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22 \maketitle |
22 \maketitle |
23 |
23 |
24 \textbf{Draft version, do not distribute.} |
24 \textbf{Draft version, do not distribute.} |
25 |
25 |
26 \versioninfo |
26 \versioninfo |
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27 |
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28 \noop{ |
27 |
29 |
28 \section*{Todo} |
30 \section*{Todo} |
29 |
31 |
30 \subsection*{What else?...} |
32 \subsection*{What else?...} |
31 |
33 |
44 \begin{itemize} |
46 \begin{itemize} |
45 \item $n=2$ examples |
47 \item $n=2$ examples |
46 \item dimension $n+1$ (generalized Deligne conjecture?) |
48 \item dimension $n+1$ (generalized Deligne conjecture?) |
47 \item should be clear about PL vs Diff; probably PL is better |
49 \item should be clear about PL vs Diff; probably PL is better |
48 (or maybe not) |
50 (or maybe not) |
49 \item say what we mean by $n$-category, $A_\infty$ or $E_\infty$ $n$-category |
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50 \item something about higher derived coend things (derived 2-coend, e.g.) |
51 \item something about higher derived coend things (derived 2-coend, e.g.) |
51 \item shuffle product vs gluing product (?) |
52 \item shuffle product vs gluing product (?) |
52 \item commutative algebra results |
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53 \item $A_\infty$ blob complex |
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54 \item connection between $A_\infty$ operad and topological $A_\infty$ cat defs |
53 \item connection between $A_\infty$ operad and topological $A_\infty$ cat defs |
55 \end{itemize} |
54 \end{itemize} |
56 \item lower priority |
55 \item lower priority |
57 \begin{itemize} |
56 \begin{itemize} |
58 \item Derive Hochschild standard results from blob point of view? |
57 \item Derive Hochschild standard results from blob point of view? |
59 \item Kh |
58 \item Kh |
60 \item Mention somewhere \cite{MR1624157} ``Skein homology''; it's not directly related, but has similar motivations. |
59 \item Mention somewhere \cite{MR1624157} ``Skein homology''; it's not directly related, but has similar motivations. |
61 \end{itemize} |
60 \end{itemize} |
62 \end{itemize} |
61 \end{itemize} |
63 |
62 |
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63 } %end \noop |
64 |
64 |
65 \section{Introduction} |
65 \section{Introduction} |
66 |
66 |
67 [Outline for intro] |
67 [Outline for intro] |
68 \begin{itemize} |
68 \begin{itemize} |
162 \todo{Say that this is just the original $n$-category?} |
162 \todo{Say that this is just the original $n$-category?} |
163 \end{property} |
163 \end{property} |
164 |
164 |
165 \begin{property}[Skein modules] |
165 \begin{property}[Skein modules] |
166 \label{property:skein-modules}% |
166 \label{property:skein-modules}% |
167 The $0$-th blob homology of $X$ is the usual skein module associated to $X$. (See \S \ref{sec:local-relations}.) |
167 The $0$-th blob homology of $X$ is the usual |
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168 (dual) TQFT Hilbert space (a.k.a.\ skein module) associated to $X$ |
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169 by $(\cF,\cU)$. (See \S \ref{sec:local-relations}.) |
168 \begin{equation*} |
170 \begin{equation*} |
169 H_0(\bc_*^{\cF,\cU}(X)) \iso A^{\cF,\cU}(X) |
171 H_0(\bc_*^{\cF,\cU}(X)) \iso A^{\cF,\cU}(X) |
170 \end{equation*} |
172 \end{equation*} |
171 \end{property} |
173 \end{property} |
172 |
174 |
196 \CD{X_1} \otimes \CD{X_2} \otimes \bc_*(X_1) \otimes \bc_*(X_2) |
198 \CD{X_1} \otimes \CD{X_2} \otimes \bc_*(X_1) \otimes \bc_*(X_2) |
197 \ar@/_4ex/[r]_{\ev_{X_1} \otimes \ev_{X_2}} \ar[u]^{\gl^{\Diff}_Y \otimes \gl_Y} & |
199 \ar@/_4ex/[r]_{\ev_{X_1} \otimes \ev_{X_2}} \ar[u]^{\gl^{\Diff}_Y \otimes \gl_Y} & |
198 \bc_*(X_1) \otimes \bc_*(X_2) \ar[u]_{\gl_Y} |
200 \bc_*(X_1) \otimes \bc_*(X_2) \ar[u]_{\gl_Y} |
199 } |
201 } |
200 \end{equation*} |
202 \end{equation*} |
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203 \nn{should probably say something about associativity here (or not?)} |
201 \end{property} |
204 \end{property} |
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205 |
202 |
206 |
203 \begin{property}[Gluing formula] |
207 \begin{property}[Gluing formula] |
204 \label{property:gluing}% |
208 \label{property:gluing}% |
205 \mbox{}% <-- gets the indenting right |
209 \mbox{}% <-- gets the indenting right |
206 \begin{itemize} |
210 \begin{itemize} |
217 \begin{equation*} |
221 \begin{equation*} |
218 \bc_*(X') \iso \bc_*(X) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y \times I)}} \!\!\!\!\!\!\xymatrix{ \ar@(ru,rd)@<-1ex>[]} |
222 \bc_*(X') \iso \bc_*(X) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y \times I)}} \!\!\!\!\!\!\xymatrix{ \ar@(ru,rd)@<-1ex>[]} |
219 \end{equation*} |
223 \end{equation*} |
220 \end{itemize} |
224 \end{itemize} |
221 \end{property} |
225 \end{property} |
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226 |
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227 \nn{add product formula? $n$-dimensional fat graph operad stuff?} |
222 |
228 |
223 Properties \ref{property:functoriality}, \ref{property:gluing-map} and \ref{property:skein-modules} will be immediate from the definition given in |
229 Properties \ref{property:functoriality}, \ref{property:gluing-map} and \ref{property:skein-modules} will be immediate from the definition given in |
224 \S \ref{sec:blob-definition}, and we'll recall them at the appropriate points there. \todo{Make sure this gets done.} |
230 \S \ref{sec:blob-definition}, and we'll recall them at the appropriate points there. \todo{Make sure this gets done.} |
225 Properties \ref{property:disjoint-union} and \ref{property:contractibility} are established in \S \ref{sec:basic-properties}. |
231 Properties \ref{property:disjoint-union} and \ref{property:contractibility} are established in \S \ref{sec:basic-properties}. |
226 Property \ref{property:hochschild} is established in \S \ref{sec:hochschild}, Property \ref{property:evaluation} in \S \ref{sec:evaluation}, |
232 Property \ref{property:hochschild} is established in \S \ref{sec:hochschild}, Property \ref{property:evaluation} in \S \ref{sec:evaluation}, |