball of dimension $k<n$, define $\cC^A(X)$ to be a point. |
1230 If $X$ is a ball of dimension $k<n$, define $\cC^A(X)$ to be a point. |
1230 In other words, the $k$-morphisms are trivial for $k<n$. |
1231 In other words, the $k$-morphisms are trivial for $k<n$. |
1231 If $X$ is an $n$-ball, we define $\cC^A(X)$ via a colimit construction. |
1232 If $X$ is an $n$-ball, we define $\cC^A(X)$ via a colimit construction. |
1232 (Plain colimit, not homotopy colimit.) |
1233 (Plain colimit, not homotopy colimit.) |
1233 Let $J$ be the category whose objects are embeddings of a disjoint union of copies of |
1234 Let $J$ be the category whose objects are embeddings of a disjoint union of copies of |
1246 $\cC(X)$ are trivial (single point) for $k<n$, gives rise to |
1247 $\cC(X)$ are trivial (single point) for $k<n$, gives rise to |
1247 an $\cE\cB_n$-algebra. |
1248 an $\cE\cB_n$-algebra. |
1248 %\nn{The paper is already long; is it worth giving details here?} |
1249 %\nn{The paper is already long; is it worth giving details here?} |
1249 % According to the referee, yes it is... |
1250 % According to the referee, yes it is... |
1250 Let $A = \cC(B^n)$, where $B^n$ is the standard $n$-ball. |
1251 Let $A = \cC(B^n)$, where $B^n$ is the standard $n$-ball. |
1251 \nn{need to finish this} |
1252 We must define maps |
|
1253 \[ |
|
1254 \cE\cB_n^k \times A \times \cdots \times A \to A , |
|
1255 \] |
|
1256 where $\cE\cB_n^k$ is the $k$-th space of the $\cE\cB_n$ operad. |
|
1257 |
1252 |
1258 |
1253 If we apply the homotopy colimit construction of the next subsection to this example, |
1259 If we apply the homotopy colimit construction of the next subsection to this example, |
1254 we get an instance of Lurie's topological chiral homology construction. |
1260 we get an instance of Lurie's topological chiral homology construction. |
1255 \end{example} |
1261 \end{example} |
1256 |
1262 |