943 Note that if we think of an ordinary 1-category as an $A_\infty$ 1-category where $k$-morphisms are identities for $k>1$, |
943 Note that if we think of an ordinary 1-category as an $A_\infty$ 1-category where $k$-morphisms are identities for $k>1$, |
944 then Axiom \ref{axiom:families} implies Axiom \ref{axiom:extended-isotopies}. |
944 then Axiom \ref{axiom:families} implies Axiom \ref{axiom:extended-isotopies}. |
945 |
945 |
946 Another variant of the above axiom would be to drop the ``up to homotopy" and require a strictly associative action. |
946 Another variant of the above axiom would be to drop the ``up to homotopy" and require a strictly associative action. |
947 In fact, the alternative construction $\btc_*(X)$ of the blob complex described in \S \ref{ss:alt-def} |
947 In fact, the alternative construction $\btc_*(X)$ of the blob complex described in \S \ref{ss:alt-def} |
948 gives $n$-categories as in Example \ref{ex:blob-complexes-of-balls} which satisfy this stronger axiom; |
948 gives $n$-categories as in Example \ref{ex:blob-complexes-of-balls} which satisfy this stronger axiom. |
949 since that construction is only homotopy equivalent to the usual one, only the weaker axiom carries across. |
949 %since that construction is only homotopy equivalent to the usual one, only the weaker axiom carries across. |
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950 For future reference we make the following definition. |
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951 |
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952 \begin{defn} |
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953 A {\em strict $A_\infty$ $n$-category} is one in which the actions of Axiom \ref{axiom:families} are strictly associative. |
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954 \end{defn} |
950 |
955 |
951 \noop{ |
956 \noop{ |
952 Note that if we take homology of chain complexes, we turn an $A_\infty$ $n$-category |
957 Note that if we take homology of chain complexes, we turn an $A_\infty$ $n$-category |
953 into a ordinary $n$-category (enriched over graded groups). |
958 into a ordinary $n$-category (enriched over graded groups). |
954 In a different direction, if we enrich over topological spaces instead of chain complexes, |
959 In a different direction, if we enrich over topological spaces instead of chain complexes, |
1218 \rm |
1223 \rm |
1219 \label{ex:e-n-alg} |
1224 \label{ex:e-n-alg} |
1220 Let $A$ be an $\cE\cB_n$-algebra. |
1225 Let $A$ be an $\cE\cB_n$-algebra. |
1221 Note that this implies a $\Diff(B^n)$ action on $A$, |
1226 Note that this implies a $\Diff(B^n)$ action on $A$, |
1222 since $\cE\cB_n$ contains a copy of $\Diff(B^n)$. |
1227 since $\cE\cB_n$ contains a copy of $\Diff(B^n)$. |
1223 We will define an $A_\infty$ $n$-category $\cC^A$. |
1228 We will define a strict $A_\infty$ $n$-category $\cC^A$. |
1224 If $X$ is a ball of dimension $k<n$, define $\cC^A(X)$ to be a point. |
1229 If $X$ is a ball of dimension $k<n$, define $\cC^A(X)$ to be a point. |
1225 In other words, the $k$-morphisms are trivial for $k<n$. |
1230 In other words, the $k$-morphisms are trivial for $k<n$. |
1226 If $X$ is an $n$-ball, we define $\cC^A(X)$ via a colimit construction. |
1231 If $X$ is an $n$-ball, we define $\cC^A(X)$ via a colimit construction. |
1227 (Plain colimit, not homotopy colimit.) |
1232 (Plain colimit, not homotopy colimit.) |
1228 Let $J$ be the category whose objects are embeddings of a disjoint union of copies of |
1233 Let $J$ be the category whose objects are embeddings of a disjoint union of copies of |
1235 The remaining data for the $A_\infty$ $n$-category |
1240 The remaining data for the $A_\infty$ $n$-category |
1236 --- composition and $\Diff(X\to X')$ action --- |
1241 --- composition and $\Diff(X\to X')$ action --- |
1237 also comes from the $\cE\cB_n$ action on $A$. |
1242 also comes from the $\cE\cB_n$ action on $A$. |
1238 %\nn{should we spell this out?} |
1243 %\nn{should we spell this out?} |
1239 |
1244 |
1240 Conversely, one can show that a disk-like $A_\infty$ $n$-category $\cC$, where the $k$-morphisms |
1245 Conversely, one can show that a disk-like strict $A_\infty$ $n$-category $\cC$, where the $k$-morphisms |
1241 $\cC(X)$ are trivial (single point) for $k<n$, gives rise to |
1246 $\cC(X)$ are trivial (single point) for $k<n$, gives rise to |
1242 an $\cE\cB_n$-algebra. |
1247 an $\cE\cB_n$-algebra. |
1243 %\nn{The paper is already long; is it worth giving details here?} |
1248 %\nn{The paper is already long; is it worth giving details here?} |
|
1249 % According to the referee, yes it is... |
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1250 Let $A = \cC(B^n)$, where $B^n$ is the standard $n$-ball. |
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1251 \nn{need to finish this} |
1244 |
1252 |
1245 If we apply the homotopy colimit construction of the next subsection to this example, |
1253 If we apply the homotopy colimit construction of the next subsection to this example, |
1246 we get an instance of Lurie's topological chiral homology construction. |
1254 we get an instance of Lurie's topological chiral homology construction. |
1247 \end{example} |
1255 \end{example} |
1248 |
1256 |