95 \newcommand{\yc}{6} |
95 \newcommand{\yc}{6} |
96 |
96 |
97 \node[box] at (-4,\yb) (tC) {$C$ \\ a `traditional' \\ weak $n$-category}; |
97 \node[box] at (-4,\yb) (tC) {$C$ \\ a `traditional' \\ weak $n$-category}; |
98 \node[box] at (\xa,\ya) (C) {$\cC$ \\ a topological \\ $n$-category}; |
98 \node[box] at (\xa,\ya) (C) {$\cC$ \\ a topological \\ $n$-category}; |
99 \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}; |
100 \node[box] at (\xa,\yb) (FU) {$(\cF, \cU)$ \\ fields and\\ local relations}; |
100 \node[box] at (\xa,\yb) (FU) {$(\cF, U)$ \\ fields and\\ local relations}; |
101 \node[box] at (\xb,\yb) (BC) {$\bc_*(M; \cF)$ \\ the blob complex}; |
101 \node[box] at (\xb,\yb) (BC) {$\bc_*(M; \cF)$ \\ the blob complex}; |
102 \node[box] at (\xa,\yc) (Cs) {$\cC_*$ \\ an $A_\infty$ \\$n$-category}; |
102 \node[box] at (\xa,\yc) (Cs) {$\cC_*$ \\ an $A_\infty$ \\$n$-category}; |
103 \node[box] at (\xb,\yc) (BCs) {$\underrightarrow{\cC_*}(M)$}; |
103 \node[box] at (\xb,\yc) (BCs) {$\underrightarrow{\cC_*}(M)$}; |
104 |
104 |
105 |
105 |
106 |
106 |
107 \draw[->] (C) -- node[above] {$\displaystyle \colim_{\cell(M)} \cC$} node[below] {\S\S \ref{sec:constructing-a-tqft} \& \ref{ss:ncat_fields}} (A); |
107 \draw[->] (C) -- node[above] {$\displaystyle \colim_{\cell(M)} \cC$} node[below] {\S\S \ref{sec:constructing-a-tqft} \& \ref{ss:ncat_fields}} (A); |
108 \draw[->] (FU) -- node[below] {blob complex \\ for $M$} (BC); |
108 \draw[->] (FU) -- node[below] {blob complex \\ for $M$} (BC); |
109 \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); |
110 |
110 |
111 \draw[->] (FU) -- node[right=10pt] {$\cF(M)/\cU$} (A); |
111 \draw[->] (FU) -- node[right=10pt] {$\cF(M)/U$} (A); |
112 |
112 |
113 \draw[->] (tC) -- node[above] {Example \ref{ex:traditional-n-categories(fields)}} (FU); |
113 \draw[->] (tC) -- node[above] {Example \ref{ex:traditional-n-categories(fields)}} (FU); |
114 |
114 |
115 \draw[->] (C.-100) -- node[left] { |
115 \draw[->] (C.-100) -- node[left] { |
116 \S \ref{ss:ncat_fields} |
116 \S \ref{ss:ncat_fields} |
117 %$\displaystyle \cF(M) = \DirectSum_{c \in\cell(M)} \cC(c)$ \\ $\displaystyle \cU(B) = \DirectSum_{c \in \cell(B)} \ker \ev: \cC(c) \to \cC(B)$ |
117 %$\displaystyle \cF(M) = \DirectSum_{c \in\cell(M)} \cC(c)$ \\ $\displaystyle U(B) = \DirectSum_{c \in \cell(B)} \ker \ev: \cC(c) \to \cC(B)$ |
118 } (FU.100); |
118 } (FU.100); |
119 \draw[->] (C) -- node[above left=3pt] {restrict to \\ standard balls} (tC); |
119 \draw[->] (C) -- node[above left=3pt] {restrict to \\ standard balls} (tC); |
120 \draw[->] (FU.80) -- node[right] {restrict \\ to balls} (C.-80); |
120 \draw[->] (FU.80) -- node[right] {restrict \\ to balls} (C.-80); |
121 \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); |
122 |
122 |