567 \draw (p1) .. controls +(1,-2) and +(-1,-1) .. (pd); |
567 \draw (p1) .. controls +(1,-2) and +(-1,-1) .. (pd); |
568 \draw (p2) .. controls +(-1,2) and +(1,1) .. (pu); |
568 \draw (p2) .. controls +(-1,2) and +(1,1) .. (pu); |
569 \draw[->, thick, orange!50!brown] (1.45,-1.1)-- node[left, black] {$f$} +(0,.7); |
569 \draw[->, thick, orange!50!brown] (1.45,-1.1)-- node[left, black] {$f$} +(0,.7); |
570 \draw[->, thick, orange!50!brown] (4.35,.4)-- node[left, black] {$g$} +(0,.7); |
570 \draw[->, thick, orange!50!brown] (4.35,.4)-- node[left, black] {$g$} +(0,.7); |
571 \draw[->, thick, blue!75!yellow] (1.5,.78) node[black, above] {$(b\cdot c)\times I$} -- (2.5,0); |
571 \draw[->, thick, blue!75!yellow] (1.5,.78) node[black, above] {$(b\cdot c)\times I$} -- (2.5,0); |
572 \end{tikzpicture}} |
572 \end{tikzpicture}} \\ |
573 \end{equation*} |
573 \;&=\; |
574 \begin{equation*} |
574 \raisebox{-2.1cm}{ |
575 \mathfig{0.6}{triangle/triangle3b} |
575 \begin{tikzpicture} |
576 \end{equation*} |
576 \draw (0,0) coordinate (p1); |
|
577 \draw (5.8,0) coordinate (p2); |
|
578 \draw (2.9,0) coordinate (pu); |
|
579 \draw (2.9,-.9) coordinate (pd); |
|
580 \begin{scope} |
|
581 \clip (p1) .. controls +(.6,-.3) and +(-.5,-.3) .. (pu) |
|
582 .. controls +(.5,-.3) and +(-.6,-.3) .. (p2) |
|
583 .. controls +(-.6,-.9) and +(.5,0) .. (pd) |
|
584 .. controls +(-.5,0) and +(.6,-.9) .. (p1); |
|
585 \foreach \t in {0,.03,...,1} { |
|
586 \draw[green!50!brown] ($(p1)!\t!(p2) + (0,2)$) -- +(0,-4); |
|
587 } |
|
588 \end{scope} |
|
589 \draw (p1) .. controls +(.6,-.3) and +(-.5,-.3) .. (pu) |
|
590 .. controls +(.5,-.3) and +(-.6,-.3) .. (p2) |
|
591 .. controls +(-.6,-.9) and +(.5,0) .. (pd) |
|
592 .. controls +(-.5,0) and +(.6,-.9) .. (p1); |
|
593 \draw (p1) .. controls +(1,1) and +(-1,1) .. (pu); |
|
594 \draw (p2) .. controls +(-1,1) and +(1,1) .. (pu); |
|
595 \draw[->, thick, orange!50!brown] (1.45,-0.1)-- node[left, black] {$f$} +(0,.7); |
|
596 \draw[->, thick, orange!50!brown] (4.35,-0.1)-- node[left, black] {$g$} +(0,.7); |
|
597 \draw[->, thick, blue!75!yellow] (4.3,-1.5) node[black, below] {$(a\cdot c)\times I$} -- (3.3,-0.5); |
|
598 \end{tikzpicture}} \\ |
|
599 \;&=\; |
|
600 \raisebox{-1.9cm}{ |
|
601 \begin{tikzpicture}[y=-1cm] |
|
602 \draw (0,0) coordinate (p1); |
|
603 \draw (5.8,0) coordinate (p2); |
|
604 \draw (2.9,.3) coordinate (pu); |
|
605 \draw (2.9,-.3) coordinate (pd); |
|
606 \begin{scope} |
|
607 \clip (p1) .. controls +(.6,.3) and +(-.5,0) .. (pu) |
|
608 .. controls +(.5,0) and +(-.6,.3) .. (p2) |
|
609 .. controls +(-.6,-.3) and +(.5,0) .. (pd) |
|
610 .. controls +(-.5,0) and +(.6,-.3) .. (p1); |
|
611 \foreach \t in {0,.03,...,1} { |
|
612 \draw[green!50!brown] ($(p1)!\t!(p2) + (0,2)$) -- +(0,-4); |
|
613 } |
|
614 \end{scope} |
|
615 \draw (p1) .. controls +(.6,.3) and +(-.5,0) .. (pu) |
|
616 .. controls +(.5,0) and +(-.6,.3) .. (p2) |
|
617 .. controls +(-.6,-.3) and +(.5,0) .. (pd) |
|
618 .. controls +(-.5,0) and +(.6,-.3) .. (p1); |
|
619 \draw (p1) .. controls +(1,-2) and +(-1,-1) .. (pd); |
|
620 \draw (p2) .. controls +(-1,2) and +(1,1) .. (pu); |
|
621 \draw[<-, thick, orange!50!brown] (1.45,-1.1)-- node[left, black] {$f$} +(0,.7); |
|
622 \draw[<-, thick, orange!50!brown] (4.35,.4)-- node[left, black] {$g$} +(0,.7); |
|
623 \draw[->, thick, blue!75!yellow] (1.5,.78) node[black, below] {$(a\cdot d)\times I$} -- (2.5,0); |
|
624 \end{tikzpicture}} \\ |
|
625 \;&=\; |
|
626 \raisebox{-1.0cm}{ |
|
627 \begin{tikzpicture}[y=-1cm] |
|
628 \draw (0,0) coordinate (p1); |
|
629 \draw (5.8,0) coordinate (p2); |
|
630 \draw (2.9,0) coordinate (pu); |
|
631 \draw (2.9,-.9) coordinate (pd); |
|
632 \begin{scope} |
|
633 \clip (p1) .. controls +(.6,-.3) and +(-.5,-.3) .. (pu) |
|
634 .. controls +(.5,-.3) and +(-.6,-.3) .. (p2) |
|
635 .. controls +(-.6,-.9) and +(.5,0) .. (pd) |
|
636 .. controls +(-.5,0) and +(.6,-.9) .. (p1); |
|
637 \foreach \t in {0,.03,...,1} { |
|
638 \draw[green!50!brown] ($(p1)!\t!(p2) + (0,2)$) -- +(0,-4); |
|
639 } |
|
640 \end{scope} |
|
641 \draw (p1) .. controls +(.6,-.3) and +(-.5,-.3) .. (pu) |
|
642 .. controls +(.5,-.3) and +(-.6,-.3) .. (p2) |
|
643 .. controls +(-.6,-.9) and +(.5,0) .. (pd) |
|
644 .. controls +(-.5,0) and +(.6,-.9) .. (p1); |
|
645 \draw (p1) .. controls +(1,1) and +(-1,1) .. (pu); |
|
646 \draw (p2) .. controls +(-1,1) and +(1,1) .. (pu); |
|
647 \draw[<-, thick, orange!50!brown] (1.45,-0.1)-- node[left, black] {$f$} +(0,.7); |
|
648 \draw[<-, thick, orange!50!brown] (4.35,-0.1)-- node[left, black] {$g$} +(0,.7); |
|
649 \draw[->, thick, blue!75!yellow] (4.3,-1.5) node[black, above] {$(b\cdot d)\times I$} -- (3.3,-0.5); |
|
650 \end{tikzpicture}} |
|
651 \end{align*} |
577 \caption{Horizontal composition of 2-morphisms} |
652 \caption{Horizontal composition of 2-morphisms} |
578 \label{fzo5} |
653 \label{fzo5} |
579 \end{figure} |
654 \end{figure} |
580 \begin{figure}[t] |
655 \begin{figure}[t] |
581 $$ |
656 $$ |