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87 \begin{figure}[!ht] |
87 \begin{figure}[!ht] |
88 {\center |
88 {\center |
89 |
89 |
90 \begin{tikzpicture}[align=center,line width = 1.5pt] |
90 \begin{tikzpicture}[align=center,line width = 1.5pt] |
91 \newcommand{\xa}{2} |
91 \newcommand{\xa}{2} |
92 \newcommand{\xb}{10} |
92 \newcommand{\xb}{8} |
93 \newcommand{\ya}{14} |
93 \newcommand{\ya}{14} |
94 \newcommand{\yb}{10} |
94 \newcommand{\yb}{10} |
95 \newcommand{\yc}{6} |
95 \newcommand{\yc}{6} |
96 |
96 |
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97 \node[box] at (-4,\yb) (tC) {$C$ \\ a `traditional' \\ weak $n$-category}; |
97 \node[box] at (\xa,\ya) (C) {$\cC$ \\ a topological \\ $n$-category}; |
98 \node[box] at (\xa,\ya) (C) {$\cC$ \\ a topological \\ $n$-category}; |
98 \node[box] at (\xb,\ya) (A) {$\underrightarrow{\cC}(M)$ \\ the (dual) TQFT \\ Hilbert space}; |
99 \node[box] at (\xb,\ya) (A) {$\underrightarrow{\cC}(M)$ \\ the (dual) TQFT \\ Hilbert space}; |
99 \node[box] at (\xa,\yb) (FU) {$(\cF, \cU)$ \\ fields and\\ local relations}; |
100 \node[box] at (\xa,\yb) (FU) {$(\cF, \cU)$ \\ fields and\\ local relations}; |
100 \node[box] at (\xb,\yb) (BC) {$\bc_*(M; \cC)$ \\ the blob complex}; |
101 \node[box] at (\xb,\yb) (BC) {$\bc_*(M; \cC)$ \\ the blob complex}; |
101 \node[box] at (\xa,\yc) (Cs) {$\cC_*$ \\ an $A_\infty$ \\$n$-category}; |
102 \node[box] at (\xa,\yc) (Cs) {$\cC_*$ \\ an $A_\infty$ \\$n$-category}; |
107 \draw[->] (FU) -- node[below] {blob complex \\ for $M$} (BC); |
108 \draw[->] (FU) -- node[below] {blob complex \\ for $M$} (BC); |
108 \draw[->] (Cs) -- node[above] {$\displaystyle \hocolim_{\cell(M)} \cC_*$} node[below] {\S \ref{ss:ncat_fields}} (BCs); |
109 \draw[->] (Cs) -- node[above] {$\displaystyle \hocolim_{\cell(M)} \cC_*$} node[below] {\S \ref{ss:ncat_fields}} (BCs); |
109 |
110 |
110 \draw[->] (FU) -- node[right=10pt] {$\cF(M)/\cU$} (A); |
111 \draw[->] (FU) -- node[right=10pt] {$\cF(M)/\cU$} (A); |
111 |
112 |
112 \draw[->] (C) -- node[left=10pt] { |
113 \draw[->] (tC) -- node[above] {Example \ref{ex:traditional-n-categories(fields)}} (FU); |
113 Example \ref{ex:traditional-n-categories(fields)} \\ and \S \ref{ss:ncat_fields} |
114 |
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115 \draw[->] (C.-100) -- node[left] { |
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116 \S \ref{ss:ncat_fields} |
114 %$\displaystyle \cF(M) = \DirectSum_{c \in\cell(M)} \cC(c)$ \\ $\displaystyle \cU(B) = \DirectSum_{c \in \cell(B)} \ker \ev: \cC(c) \to \cC(B)$ |
117 %$\displaystyle \cF(M) = \DirectSum_{c \in\cell(M)} \cC(c)$ \\ $\displaystyle \cU(B) = \DirectSum_{c \in \cell(B)} \ker \ev: \cC(c) \to \cC(B)$ |
115 } (FU); |
118 } (FU.100); |
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119 \draw[->] (C) -- node[above left=3pt] {restrict to \\ standard balls} (tC); |
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120 \draw[->] (FU.80) -- node[right] {restrict \\ to balls} (C.-80); |
116 \draw[->] (BC) -- node[left] {$H_0$} node[right] {c.f. Theorem \ref{thm:skein-modules}} (A); |
121 \draw[->] (BC) -- node[left] {$H_0$} node[right] {c.f. Theorem \ref{thm:skein-modules}} (A); |
117 |
122 |
118 \draw[->] (FU) -- node[left] {blob complex \\ for balls} (Cs); |
123 \draw[->] (FU) -- node[left] {blob complex \\ for balls} (Cs); |
119 \draw (BC) -- node[right] {$\iso$ by \\ Corollary \ref{cor:new-old}} (BCs); |
124 \draw (BC) -- node[right] {$\iso$ by \\ Corollary \ref{cor:new-old}} (BCs); |
120 \end{tikzpicture} |
125 \end{tikzpicture} |
203 \cdots\to \bc_2(W, L) \to \bc_1(W, L) \to \bc_0(W, L) . |
208 \cdots\to \bc_2(W, L) \to \bc_1(W, L) \to \bc_0(W, L) . |
204 \] |
209 \] |
205 Here $\bc_0$ is linear combinations of fields on $W$, |
210 Here $\bc_0$ is linear combinations of fields on $W$, |
206 $\bc_1$ is linear combinations of local relations on $W$, |
211 $\bc_1$ is linear combinations of local relations on $W$, |
207 $\bc_2$ is linear combinations of relations amongst relations on $W$, |
212 $\bc_2$ is linear combinations of relations amongst relations on $W$, |
208 and so on. |
213 and so on. We now have a short exact sequence of chain complexes relating resolutions of the link $L$ (c.f. Lemma \ref{lem:hochschild-exact} which shows exactness with respect to boundary conditions in the context of Hochschild homology). |
209 |
214 |
210 |
215 |
211 \subsection{Formal properties} |
216 \subsection{Formal properties} |
212 \label{sec:properties} |
217 \label{sec:properties} |
213 The blob complex enjoys the following list of formal properties. |
218 The blob complex enjoys the following list of formal properties. |
224 complexes and isomorphisms between them. |
229 complexes and isomorphisms between them. |
225 \end{property} |
230 \end{property} |
226 As a consequence, there is an action of $\Homeo(X)$ on the chain complex $\bc_*(X; \cF)$; |
231 As a consequence, there is an action of $\Homeo(X)$ on the chain complex $\bc_*(X; \cF)$; |
227 this action is extended to all of $C_*(\Homeo(X))$ in Theorem \ref{thm:evaluation} below. |
232 this action is extended to all of $C_*(\Homeo(X))$ in Theorem \ref{thm:evaluation} below. |
228 |
233 |
229 The blob complex is also functorial (indeed, exact) with respect to $\cF$, |
234 The blob complex is also functorial with respect to $\cF$, |
230 although we will not address this in detail here. |
235 although we will not address this in detail here. |
231 \nn{KW: what exactly does ``exact in $\cF$" mean? |
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232 Do we mean a similar statement for module labels?} |
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233 |
236 |
234 \begin{property}[Disjoint union] |
237 \begin{property}[Disjoint union] |
235 \label{property:disjoint-union} |
238 \label{property:disjoint-union} |
236 The blob complex of a disjoint union is naturally isomorphic to the tensor product of the blob complexes. |
239 The blob complex of a disjoint union is naturally isomorphic to the tensor product of the blob complexes. |
237 \begin{equation*} |
240 \begin{equation*} |