25 \medskip |
25 \medskip |
26 |
26 |
27 There are many existing definitions of $n$-categories, with various intended uses. |
27 There are many existing definitions of $n$-categories, with various intended uses. |
28 In any such definition, there are sets of $k$-morphisms for each $0 \leq k \leq n$. |
28 In any such definition, there are sets of $k$-morphisms for each $0 \leq k \leq n$. |
29 Generally, these sets are indexed by instances of a certain typical shape. |
29 Generally, these sets are indexed by instances of a certain typical shape. |
30 Some $n$-category definitions model $k$-morphisms on the standard bihedrons (interval, bigon, and so on). |
30 Some $n$-category definitions model $k$-morphisms on the standard bihedron (interval, bigon, and so on). |
31 Other definitions have a separate set of 1-morphisms for each interval $[0,l] \sub \r$, |
31 Other definitions have a separate set of 1-morphisms for each interval $[0,l] \sub \r$, |
32 a separate set of 2-morphisms for each rectangle $[0,l_1]\times [0,l_2] \sub \r^2$, |
32 a separate set of 2-morphisms for each rectangle $[0,l_1]\times [0,l_2] \sub \r^2$, |
33 and so on. |
33 and so on. |
34 (This allows for strict associativity.) |
34 (This allows for strict associativity.) |
35 Still other definitions (see, for example, \cite{MR2094071}) |
35 Still other definitions (see, for example, \cite{MR2094071}) |
747 Note that this implies a $\Diff(B^n)$ action on $A$, |
747 Note that this implies a $\Diff(B^n)$ action on $A$, |
748 since $\cE\cB_n$ contains a copy of $\Diff(B^n)$. |
748 since $\cE\cB_n$ contains a copy of $\Diff(B^n)$. |
749 We will define an $A_\infty$ $n$-category $\cC^A$. |
749 We will define an $A_\infty$ $n$-category $\cC^A$. |
750 If $X$ is a ball of dimension $k<n$, define $\cC^A(X)$ to be a point. |
750 If $X$ is a ball of dimension $k<n$, define $\cC^A(X)$ to be a point. |
751 In other words, the $k$-morphisms are trivial for $k<n$. |
751 In other words, the $k$-morphisms are trivial for $k<n$. |
752 %If $X$ is an $n$-ball, we define $\cC^A(X)$ via a colimit construction. |
752 If $X$ is an $n$-ball, we define $\cC^A(X)$ via a colimit construction. |
753 %(Plain colimit, not homotopy colimit.) |
753 (Plain colimit, not homotopy colimit.) |
754 %Let $J$ be the category whose objects are embeddings of a disjoint union of copies of |
754 Let $J$ be the category whose objects are embeddings of a disjoint union of copies of |
755 %the standard ball $B^n$ into $X$, and who morphisms are given by engu |
755 the standard ball $B^n$ into $X$, and who morphisms are given by engulfing some of the |
756 |
756 embedded balls into a single larger embedded ball. |
757 \nn{...} |
757 To each object of $J$ we associate $A^{\times m}$ (where $m$ is the number of balls), and |
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758 to each morphism of $J$ we associate a morphism coming from the $\cE\cB_n$ action on $A$. |
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759 Alternatively and more simply, we could define $\cC^A(X)$ to be |
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760 $\Diff(B^n\to X)\times A$ modulo the diagonal action of $\Diff(B^n)$. |
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761 The remaining data for the $A_\infty$ $n$-category |
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762 --- composition and $\Diff(X\to X')$ action --- |
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763 also comes from the $\cE\cB_n$ action on $A$. |
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764 \nn{should we spell this out?} |
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765 |
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766 \nn{Should remark that this is just Lurie's topological chiral homology construction |
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767 applied to $n$-balls (check this).} |
758 \end{example} |
768 \end{example} |
759 |
769 |
760 |
770 |
761 |
771 |
762 |
772 |