135 on $C^2$ (Figure \ref{fzo1}). |
135 on $C^2$ (Figure \ref{fzo1}). |
136 Isotopy invariance implies that this is associative. |
136 Isotopy invariance implies that this is associative. |
137 We will define a ``horizontal" composition later. |
137 We will define a ``horizontal" composition later. |
138 |
138 |
139 \begin{figure}[t] |
139 \begin{figure}[t] |
140 \begin{equation*} |
140 \begin{center} |
141 \mathfig{.73}{tempkw/zo1} |
141 \begin{tikzpicture} |
142 \end{equation*} |
142 |
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143 \newcommand{\vertex}{node[circle,fill=black,inner sep=1pt] {}} |
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144 \newcommand{\nsep}{1.8} |
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145 |
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146 \node[outer sep=\nsep](A) at (0,0) { |
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147 \begin{tikzpicture} |
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148 \draw (0,0) coordinate (p1); |
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149 \draw (4,0) coordinate (p2); |
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150 \draw (2,1.2) coordinate (pu); |
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151 \draw (2,-1.2) coordinate (pd); |
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152 |
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153 \draw (p1) .. controls (pu) .. (p2) .. controls (pd) .. (p1); |
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154 \draw (p1)--(p2); |
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155 |
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156 \draw (p1) \vertex; |
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157 \draw (p2) \vertex; |
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158 |
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159 \node at (2.1, .44) {$B^2$}; |
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160 \node at (2.1, -.44) {$B^2$}; |
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161 |
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162 \end{tikzpicture} |
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163 }; |
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164 |
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165 \node[outer sep=\nsep](B) at (6,0) { |
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166 \begin{tikzpicture} |
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167 \draw (0,0) coordinate (p1); |
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168 \draw (4,0) coordinate (p2); |
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169 \draw (2,.6) coordinate (pu); |
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170 \draw (2,-.6) coordinate (pd); |
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171 |
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172 \draw (p1) .. controls (pu) .. (p2) .. controls (pd) .. (p1); |
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173 \draw[help lines, dashed] (p1)--(p2); |
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174 |
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175 \draw (p1) \vertex; |
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176 \draw (p2) \vertex; |
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177 |
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178 \node at (2.1,0) {$B^2$}; |
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179 |
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180 \end{tikzpicture} |
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181 }; |
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182 |
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183 \draw[->, thick, blue!50!green] (A) -- node[black, above] {$\cong$} (B); |
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184 |
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185 \end{tikzpicture} |
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186 \end{center} |
143 \caption{Vertical composition of 2-morphisms} |
187 \caption{Vertical composition of 2-morphisms} |
144 \label{fzo1} |
188 \label{fzo1} |
145 \end{figure} |
189 \end{figure} |
146 |
190 |
147 Given $a\in C^1$, define $\id_a = a\times I \in C^2$ (pinched boundary). |
191 Given $a\in C^1$, define $\id_a = a\times I \in C^2$ (pinched boundary). |
369 \begin{tikzpicture} |
411 \begin{tikzpicture} |
370 |
412 |
371 \newcommand{\vertex}{node[circle,fill=black,inner sep=1pt] {}} |
413 \newcommand{\vertex}{node[circle,fill=black,inner sep=1pt] {}} |
372 \newcommand{\nsep}{1.8} |
414 \newcommand{\nsep}{1.8} |
373 |
415 |
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416 \clip (-4,-1.25)--(12,-1.25)--(16,1.25)--(-1,1.25)--cycle; |
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417 |
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418 |
374 \node[outer sep=\nsep](A) at (0,0) { |
419 \node[outer sep=\nsep](A) at (0,0) { |
375 \begin{tikzpicture} |
420 \begin{tikzpicture} |
376 \draw (0,0) coordinate (p1); |
421 \draw (0,0) coordinate (p1); |
377 \draw (4,0) coordinate (p2); |
422 \draw (4,0) coordinate (p2); |
378 \draw (2.4,0) coordinate (p2a); |
423 \draw (2.4,0) coordinate (p2a); |
461 \caption{Composition of weak identities, 2} |
506 \caption{Composition of weak identities, 2} |
462 \label{fzo4} |
507 \label{fzo4} |
463 \end{figure} |
508 \end{figure} |
464 We identify a product region and remove it. |
509 We identify a product region and remove it. |
465 |
510 |
466 We define horizontal composition of 2-morphisms as shown in Figure \ref{fzo5}. |
511 We define horizontal composition $f *_h g$ of 2-morphisms $f$ and $g$ as shown in Figure \ref{fzo5}. |
467 It is not hard to show that this is independent of the arbitrary (left/right) |
512 It is not hard to show that this is independent of the arbitrary (left/right) |
468 choice made in the definition, and that it is associative. |
513 choice made in the definition, and that it is associative. |
469 \begin{figure}[t] |
514 \begin{figure}[t] |
470 \begin{equation*} |
515 \begin{equation*} |
471 \mathfig{.83}{tempkw/zo5} |
516 \raisebox{-.9cm}{ |
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517 \begin{tikzpicture} |
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518 \draw (0,0) .. controls +(1,.8) and +(-1,.8) .. node[above] {$b$} (2.9,0) |
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519 .. controls +(-1,-.8) and +(1,-.8) .. node[below] {$a$} (0,0); |
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520 \draw[->, thick, orange!50!brown] (1.45,-.4)-- node[left, black] {$f$} +(0,.8); |
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521 \end{tikzpicture}} |
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522 \;\;\;*_h\;\; |
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523 \raisebox{-.9cm}{ |
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524 \begin{tikzpicture} |
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525 \draw (0,0) .. controls +(1,.8) and +(-1,.8) .. node[above] {$d$} (2.9,0) |
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526 .. controls +(-1,-.8) and +(1,-.8) .. node[below] {$c$} (0,0); |
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527 \draw[->, thick, orange!50!brown] (1.45,-.4)-- node[left, black] {$g$} +(0,.8); |
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528 \end{tikzpicture}} |
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529 \;=\; |
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530 \raisebox{-1.9cm}{ |
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531 \begin{tikzpicture} |
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532 \draw (0,0) coordinate (p1); |
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533 \draw (5.8,0) coordinate (p2); |
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534 \draw (2.9,.3) coordinate (pu); |
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535 \draw (2.9,-.3) coordinate (pd); |
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536 \begin{scope} |
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537 \clip (p1) .. controls +(.6,.3) and +(-.5,0) .. (pu) |
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538 .. controls +(.5,0) and +(-.6,.3) .. (p2) |
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539 .. controls +(-.6,-.3) and +(.5,0) .. (pd) |
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540 .. controls +(-.5,0) and +(.6,-.3) .. (p1); |
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541 \foreach \t in {0,.03,...,1} { |
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542 \draw[green!50!brown] ($(p1)!\t!(p2) + (0,2)$) -- +(0,-4); |
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543 } |
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544 \end{scope} |
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545 \draw (p1) .. controls +(.6,.3) and +(-.5,0) .. (pu) |
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546 .. controls +(.5,0) and +(-.6,.3) .. (p2) |
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547 .. controls +(-.6,-.3) and +(.5,0) .. (pd) |
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548 .. controls +(-.5,0) and +(.6,-.3) .. (p1); |
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549 \draw (p1) .. controls +(1,-2) and +(-1,-1) .. (pd); |
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550 \draw (p2) .. controls +(-1,2) and +(1,1) .. (pu); |
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551 \draw[->, thick, orange!50!brown] (1.45,-1.1)-- node[left, black] {$f$} +(0,.7); |
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552 \draw[->, thick, orange!50!brown] (4.35,.4)-- node[left, black] {$g$} +(0,.7); |
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553 \draw[->, thick, blue!75!yellow] (1.5,.78) node[black, above] {$(b\cdot c)\times I$} -- (2.5,0); |
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554 \end{tikzpicture}} |
472 \end{equation*} |
555 \end{equation*} |
473 \caption{Horizontal composition of 2-morphisms} |
556 \caption{Horizontal composition of 2-morphisms} |
474 \label{fzo5} |
557 \label{fzo5} |
475 \end{figure} |
558 \end{figure} |
476 |
559 |