655 \label{fzo5} |
661 \label{fzo5} |
656 \end{figure} |
662 \end{figure} |
657 \begin{figure}[t] |
663 \begin{figure}[t] |
658 $$ |
664 $$ |
659 \mathfig{0.6}{triangle/triangle3c} |
665 \mathfig{0.6}{triangle/triangle3c} |
|
666 $$ |
|
667 $$ |
|
668 \begin{tikzpicture} |
|
669 \node (fg1) at (0,0) { |
|
670 \begin{tikzpicture}[baseline=-0.6cm] |
|
671 \path (0,0) coordinate (f1); |
|
672 \path (3,0) coordinate (f2); |
|
673 \path (3,-0.5) coordinate (g1); |
|
674 \path (6,-0.5) coordinate (g2); |
|
675 \node at (1.5,0.125) {$f$}; |
|
676 \node at (4.5,-0.625) {$g$}; |
|
677 \draw (f1) .. controls +(1,.8) and +(-1,.8) .. (f2); |
|
678 \draw[dashed] (f1) .. controls +(1,-.4) and +(-1,-.4) .. (f2); |
|
679 \draw (f1) .. controls +(1,-1) and +(-1,-.4) .. (g1); |
|
680 \draw (g1) .. controls +(1,-.8) and +(-1,-.8) .. (g2); |
|
681 \draw[dashed] (g1) .. controls +(1,.4) and +(-1,.4) .. (g2); |
|
682 \draw (f2) .. controls +(1,.4) and +(-1,1) .. (g2); |
|
683 % |
|
684 \draw[blue,->] (-0.8,-1.2) node[below] {$(a \circ d) \times I$} -- (1,-0.5) ; |
|
685 \path[clip] (f1) .. controls +(1,-.4) and +(-1,-.4) .. (f2) |
|
686 .. controls +(1,.4) and +(-1,1) .. (g2) |
|
687 .. controls +(-1,.4) and +(1,.4) .. (g1) |
|
688 .. controls +(-1,-.4) and +(1,-1) .. (f1); |
|
689 \foreach \x in {0,0.1, ..., 6} { |
|
690 \draw[green!50!brown] (\x,-2) -- + (0,4); |
|
691 } |
|
692 \end{tikzpicture} |
|
693 }; |
|
694 \node (fg2) at (4,-4) { |
|
695 \begin{tikzpicture}[baseline=-0.1cm] |
|
696 \path (0,0) coordinate (f1); |
|
697 \path (3,0) coordinate (f2); |
|
698 \path (3,-0.5) coordinate (g1); |
|
699 \path (6,-0.5) coordinate (g2); |
|
700 \node at (1.5,0.125) {$f$}; |
|
701 \node at (4.5,-0.625) {$g$}; |
|
702 \draw[dashed] (f1) .. controls +(1,.8) and +(-1,.8) .. (f2); |
|
703 \draw[dashed] (f1) .. controls +(1,-.4) and +(-1,-.4) .. (f2); |
|
704 \draw (f1) .. controls +(1,-1) and +(-1,-.4) .. (g1); |
|
705 \draw (g1) .. controls +(1,-.8) and +(-1,-.8) .. (g2); |
|
706 \draw[dashed] (g1) .. controls +(1,.4) and +(-1,.4) .. (g2); |
|
707 \draw[dashed] (f2) .. controls +(1,.4) and +(-1,1) .. (g2); |
|
708 \draw (f1) .. controls +(1,1.5) and +(-1,2)..(g2); |
|
709 % |
|
710 \begin{scope} |
|
711 \path[clip] (f1) .. controls +(1,-.4) and +(-1,-.4) .. (f2) |
|
712 .. controls +(1,.4) and +(-1,1) .. (g2) |
|
713 .. controls +(-1,.4) and +(1,.4) .. (g1) |
|
714 .. controls +(-1,-.4) and +(1,-1) .. (f1); |
|
715 \foreach \x in {0,0.1, ..., 6} { |
|
716 \draw[green!50!brown] (\x,-2) -- + (0,4); |
|
717 } |
|
718 \end{scope} |
|
719 \begin{scope} |
|
720 \path[clip] (f1) .. controls +(1,1.5) and +(-1,2).. (g2) |
|
721 .. controls +(-1,1) and +(1,.4) .. (f2) |
|
722 .. controls +(-1,.8) and + (1,.8) .. (f1); |
|
723 \foreach \x in {0,0.1, ..., 6} { |
|
724 \draw[green!50!brown] (\x,-2) -- + (0,4); |
|
725 } |
|
726 \end{scope} |
|
727 \end{tikzpicture} |
|
728 }; |
|
729 \node (fg3) at (8,0) { |
|
730 \begin{tikzpicture}[baseline=-2.45cm] |
|
731 \path (0,0) coordinate (f1); |
|
732 \path (3,0) coordinate (f2); |
|
733 \path (3,0) coordinate (g1); |
|
734 \path (6,0) coordinate (g2); |
|
735 \node at (1.5,0) {$f$}; |
|
736 \node at (4.5,0) {$g$}; |
|
737 \draw[dashed] (f1) .. controls +(1,.8) and +(-1,.8) .. (f2); |
|
738 \draw (f1) .. controls +(1,-.8) and +(-1,-.8) .. (f2); |
|
739 \draw (g1) .. controls +(1,-.8) and +(-1,-.8) .. (g2); |
|
740 \draw[dashed] (g1) .. controls +(1,.8) and +(-1,.8) .. (g2); |
|
741 \draw (f1) .. controls +(1,1.5) and +(-1,1.5)..(g2); |
|
742 % |
|
743 \draw[blue,->] (4,1.75) node[above] {$(b \circ d) \times I$}-- + (0,-1); |
|
744 \begin{scope} |
|
745 \path[clip] (f1) .. controls +(1,1.5) and +(-1,1.5).. (g2) |
|
746 .. controls +(-1,.8) and +(1,.8) .. (f2) |
|
747 .. controls +(-1,.8) and + (1,.8) .. (f1); |
|
748 \foreach \x in {0,0.1, ..., 6} { |
|
749 \draw[green!50!brown] (\x,-2) -- + (0,4); |
|
750 } |
|
751 \end{scope} |
|
752 \end{tikzpicture} |
|
753 }; |
|
754 \draw[->] ($(fg1.south)+(0,0.5)$) -- node[left=0.5cm] {add $(b \circ d) \times I$} (fg2); |
|
755 \draw[->] (fg2) -- node[right=0.5cm] {remove $(a \circ d) \times I$} ($(fg3.south)+(0,1.75)$); |
|
756 \path (fg1) -- node {$=$} (fg3); |
|
757 \end{tikzpicture} |
660 $$ |
758 $$ |
661 \caption{Part of the proof that the four different horizontal compositions of 2-morphisms are equal.} |
759 \caption{Part of the proof that the four different horizontal compositions of 2-morphisms are equal.} |
662 \label{fig:horizontal-compositions-equal} |
760 \label{fig:horizontal-compositions-equal} |
663 \end{figure} |
761 \end{figure} |
664 |
762 |