869 %\nn{add citation for this operad if we can find one} |
869 %\nn{add citation for this operad if we can find one} |
870 |
870 |
871 \begin{example}[$E_n$ algebras] |
871 \begin{example}[$E_n$ algebras] |
872 \rm |
872 \rm |
873 \label{ex:e-n-alg} |
873 \label{ex:e-n-alg} |
874 |
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875 Let $A$ be an $\cE\cB_n$-algebra. |
874 Let $A$ be an $\cE\cB_n$-algebra. |
876 Note that this implies a $\Diff(B^n)$ action on $A$, |
875 Note that this implies a $\Diff(B^n)$ action on $A$, |
877 since $\cE\cB_n$ contains a copy of $\Diff(B^n)$. |
876 since $\cE\cB_n$ contains a copy of $\Diff(B^n)$. |
878 We will define an $A_\infty$ $n$-category $\cC^A$. |
877 We will define an $A_\infty$ $n$-category $\cC^A$. |
879 If $X$ is a ball of dimension $k<n$, define $\cC^A(X)$ to be a point. |
878 If $X$ is a ball of dimension $k<n$, define $\cC^A(X)$ to be a point. |
890 The remaining data for the $A_\infty$ $n$-category |
889 The remaining data for the $A_\infty$ $n$-category |
891 --- composition and $\Diff(X\to X')$ action --- |
890 --- composition and $\Diff(X\to X')$ action --- |
892 also comes from the $\cE\cB_n$ action on $A$. |
891 also comes from the $\cE\cB_n$ action on $A$. |
893 \nn{should we spell this out?} |
892 \nn{should we spell this out?} |
894 |
893 |
895 \nn{Should remark that the associated hocolim for manifolds |
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896 agrees with Lurie's topological chiral homology construction; maybe wait |
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897 until next subsection to say that?} |
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898 |
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899 Conversely, one can show that a topological $A_\infty$ $n$-category $\cC$, where the $k$-morphisms |
894 Conversely, one can show that a topological $A_\infty$ $n$-category $\cC$, where the $k$-morphisms |
900 $\cC(X)$ are trivial (single point) for $k<n$, gives rise to |
895 $\cC(X)$ are trivial (single point) for $k<n$, gives rise to |
901 an $\cE\cB_n$-algebra. |
896 an $\cE\cB_n$-algebra. |
902 \nn{The paper is already long; is it worth giving details here?} |
897 \nn{The paper is already long; is it worth giving details here?} |
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898 |
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899 If we apply the homotopy colimit construction of the next subsection to this example, |
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900 we get an instance of Lurie's topological chiral homology construction. |
903 \end{example} |
901 \end{example} |
904 |
902 |
905 |
903 |
906 \subsection{From balls to manifolds} |
904 \subsection{From balls to manifolds} |
907 \label{ss:ncat_fields} \label{ss:ncat-coend} |
905 \label{ss:ncat_fields} \label{ss:ncat-coend} |
2353 It is easy to show that this is independent of the choice of $E$. |
2351 It is easy to show that this is independent of the choice of $E$. |
2354 Note also that this map depends only on the restriction of $f$ to $\bd X$. |
2352 Note also that this map depends only on the restriction of $f$ to $\bd X$. |
2355 In particular, if $F: X\to X$ is the identity on $\bd X$ then $f$ acts trivially, as required by |
2353 In particular, if $F: X\to X$ is the identity on $\bd X$ then $f$ acts trivially, as required by |
2356 Axiom \ref{axiom:extended-isotopies} of \S\ref{ss:n-cat-def}. |
2354 Axiom \ref{axiom:extended-isotopies} of \S\ref{ss:n-cat-def}. |
2357 |
2355 |
2358 |
2356 We define product $n{+}1$-morphisms to be identity maps of modules. |
2359 \nn{still to do: gluing, associativity, collar maps} |
2357 |
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2358 To define (binary) composition of $n{+}1$-morphisms, choose the obvious common equator |
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2359 then compose the module maps. |
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2360 |
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2361 |
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2362 \nn{still to do: associativity} |
2360 |
2363 |
2361 \medskip |
2364 \medskip |
2362 \hrule |
2365 |
2363 \medskip |
2366 \nn{Stuff that remains to be done (either below or in an appendix or in a separate section or in |
2364 |
2367 a separate paper): discuss Morita equivalence; functors} |
2365 |
2368 |
2366 |
2369 |
2367 Stuff that remains to be done (either below or in an appendix or in a separate section or in |
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2368 a separate paper): |
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2369 \begin{itemize} |
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2370 \item discuss Morita equivalence |
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2371 \item functors |
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2372 \end{itemize} |
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2373 |
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2374 |
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