111 \newcommand{\yyb}{10} |
111 \newcommand{\yyb}{10} |
112 \newcommand{\yyc}{6} |
112 \newcommand{\yyc}{6} |
113 |
113 |
114 \node[box] at (-4,\yyb) (tC) {$C$ \\ a `traditional' \\ weak $n$-category}; |
114 \node[box] at (-4,\yyb) (tC) {$C$ \\ a `traditional' \\ weak $n$-category}; |
115 \node[box] at (\xxa,\yya) (C) {$\cC$ \\ a disk-like \\ $n$-category}; |
115 \node[box] at (\xxa,\yya) (C) {$\cC$ \\ a disk-like \\ $n$-category}; |
116 \node[box] at (\xxb,\yya) (A) {$\underrightarrow{\cC}(M)$ \\ the (dual) TQFT \\ Hilbert space}; |
116 \node[box] at (\xxb,\yya) (A) {$A(M)$ \\ the (dual) TQFT \\ Hilbert space}; |
117 \node[box] at (\xxa,\yyb) (FU) {$(\cF, U)$ \\ fields and\\ local relations}; |
117 \node[box] at (\xxa,\yyb) (FU) {$(\cF, U)$ \\ fields and\\ local relations}; |
118 \node[box] at (\xxb,\yyb) (BC) {$\bc_*(M; \cF)$ \\ the blob complex}; |
118 \node[box] at (\xxb,\yyb) (BC) {$\bc_*(M; \cF)$ \\ the blob complex}; |
119 \node[box] at (\xxa,\yyc) (Cs) {$\cC_*$ \\ an $A_\infty$ \\$n$-category}; |
119 \node[box] at (\xxa,\yyc) (Cs) {$\cC_*$ \\ an $A_\infty$ \\$n$-category}; |
120 \node[box] at (\xxb,\yyc) (BCs) {$\underrightarrow{\cC_*}(M)$}; |
120 \node[box] at (\xxb,\yyc) (BCs) {$\underrightarrow{\cC_*}(M)$}; |
121 |
121 |
122 |
122 |
123 |
123 |
124 \draw[->] (C) -- node[above] {$\displaystyle \colim_{\cell(M)} \cC$} node[below] {\S\S \ref{sec:constructing-a-tqft} \& \ref{ss:ncat_fields}} (A); |
124 \draw[->] (C) -- node[above] {$\displaystyle \colim_{\cell(M)} \cC$} node[below] {\S\S \ref{sec:constructing-a-tqft} \& \ref{ss:ncat_fields}} (A); |
125 \draw[->] (FU) -- node[below] {blob complex \\ for $M$} (BC); |
125 |
|
126 \draw[->] (FU) -- node[above] {blob complex \\ for $M$} node[below]{\S \ref{sec:blob-definition}} (BC); |
126 \draw[->] (Cs) -- node[above] {$\displaystyle \hocolim_{\cell(M)} \cC_*$} node[below] {\S \ref{ss:ncat_fields}} (BCs); |
127 \draw[->] (Cs) -- node[above] {$\displaystyle \hocolim_{\cell(M)} \cC_*$} node[below] {\S \ref{ss:ncat_fields}} (BCs); |
127 |
128 |
128 \draw[->] (FU) -- node[right=10pt] {$\cF(M)/U$} (A); |
129 \draw[->] (FU) -- node[right=10pt] {$\cF(M)/U$ \\ Defn \ref{defn:TQFT-invariant}} (A); |
129 |
130 |
130 \draw[->] (tC) -- node[above] {Example \ref{ex:traditional-n-categories(fields)}} (FU); |
131 \draw[->] (tC) -- node[below] {Example \ref{ex:traditional-n-categories(fields)}\\ and \S \ref{sec:example:traditional-n-categories(fields)}} (FU); |
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132 |
131 |
133 |
132 \draw[->] (C.-100) -- node[left] { |
134 \draw[->] (C.-100) -- node[left] { |
133 \S \ref{ss:ncat_fields} |
135 \S \ref{ss:ncat_fields} |
134 %$\displaystyle \cF(M) = \DirectSum_{c \in\cell(M)} \cC(c)$ \\ $\displaystyle U(B) = \DirectSum_{c \in \cell(B)} \ker \ev: \cC(c) \to \cC(B)$ |
136 %$\displaystyle \cF(M) = \DirectSum_{c \in\cell(M)} \cC(c)$ \\ $\displaystyle U(B) = \DirectSum_{c \in \cell(B)} \ker \ev: \cC(c) \to \cC(B)$ |
135 } (FU.100); |
137 } (FU.100); |
136 \draw[->] (C) -- node[above left=3pt] {restrict to \\ standard balls} (tC); |
138 \draw[->] (C.210) -- node[above left=3pt] {restrict to \\ standard balls} (tC.42); |
|
139 \draw[->] (tC) -- node[below=4.5pt] {c.f. \S \ref{sec:comparing-defs}} (C.220); |
137 \draw[->] (FU.80) -- node[right] {restrict \\ to balls} (C.-80); |
140 \draw[->] (FU.80) -- node[right] {restrict \\ to balls} (C.-80); |
138 \draw[->] (BC) -- node[right] {$H_0$ \\ c.f. Proposition \ref{thm:skein-modules}} (A); |
141 \draw[->] (BC) -- node[right] {$H_0$ \\ c.f. Proposition \ref{thm:skein-modules}} (A); |
139 |
142 |
140 \draw[->] (FU) -- node[left] {blob complex \\ for balls} (Cs); |
143 \draw[->] (FU) -- node[left] {blob complex \\ for balls \\ Example \ref{ex:blob-complexes-of-balls}} (Cs); |
141 \draw[<->] (BC) -- node[right] {$\iso$ by \\ Corollary \ref{cor:new-old}} (BCs); |
144 \draw[<->] (BC) -- node[right] {$\iso$ by \\ Corollary \ref{cor:new-old}} (BCs); |
142 \end{tikzpicture} |
145 \end{tikzpicture} |
143 \endpgfgraphicnamed% |
146 \endpgfgraphicnamed% |
144 \mbox{} % <-- weird, doesn't compile unless I put something here after the \endpgfgraphicnamed...? -S |
147 \mbox{} % <-- weird, doesn't compile unless I put something here after the \endpgfgraphicnamed...? -S |
145 } |
148 } |