422 \caption{Composition of weak identities, 1} |
422 \caption{Composition of weak identities, 1} |
423 \label{fzo3} |
423 \label{fzo3} |
424 \end{figure} |
424 \end{figure} |
425 In the first step we have inserted a copy of $(x\times I)\times I$. |
425 In the first step we have inserted a copy of $(x\times I)\times I$. |
426 Figure \ref{fzo4} shows the other case. |
426 Figure \ref{fzo4} shows the other case. |
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427 |
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428 \newcommand{\vertex}{node[circle,fill=black,inner sep=1pt] {}} |
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429 |
427 \begin{figure}[t] |
430 \begin{figure}[t] |
428 \centering |
431 \centering |
429 \begin{tikzpicture} |
432 \begin{tikzpicture} |
430 |
433 |
431 \newcommand{\vertex}{node[circle,fill=black,inner sep=1pt] {}} |
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432 \newcommand{\nsep}{1.8} |
434 \newcommand{\nsep}{1.8} |
433 |
435 |
434 \clip (-4,-1.25)--(12,-1.25)--(16,1.25)--(-1,1.25)--cycle; |
436 \clip (-4,-1.25)--(12,-1.25)--(16,1.25)--(-1,1.25)--cycle; |
435 |
437 |
436 |
438 |
665 This is just a reparameterization of the pinched product $(a\bullet b\bullet c)\times I$ of $\cC$. |
667 This is just a reparameterization of the pinched product $(a\bullet b\bullet c)\times I$ of $\cC$. |
666 \begin{figure}[t] |
668 \begin{figure}[t] |
667 $$ |
669 $$ |
668 \mathfig{0.4}{triangle/triangle4a} |
670 \mathfig{0.4}{triangle/triangle4a} |
669 $$ |
671 $$ |
670 \nn{remember to change the labels} |
672 $$ |
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673 \begin{tikzpicture} |
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674 \node[circle,fill=black,inner sep=1pt] at (1.73,0) {}; |
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675 \node[circle,fill=black,inner sep=1pt] at (-1.73,0) {}; |
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676 \begin{scope}[yshift=-1cm] |
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677 \path[clip] (0,0) circle (2); |
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678 \begin{scope}[yshift=2cm] |
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679 \draw (0,0) circle (2); |
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680 \node[circle,fill=black,inner sep=1pt] (L2) at (-90:2) {}; |
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681 \node[circle,fill=black,inner sep=1pt] (L1) at (-120:2) {}; |
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682 \node at (-60:2.25) {$c$}; |
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683 \node at (-105:2.25) {$b$}; |
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684 \node at (-135:2.25) {$a$}; |
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685 \end{scope} |
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686 \end{scope} |
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687 \begin{scope}[yshift=1cm] |
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688 \path[clip] (0,0) circle (2); |
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689 \begin{scope}[yshift=-2cm] |
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690 \node at (120:2.25) {$a$}; |
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691 \node at (75:2.25) {$b$}; |
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692 \node at (45:2.25) {$c$}; |
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693 \draw (0,0) circle (2); |
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694 \node[circle,fill=black,inner sep=1pt] (U1) at (90:2) {}; |
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695 \node[circle,fill=black,inner sep=1pt] (U2) at (60:2) {}; |
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696 \end{scope} |
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697 \end{scope} |
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698 \draw[dashed] (L1) -- (U1); |
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699 \draw[dashed] (L2) -- (U2); |
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700 \begin{scope} |
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701 \path[clip] (0,1) circle (2); |
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702 \path[clip] (0,-1) circle (2); |
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703 \foreach \t in {-2.5,-2.4,...,-1.2} { |
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704 \draw[green!50!brown] (\t,-1) -- +(1.19,2); |
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705 } |
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706 \foreach \t in {-1.1,-1.0,...,0} { |
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707 \draw[blue!50!brown] (\t,-1) -- +(1.19,2); |
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708 } |
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709 \foreach \t in {0.1,0.2,...,2.5} { |
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710 \draw[red!50!brown] (\t,-1) -- +(1.19,2); |
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711 } |
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712 \end{scope} |
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713 \draw[->, thick, blue!75!yellow] (-2,0.5) node[black, left] {$a\times I$} -- (-1,0.5); |
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714 \draw[->, thick, blue!75!yellow] (-0.25,-1.5) node[black, below] {$b\times I$} -- (-0.25,-0.5); |
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715 \draw[->, thick, blue!75!yellow] (2,-0.5) node[black, right] {$c\times I$} -- (1,-0.5); |
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716 \end{tikzpicture} |
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717 $$ |
671 \caption{An associator.} |
718 \caption{An associator.} |
672 \label{fig:associator} |
719 \label{fig:associator} |
673 \end{figure} |
720 \end{figure} |
674 |
721 |
675 Let $x,y,z$ be objects of $D$ and let $a:x\to y$ and $b:y\to z$ be 1-morphisms of $D$. |
722 Let $x,y,z$ be objects of $D$ and let $a:x\to y$ and $b:y\to z$ be 1-morphisms of $D$. |