String Diagram Tutorial

by Aaron Lauda

\[
\xy
(0,0)*{f};
(0,0)*\xycircle(2.65,2.65){-}="f";
(0,10)**\dir{-} ?(.5)*\dir{<}+(3,0)*{\scriptstyle x};
"f";(0,-10)**\dir{-} ?(.75)*\dir{>}+(3,0)*{\scriptstyle y};
\endxy
\]

Here I am using the \xycircle command to create a circle in this diagram. There are other ways to make circles but I kept having trouble with the pdf version of the document so now I just use this one. I simply gave the circle the name f and made directionals coming out both ends. The downside with making the circles this way is that I had to include the extra command to stick the label inside the circle. The nice thing about using \xycircle is that you can adjust the size of the circle to accomodate any size label. The +(3,0)*{\scriptstyle y} included after the directional is a quick way to stick on labels. Sometimes I have trouble getting labels to line up properly when I label things in this way, so I ofter just stick the label in using something like \(5,5)*{\scriptstyle y}.

\xy
(-5,0)*{f};
(-5,12)*{}; (-5,0)*\xycircle(2.65,2.65){-}="1_x"; **\dir{-}
?(.5)*\dir{<}+(3,0)*{\scriptstyle x}; "1_x";(-5,-12)*{}; **\dir{-}
?(.4)*\dir{<}+(3,0)*{\scriptstyle y}; (5,12)*{};
(5,0)*\xycircle(2.65,2.65){-}="1_x"; **\dir{-}
?(.5)*\dir{<}+(3,0)*{\scriptstyle x'}; "1_x";(5,-12)*{}; **\dir{-}
?(.4)*\dir{<}+(3,0)*{\scriptstyle y'}; (5,0)*{g};
\endxy
\qquad = \qquad \xy (0,12)*{};
(0,0)*\xycircle(2.95,2.95){-}="1_x"; **\dir{-}
?(.5)*\dir{<}+(4,3)*{\scriptstyle x \ten x'}; "1_x";(0,-12)*{};
**\dir{-} ?(.4)*\dir{<}+(4,0)*{\scriptstyle y \ten y'}; (0,0)*{
\scriptstyle f \ten g};
\endxy

\xy (0,12)*{};(0,-12)*{}; **\dir{-}
?(.47)*\dir{<}+(3,1)*{\scriptstyle x}
\endxy
\qquad = \qquad \xy (0,12)*{};
(0,0)*\xycircle(2.65,2.65){-}="1_x"; **\dir{-}
?(.5)*\dir{<}+(3,0)*{\scriptstyle x}; "1_x";(0,-12)*{}; **\dir{-}
?(.4)*\dir{<}+(3,0)*{\scriptstyle x}; (0,0)*{1_x};
\endxy

\xy
(0,10)*{};
(0,-10)*{}
**\dir{-}?(.54)*\dir{>}+(-2.5,1)*{\xb};
\endxy
\qquad = \qquad
\xy
(0,10)*{};
(0,-10)*{}
**\dir{-}?(.45)*\dir{<}+(2.5,-1)*{x};
\endxy

\xy
(-6,0)*{};
(6,0)*{};
**\crv{(0,18)} ?(.16)*\dir{>} ?(.9)*\dir{>};
(1,11)*{\scriptstyle i_{x}};
\endxy

\xy
(-6,4)*{};(6,4)*{};
**\crv{(0,-12)}
?(.20)*\dir{>}+(2,-1)
?(.89)*\dir{>}+(-2,-1);
(1,-6)*{\scriptstyle e_{x}};
\endxy

\xy
(0,-10)*++{}="g";
(10,0)*{}="mid";
**\crv{(5,20)} ?(.83)*\dir{>};
(6.5,10)*{\scriptstyle i_{x}};
(20,10)*++{}="f";
"f";"mid"; **\crv{(15,-20)} ?(0)*\dir{<} ?(.76)*\dir{<};
(15.5,-10)*{\scriptstyle e_{x}};
\endxy
\quad = \quad \xy (0,10)*{};(0,-10)*{}; **\dir{-}
?(.47)*\dir{<}+(3,1)*{\scriptstyle x}
\endxy
\qquad \qquad \xy
(0,10)*++{}="g";
(10,0)*{}="mid";
**\crv{(5,-20)} ?(.02)*\dir{>} ?(.83)*\dir{>};
(6,-10)*{\scriptstyle e_{x}};
(20,-10)*++{}="f";
"f";"mid"; **\crv{(15,20)} ?(.78)*\dir{<};
(15,10)*{\scriptstyle i_{x}};
\endxy
\quad = \quad \xy (0,10)*{};(0,-10)*{}; **\dir{-}
?(.53)*\dir{>}+(3,1)*{\scriptstyle x}
\endxy

\xy
(-6,0)*{};
(6,0)*{};
**\crv{(0,18)} ?(.16)*\dir{>} ?(.9)*\dir{>};
(1,11)*{};
\endxy

\xy
(-6,4)*{};(6,4)*{};
**\crv{(0,-12)}
?(.20)*\dir{>}+(2,-1)
?(.89)*\dir{>}+(-2,-1);
(1,-6)*{};
\endxy

\xy
(0,-10)*++{}="g";
(10,0)*{}="mid";
**\crv{(5,20)} ?(.83)*\dir{>};
(6.5,10)*{};
(20,10)*++{}="f";
"f";"mid"; **\crv{(15,-20)} ?(0)*\dir{<} ?(.76)*\dir{<};
(15.5,-10)*{};
\endxy
\quad = \quad \xy (0,10)*{};(0,-10)*{}; **\dir{-}
?(.47)*\dir{<}+(3,1)*{}
\endxy
\qquad \qquad \xy
(0,10)*++{}="g";
(10,0)*{}="mid";
**\crv{(5,-20)} ?(.02)*\dir{>} ?(.83)*\dir{>};
(20,-10)*++{}="f";
"f";"mid"; **\crv{(15,20)} ?(.78)*\dir{<};
(15,10)*{};
\endxy
\quad = \quad \xy (0,10)*{};(0,-10)*{}; **\dir{-}
?(.53)*\dir{>}+(3,1)*{}
\endxy

\xy
(-6,4)*{};(6,4)*{};
**\crv{(0,-12)}
?(.1)*\dir{<}+(2,-1)
?(.80)*\dir{<}+(-2,-1);
(1,-6)*{};
\endxy

\xy
(-6,0)*{};
(6,0)*{};
**\crv{(0,18)}
?(.10)*\dir{<}
?(.85)*\dir{<};
(1,11)*{};
\endxy

\xy
(6,0)*{}="1";
(-6,0)*{}="2";
"1";"2" **\crv{(7,15)&(-7,15)}
?(0)*\dir{>} ;
"2";"1" **\crv{ (-7,-15)& (7,-15)}
?(.05)*\dir{>} ;
\endxy
\quad = \quad \qquad \qquad \qquad \qquad \qquad \xy
(6,0)*{}="1";
(-6,0)*{}="2";
"2";"1" **\crv{(-7,15)&(7,15)}
?(0)*\dir{>} ;
"1";"2" **\crv{(7,-15) & (-7,-15)}
?(.05)*\dir{>} ;
\endxy
\quad = \quad \qquad \qquad

\xy
(-6,10)*{};
(6,10)*{};
**\crv{(0,-8)} ?(.22)*\dir{>} ?(.87)*\dir{>};
(6,-10)*{};
(-6,-10)*{};
**\crv{(0,8)} ?(.20)*\dir{>} ?(.87)*\dir{>};
\endxy
\quad = \quad \xy (-3,-10)*{};(-3,10)**\dir{-} ?(.57)*\dir{>};
(3,10)*{};(3,-10)**\dir{-} ?(.57)*\dir{>};
\endxy
\quad \qquad \qquad \qquad \quad \xy
(6,10)*{};
(-6,10)*{};
**\crv{(0,-8)} ?(.22)*\dir{>} ?(.87)*\dir{>};
(-6,-10)*{};
(6,-10)*{};
**\crv{(0,8)} ?(.20)*\dir{>} ?(.87)*\dir{>};
\endxy
\quad = \quad \xy (-3,10)*{};(-3,-10)**\dir{-} ?(.57)*\dir{>};
(3,-10)*{};(3,10)**\dir{-} ?(.57)*\dir{>};
\endxy

\xy
(0,-10)*++{}="g";
(10,0)*{}="mid";
**\crv{(5,20)} ?(.8)*\dir{<};
(6.5,10)*{};
(20,10)*++{}="f";
"f";"mid"; **\crv{(15,-20)} ?(.08)*\dir{>} ?(.8)*\dir{>};
(15.5,-10)*{};
\endxy
\quad = \quad \xy (0,10)*{};(0,-10)*{}; **\dir{-}
?(.53)*\dir{>}+(3,1)*{}
\endxy
\qquad \qquad \xy
(0,10)*++{}="g";
(10,0)*{}="mid";
**\crv{(5,-20)} ?(.02)*\dir{<} ?(.75)*\dir{<};
(6,-10)*{};
(20,-10)*++{}="f";
"f";"mid"; **\crv{(15,20)} ?(.83)*\dir{>};
(15,10)*{};
\endxy
\quad = \quad \xy (0,10)*{};(0,-10)*{}; **\dir{-}
?(.47)*\dir{<}+(3,1)*{}
\endxy

\xy
(-6,4)*{};(6,4)*{};
**\crv{(0,-12)}
?(.20)*\dir{>}+(-3,-5)*{\scriptstyle e_x};
?(.85)*\dir{>}+(-2,-1);

\endxy
\quad \xy
(6,-4)*{}; (-6,-4)*{};
**\crv{(0,12)}
?(.20)*\dir{>}+(5,5)*{\scriptstyle e_y^{-1}};
?(.85)*\dir{>}+(-2,1);
\endxy
\quad = \quad \xy
(-6,14)*{};
(6,14)*{};
**\crv{(0,-4)} ?(.22)*\dir{>} ?(.87)*\dir{>};
(6,-14)*{};
(-6,-14)*{};
**\crv{(0,4)} ?(.20)*\dir{>} ?(.87)*\dir{>};
(0,3)*{\scriptstyle e_x};
(1,-3)*{\scriptstyle e_y^{-1}};
\endxy

\xy (-6,-8)*{}="1E"; (-6,0)*{}="1"; (0,0)*{}="2"; (6,0)*{}="3";
(6,8)*{}="3B";
"2";"1" **\crv{(0,10)& (-6,10)}
?(.03)*\dir{>} ?(1)*\dir{>};
"3";"2" **\crv{(6,-10)& (0,-10)}
?(.03)*\dir{>} ;
"1";"1E" **\dir{-};
"3B";"3" **\dir{-};
\endxy
\qquad = \xy (-6,8)*{}; (0,8)*{}; (0,-8)*{}; **\dir{-}
?(.47)*\dir{<}; (6,-8)*{};
\endxy

\xy (-6,8)*{}="1E"; (-6,0)*{}="1"; (0,0)*{}="2"; (6,0)*{}="3";
(6,-8)*{}="3B";
"2";"1" **\crv{(0,-10)& (-6,-10)}
?(.03)*\dir{>} ?(1)*\dir{>};
"3";"2" **\crv{(6,10)& (0,10)}
?(.03)*\dir{>} ;
"1";"1E" **\dir{-};
"3B";"3" **\dir{-};
\endxy
\qquad = \qquad \xy (-6,8)*{}="1E"; (-6,0)*{}="1"; (0,0)*{}="2";
(6,0)*{}="3"; (6,-8)*{}="3B";
"2";"1" **\crv{(0,-10)& (-6,-10)}
?(.03)*\dir{>} ?(1)*\dir{>};
"3";"2" **\crv{(6,10)& (0,10)}
?(.03)*\dir{>} ;
"1";"1E" **\dir{-};
"3B";"3" **\dir{-};
(12,4)*{}="1";
(12,-4)*{}="A";
(18,-4)*{}="B";
(18,4)*{}="B'";
"B'";"1" **\crv{(18,10)& (12,10)};
"A";"B" **\crv{(12,-10)&(18,-10)};
"B";"B'" **\dir{-} ?(.60)*\dir{>};
"1";"A" **\dir{-}; ?(.57)*\dir{>};
\endxy

\xy (-6,0)*{}="1E"; (6,0)*{}="3B";
\endxy
\qquad = \qquad \xy (-6,8)*{}="1E"; (-6,0)*{}="1"; (0,0)*{}="2";
(6,0)*{}="3"; (6,-8)*{}="3B";
"2";"1" **\crv{(0,-10)& (-6,-10)}
?(.03)*\dir{>} ?(1)*\dir{>};
"3";"2" **\crv{(6,10)& (0,10)};
"1";"1E" **\dir{-};
"3B";"3" **\dir{-};
(12,4)*{}="1";
(12,-4)*{}="A";
(18,-4)*{}="B";
(18,4)*{}="B'";
"B'";"1" **\crv{(18,10)& (12,10)};
"A";"B" **\crv{(12,-10)&(18,-10)};
"B";"B'" **\dir{-} ?(.60)*\dir{>};
"1";"A" **\dir{-};
(6,4)*{}="D1";
(12,4)*{}="D2";
(6,-4)*{}="F1";
(12,-4)*{}="F2";
"D1";"D2" **\crv{~*=<2pt>{.}(6,0)&(12,0)};
"F1";"F2" **\crv{~*=<2pt>{.}(6,0)&(12,0)};
\endxy

\xy (-6,0)*{}="1E"; (6,0)*{}="3B";
\endxy
\qquad = \qquad \xy (-6,8)*{}="1E"; (-6,0)*{}="1"; (0,0)*{}="2";
(6,4)*{}="3"; (6,-8)*{}="3B";
"2";"1" **\crv{(0,-10)& (-6,-10)}
?(.03)*\dir{>} ?(1)*\dir{>};
"3";"2" **\crv{(6,10)& (-1,12)} ?(.05)*\dir{>};
"1";"1E" **\dir{-};
(12,4)*{}="1";
(12,-4)*{}="A";
(18,-4)*{}="B";
(18,4)*{}="B'";
"B'";"1" **\crv{(18,10)& (12,10)};
"A";"B" **\crv{(12,-10)&(18,-10)} ?(.07)*\dir{>};
"B";"B'" **\dir{-} ?(.60)*\dir{>};
(6,4)*{}="D1";
(12,4)*{}="D2";
(6,-4)*{}="F1";
(12,-4)*{}="F2";
"D1";"D2" **\crv{(6,0)&(12,0)} ?(1)*\dir{<};
"F1";"F2" **\crv{(6,0)&(12,0)}
?(.03)*\dir{>};
"3B";"F1" **\dir{-};
\endxy

\xy (-6,0)*{}="1E"; (6,0)*{}="3B";
\endxy
\qquad = \qquad \xy (-6,8)*{}="1E"; (-6,4)*{}="1"; (0,4)*{}="2";
(6,4)*{}="3"; (-6,-8)*{}="3B";
"2";"1" **\crv{(0,0)& (-6,0)};
?(.0)*\dir{>};
"3";"2" **\crv{(6,10)& (-1,12)} ?(.05)*\dir{>};
"1";"1E" **\dir{-}; ?(.25)*\dir{>};
(12,4)*{}="1";
(0,-4)*{}="A";
(18,-4)*{}="B";
(18,4)*{}="B'";
"B'";"1" **\crv{(18,10)& (12,10)};
"A";"B" **\crv{(0,-14)&(18,-14)} ?(.04)*\dir{>};
"B";"B'" **\dir{-} ?(.60)*\dir{>};
(6,4)*{}="D1";
(12,4)*{}="D2";
(-6,-4)*{}="F1";
(0,-4)*{}="F2";
"D1";"D2" **\crv{(6,0)&(12,0)} ?(1)*\dir{<};
"F1";"F2" **\crv{(-6,0)&(0,0)}
?(.03)*\dir{>};
"3B";"F1" **\dir{-};
\endxy

\xy (-6,0)*{}="1E"; (6,0)*{}="3B";
\endxy
\qquad = \qquad \xy (-6,8)*{}="1E"; (-6,4)*{}="1"; (0,4)*{}="2";
(6,4)*{}="3"; (-6,-8)*{}="3B";
"2";"1" **\crv{(0,0)& (-6,0)}
?(.0)*\dir{>};
"3";"2" **\crv{(6,10)& (-1,12)} ?(.05)*\dir{>};
"1";"1E" **\dir{-}; ?(.25)*\dir{>};
(12,4)*{}="1A";
(0,-4)*{}="A";
(18,-4)*{}="B";
(18,4)*{}="B'";
"B'";"1A" **\crv{(18,10)& (12,10)};
"A";"B" **\crv{(0,-14)&(18,-14)} ?(.04)*\dir{>};
"B";"B'" **\dir{-} ?(.60)*\dir{>};
(6,4)*{}="D1";
(12,4)*{}="D2";
(-6,-4)*{}="F1";
(0,-4)*{}="F2";
"D1";"D2" **\crv{(6,0)&(12,0)} ?(1)*\dir{<};
"F1";"F2" **\crv{(-6,0)&(0,0)}
?(.03)*\dir{>};
"3B";"F1" **\dir{-};
"1";"F1" **\dir{.};
"2";"F2" **\dir{.};
\endxy

\xy (-6,0)*{}="1E"; (6,0)*{}="3B";
\endxy
\qquad = \qquad \xy (-6,8)*{}="1E"; (-6,4)*{}="1"; (0,4)*{}="2";
(6,4)*{}="3"; (-6,-8)*{}="3B";
"3";"2" **\crv{(6,10)& (-1,12)}
?(.05)*\dir{>} ?(1)*\dir{>};
"1";"1E" **\dir{-};
(12,4)*{}="1A";
(0,-4)*{}="A";
(18,-4)*{}="B";
(18,4)*{}="B'";
"B'";"1A" **\crv{(18,10)& (12,10)};
"A";"B" **\crv{(0,-14)&(18,-14)};
"B";"B'" **\dir{-} ?(.60)*\dir{>};
(6,4)*{}="D1";
(12,4)*{}="D2";
(-6,-4)*{}="F1";
(0,-4)*{}="F2";
"D1";"D2" **\crv{(6,0)&(12,0)} ?(.95)*\dir{<};
"3B";"F1" **\dir{-};
"1";"F1" **\dir{-} ?(.47)*\dir{<};
"2";"F2" **\dir{-};
\endxy

\xy (-6,0)*{}="1E"; (6,0)*{}="3B";
\endxy
\qquad = \qquad \xy (-6,8)*{}="1E"; (-6,-8)*{}="1"; (18,0)*{};
"1";"1E" **\dir{-} ?(.53)*\dir{>};
(6,4)*{}="1";
(6,-4)*{}="A";
(12,-4)*{}="B";
(12,4)*{}="B'";
"B'";"1" **\crv{(12,10)& (6,10)};
"A";"B" **\crv{(6,-10)&(12,-10)};
"B";"B'" **\dir{-} ?(.60)*\dir{>};
"1";"A" **\dir{-}; ?(.57)*\dir{>};
\endxy

\xy (-6,0)*{}="1E"; (6,0)*{}="3B";
\endxy
\qquad = \qquad \xy (-6,8)*{}="1E"; (-6,-8)*{}="1"; (18,0)*{};
"1";"1E" **\dir{-} ?(.53)*\dir{>};
\endxy

\xy
(-6,-0)*{}="1";
(0,0)*{}="2";
(0,-5)*{}="2'";
(6,0)*{}="3";
(-6,-5)*{}="0'";
(6,0)*{}="A";
(12,0)="B";
"1";"0'" **\dir{-} ?(.05)*\dir{>};
"B";"1" **\crv{(12,18)& (-6,18)};
?(.03)*\dir{>};
"A";"B" **\crv{(6,-6)&(12,-6)}; ?(.02)*\dir{>};
"2";"A" **\crv{(0,6) & (6,6)};
?(0.05)*\dir{>};
"2'";"2" **\dir{-};
\endxy

\xy
(-6,-0)*{}="1";
(0,0)*{}="2";
(6,0)*{}="3";
(6,-5)*{}="3'";
(12,0)*{}="4";
(12,-5)*{}="4'";
"4";"1" **\crv{(12,18)& (-6,18)}
?(.03)*\dir{>}
?(1)*\dir{>};
"1";"2" **\crv{(-6,-6)&(0,-6)};
"2";"3" **\crv{(0,6)&(6,6)};
?(.04)*\dir{>} ;
"3";"3'" **\dir{-}
?(.1)*\dir{>};
"4";"4'" **\dir{-};
\endxy
\qquad = \qquad
\xy
(-6,-0)*{}="1";
(0,0)*{}="2";
(0,-5)*{}="2'";
(6,0)*{}="3";
(-6,-5)*{}="0'";
(6,0)*{}="A";
(12,0)="B";
"1";"0'" **\dir{-}
?(.05)*\dir{>};
"B";"1" **\crv{(12,18)& (-6,18)};
?(.03)*\dir{>};
"A";"B" **\crv{(6,-6)&(12,-6)};
?(.02)*\dir{>};
"2";"A" **\crv{(0,6) & (6,6)};
?(0.05)*\dir{>};
"2'";"2" **\dir{-};
\endxy

\xy
(-6,-0)*{}="1";
(0,0)*{}="2";
(6,0)*{}="3";
(12,0)*{}="4";
(12,-8)*{}="4'";
(6,-8)*{}="0";
"4";"1" **\crv{(12,18)& (-6,18)};
?(.02)*\dir{>} ;
"1";"2" **\crv{(-6,-6) & (0,-6)}
?(.04)*\dir{>};
"2";"3" **\crv{(0,6) & (6,6)}; ?(0.02)*\dir{>} ;
"4'";"4" **\dir{-};
"3";"0" **\dir{-}
?(.1)*\dir{>} ;
\endxy
\qquad = \qquad \xy
(-6,-0)*{}="1";
(0,0)*{}="2";
(6,0)*{}="3";
(12,0)*{}="4";
(12,-8)*{}="4'";
(6,-8)*{}="0";
"4";"1" **\crv{(12,18)& (-6,18)};
"1";"2" **\crv{(-6,-6) & (0,-6)};
?(.04)*\dir{>};
"2";"3" **\crv{(0,6) & (6,6)};
?(.02)*\dir{>} ;
"4'";"4" **\dir{-};
"3";"0" **\dir{-};
(6,2)*{}="A";
(6,-6)*{}="A'";
(12,2)="B";
(12,-6)="B'";
"A";"B" **\crv{~*=<2pt>{.}(6,-2)&(12,-2)};
"A'";"B'" **\crv{~*=<2pt>{.}(6,-2)&(12,-2)};
\endxy

\xy
(12,0)*{};
(-12,0)*{};
\endxy
\qquad = \qquad \xy
(-6,-0)*{}="1";
(0,0)*{}="2";
(6,0)*{}="3";
(12,0)*{}="4";
(12,-10)*{}="4'";
(6,-10)*{}="0'";
(6,2)*{}="A";
(6,-6)*{}="A'";
(12,2)="B";
(12,-6)="B'";
"B";"1" **\crv{(12,18)& (-6,18)}
?(.03)*\dir{>};
"1";"2" **\crv{(-6,-6) & (0,-6)}
?(.04)*\dir{>};
"2";"A" **\crv{(0,6) & (6,6)};
?(.02)*\dir{>} ;
"4'";"B'" **\dir{-};
?(.96)*\dir{>};
"A'";"0'" **\dir{-}; ?(.5)*\dir{>};
"A";"B" **\crv{(6,-2)&(12,-2)}; ?(0.05)*\dir{>} ;
"B'";"A'" **\crv{(12,-2)&(6,-2)};
\endxy

\xy
(12,0)*{};
(-12,0)*{};
\endxy
\qquad = \qquad \xy
(-6,-0)*{}="1";
(0,0)*{}="2";
(6,0)*{}="3";
(12,0)*{}="4";
(12,-10)*{}="4'";
(6,-10)*{}="0'";
(6,2)*{}="A";
(6,-6)*{}="A'";
(12,2)="B";
(12,-6)="B'";
"B";"1" **\crv{(12,18)& (-6,18)}
?(.03)*\dir{>};
"1";"2" **\crv{(-6,-6) & (0,-6)}
?(.04)*\dir{>};
"2";"A" **\crv{(0,6) & (6,6)};
?(.02)*\dir{>} ;
"4'";"B'" **\dir{-}
?(.96)*\dir{>};
"A'";"0'" **\dir{-} ?(.5)*\dir{>};
"A";"B" **\crv{(6,-2)&(12,-2)}
?(.05)*\dir{>} ;
"B'";"A'" **\crv{(12,-2)&(6,-2)};
(-3,0)*{}="X";
(9,-6)*{}="XX";
"X";"XX" **\dir{--};
\endxy

\xy
(12,0)*{};
(-12,0)*{};
\endxy
\qquad = \qquad \xy
(-6,2)*{}="1";
(0,2)*{}="2";
(6,0)*{}="3";
(12,0)*{}="4";
(0,-10)*{}="4'";
(-6,-10)*{}="0'";
(6,2)*{}="A";
(-6,-6)*{}="A'";
(12,2)="B";
(0,-6)="B'";
"B";"1" **\crv{(12,18)& (-6,18)}
?(.03)*\dir{>};
"1";"2" **\crv{(-6,-2) & (0,-2)}
?(.04)*\dir{>};
"2";"A" **\crv{(0,6) & (6,6)};
?(.05)*\dir{>} ;
"4'";"B'" **\dir{-}
?(.96)*\dir{>};
"A'";"0'" **\dir{-} ?(.5)*\dir{>};
"A";"B" **\crv{(6,-2)&(12,-2)}
?(.05)*\dir{>} ;
"B'";"A'" **\crv{(0,-2)&(-6,-2)};
\endxy

\xy
(-12,-0)*{};
(12,0)*{};
\endxy
\qquad = \qquad \xy
(-6,-0)*{}="1";
(0,0)*{}="2";
(0,-8)*{}="2'";
(6,0)*{}="3";
(-6,-8)*{}="0'";
(6,2)*{}="A";
(12,2)="B";
"1";"0'" **\dir{-} ?(.05)*\dir{>};
"B";"1" **\crv{(12,18)& (-6,18)};
?(.03)*\dir{>};
"A";"B" **\crv{(6,-2)&(12,-2)};
?(.05)*\dir{>};
"2";"A" **\crv{(0,6) & (6,6)};
?(.03)*\dir{>};
"2'";"2" **\dir{-};
\endxy

\xy
(0,-10)*++{}="g";
(10,0)*{}="mid";
**\crv{(5,20)} ?(.83)*\dir{>};
(6.5,10)*{\scriptstyle i'_x};
(20,10)*++{}="f";
"f";"mid"; **\crv{(15,-20)} ?(0)*\dir{<} ?(.76)*\dir{<};
(15.5,-10)*{\scriptstyle e'_x};
\endxy
\quad = \quad \xy (0,10)*{};(0,-10)*{}; **\dir{-}
?(.47)*\dir{<}+(3,1)
\endxy

\xy
(0,10)*++{}="g";
(10,0)*{}="mid";
**\crv{(5,-20)} ?(.02)*\dir{>} ?(.83)*\dir{>};
(6,-10)*{\scriptstyle e'_x};
(20,-10)*++{}="f";
"f";"mid"; **\crv{(15,20)} ?(.78)*\dir{<};
(15,10)*{\scriptstyle i'_x};
\endxy
\quad = \quad \xy (0,10)*{};(0,-10)*{}; **\dir{-}
?(.53)*\dir{>}+(3,1)
\endxy

\vcenter{ \xy
(-6,-0)*{}="1";
(0,0) *{}="2";
(6,0) *{}="3";
(6,-5) *{}="3'";
(12,0) *{}="4";
(18,0) *={}="5";
(18,12)*={}="5'";
"5";"4" **\crv{(18,-6)& (12,-6)};
"5'";"5" **\dir{-} ?(1)*\dir{>};
"4";"1" **\crv{(12,18)& (-6,18)};
?(.03)*\dir{>} ?(1)*\dir{>};
"1";"2" **\crv{(-6,-6)&(0,-6)};
"2";"3" **\crv{(0,6)&(6,6)};
?(.08)*\dir{>} ;
"3";"3'" **\dir{-}
?(0)*\dir{>};
(16,-7)*{\scriptstyle e_{x}}; %\
(4,7)*{\scriptstyle e_{x}^{-1}}; % LABELS FOR STRINGS
(4,16)*{\scriptstyle i_{x}}; %/
(-2,-7)*{\scriptstyle i_{x}^{-1}};
\endxy}
\qquad = \qquad \xy (0,8)*{};(0,-10)**\dir{-} ?(.53)*\dir{>};
\endxy

\vcenter{ \xy
(-6,-0)*{}="1";
(0,0)*{}="2";
(6,0)*{}="3";
(12,0)*{}="4";
(12,-12)*{}="4'";
(-12,12)*{}="0'";
(-12,0)*{}="0";
"4";"1" **\crv{(12,18)& (-6,18)};
?(.02)*\dir{>} ;
"1";"2" **\crv{(-6,-6) & (0,-6)}
?(.04)*\dir{>};
"2";"3" **\crv{(0,6) & (6,6)};
?(.08)*\dir{>} ;
"4'";"4" **\dir{-};
"3";"0" **\crv{(6,-18)&(-12,-18)}
?(.02)*\dir{>} ;
"0";"0'" **\dir{-} ?(.05)*\dir{>};
(3,16)*{\scriptstyle i_{x}};
(-3,-16)*{\scriptstyle e_{x}};
(-3,-7)*{\scriptstyle i_{x}^{-1}};
(4,7)*{\scriptstyle e_{x}^{-1}};
\endxy}
\qquad = \qquad \xy (0,12)*{};(0,-12)**\dir{-} ?(.47)*\dir{<};
\endxy

\xy
(-6,-0)*{}="1";
(0,0) *{}="2";
(6,0) *{}="3";
(6,-5) *{}="3'";
(12,0) *{}="4";
(18,0) *={}="5";
(18,12)*={}="5'";
"5";"4" **\crv{(18,-6)& (12,-6)};
"5'";"5" **\dir{-} ?(1)*\dir{>};
"4";"1" **\crv{(12,18)& (-6,18)};
?(.03)*\dir{>} ?(1)*\dir{>};
"1";"2" **\crv{(-6,-6)&(0,-6)};
"2";"3" **\crv{(0,6)&(6,6)};
?(.08)*\dir{>} ;
"3";"3'" **\dir{-}
?(0)*\dir{>};
\endxy
\qquad = \qquad \xy
(-6,-0)*{}="1";
(0,0)*{}="2";
(6,0)*{}="3";
(6,-5)*{}="3'";
(12,0)*{}="4";
(18,0)*={}="5";
(18,12)*={}="5'";
"5";"4" **\crv{(18,-6)& (12,-6)};
"5'";"5" **\dir{-} ?(1)*\dir{>};
"4";"1" **\crv{(12,18)& (-6,18)}
?(.03)*\dir{>} ?(1)*\dir{>};
"1";"2" **\crv{(-6,-6)&(0,-6)};
"2";"3" **\crv{(0,6)&(6,6)};
?(.08)*\dir{>} ;
"3";"3'" **\dir{-}
?(0)*\dir{>} ;
(3,0)*{}="A";
(15,0)*{}="B";
"A";"B" **\dir{--};
\endxy

\xy
(-6,-0)*{}="1";
(0,0) *{}="2";
(6,0) *{}="3";
(6,-5) *{}="3'";
(12,0) *{}="4";
(18,0) *={}="5";
(18,12)*={}="5'";
\endxy
\qquad = \qquad \xy
(-6,-0)*{}="1";
(6,-5) *{}="2";
(12,-5) *{}="3";
(12,-10) *{}="3'";
(6,5) *{}="4";
(12,5) *={}="5";
(12,10)*={}="5'";
(18,0) *={}="";
"5";"4" **\crv{(12,-1)& (6,-1)}
?(1)*\dir{>};
"5'";"5" **\dir{-} ?(1)*\dir{>};
"4";"1" **\crv{(4,12)& (-6,12)};
?(1)*\dir{>};
"1";"2" **\crv{(-6,-12)&(4,-12)};
"2";"3" **\crv{(6,1)&(12,1)};
?(.08)*\dir{>} ;
"3";"3'" **\dir{-}
?(0)*\dir{>};
\endxy

\xy
(-6,-0)*{}="";
(18,0) *={}="";
\endxy
\qquad = \qquad \xy
(-6,-0)*{}="1";
(6,-5) *{}="2";
(12,-5) *{}="3";
(12,-10) *{}="3'";
(6,5) *{}="4";
(12,5) *={}="5";
(12,10)*={}="5'";
(18,0) *={}="";
"5";"4" **\crv{(12,-1)& (6,-1)}
?(1)*\dir{>};
"5'";"5" **\dir{-} ?(1)*\dir{>};
"4";"1" **\crv{(4,12)& (-6,12)};
?(1)*\dir{>};
"1";"2" **\crv{(-6,-12)&(4,-12)};
"2";"3" **\crv{(6,1)&(12,1)};
?(.08)*\dir{>} ;
"3";"3'" **\dir{-}
?(0)*\dir{>};
"2";"4" **\dir{.};
"3";"5" **\dir{.};
\endxy

\xy
(-6,-0)*{}="";
(18,0) *={}="";
\endxy
\qquad = \qquad \xy
(6,0)*{}="1";
(-6,0)*{}="2";
"1";"2" **\crv{(7,15)&(-7,15)}
?(0)*\dir{>} ;
"2";"1" **\crv{ (-7,-15)& (7,-15)}
?(.05)*\dir{>} ;
\endxy
\quad \xy (0,10)*{};(0,-10)**\dir{-} ?(.57)*\dir{>};
\endxy \qquad

\xy
(-6,-0)*{}="1";
(18,0) *={}="5";
\endxy
\qquad = \qquad \xy
(6,0)*{}="";
(-6,0)*{}="";
\endxy
\quad \xy (0,10)*{};(0,-10)**\dir{-} ?(.57)*\dir{>};
\endxy \qquad

\xy
(-6,-0)*{}="1";
(0,0)*{}="2";
(6,0)*{}="3";
(12,0)*{}="4";
(12,-12)*{}="4'";
(-12,12)*{}="0'";
(-12,0)*{}="0";
"4";"1" **\crv{(12,18)& (-6,18)};
?(.02)*\dir{>} ;
"1";"2" **\crv{(-6,-6) & (0,-6)}
?(.04)*\dir{>};
"2";"3" **\crv{(0,6) & (6,6)};
?(.08)*\dir{>} ;
"4'";"4" **\dir{-};
"3";"0" **\crv{(6,-18)&(-12,-18)}
?(.02)*\dir{>} ;
"0";"0'" **\dir{-} ?(.05)*\dir{>};
\endxy
\qquad = \qquad \xy
(-6,-0)*{}="1";
(0,0)*{}="2";
(0,-8)*{}="2'";
(6,0)*{}="3";
(-12,12)*{}="0'";
(-12,0)*{}="0";
(6,2)*{}="A";
(12,2)="B";
"0";"0'" **\dir{-} ?(.05)*\dir{>};
"1";"0" **\crv{(-6,-12) &(-12,-12)}
?(.05)*\dir{>};
"B";"1" **\crv{(12,18)& (-6,18)};
?(.03)*\dir{>};
"A";"B" **\crv{(6,-2)&(12,-2)};
?(.08)*\dir{>};
"2";"A" **\crv{(0,6) & (6,6)};
?(.03)*\dir{>};
"2'";"2" **\dir{-};
\endxy

\xy
(-12,-0)*{};
(12,0)*{};
\endxy
\qquad = \qquad \xy
(-6,-0)*{}="1";
(0,0)*{}="2";
(0,-8)*{}="2'";
(6,0)*{}="3";
(-12,12)*{}="0'";
(-12,0)*{}="0";
(6,2)*{}="A";
(12,2)="B";
"0";"0'" **\dir{-} ?(.05)*\dir{>};
"1";"0" **\crv{(-6,-12) &(-12,-12)}
?(.05)*\dir{>};
"B";"1" **\crv{(12,18)& (-6,18)};
?(.03)*\dir{>};
"A";"B" **\crv{(6,-2)&(12,-2)};
?(.08)*\dir{>};
"2";"A" **\crv{(0,6) & (6,6)};
?(.03)*\dir{>};
"2'";"2" **\dir{-};
(-9,-6)*{}="X";
(3,1)*{}="XX";
"X";"XX" **\dir{--};
\endxy

\xy
(-12,-0)*{};
(12,0)*{};
\endxy
\qquad = \qquad \xy
(-12,12)*{}="0'";
(-12,8)*{}="0";
(-6,8)*{}="1";
(-12,-2)*{}="2";
(-12,-6)*{}="2'";
(6,-2)*{}="3";
(-6,-2)*{}="A";
(0,-2)*{}="B";
(0,8)*{}="B'";
(12,0)*{}="";
"0";"0'" **\dir{-} ?(.05)*\dir{>};
"1";"0" **\crv{(-6,2) &(-12,2)}
?(.05)*\dir{>};
"B'";"1" **\crv{(0,14)& (-6,14)};
?(.03)*\dir{>};
"A";"B" **\crv{(-6,-6)&(0,-6)};
?(.04)*\dir{>};
"2";"A" **\crv{(-12,4) & (-6,4)};
?(.03)*\dir{>};
"2'";"2" **\dir{-};
"B";"B'" **\dir{-} ?(.06)*\dir{>};;
\endxy

\xy
(-12,-0)*{};
(12,0)*{};
\endxy
\qquad = \qquad \xy
(-12,12)*{}="0'";
(-6,8)*{}="1";
(-12,-5)*{}="2'";
(6,-2)*{}="3";
(-6,-2)*{}="A";
(0,-2)*{}="B";
(0,8)*{}="B'";
(12,0)*{}="";
"B'";"1" **\crv{(0,14)& (-6,14)};
"A";"B" **\crv{(-6,-6)&(0,-6)};
"B";"B'" **\dir{-} ?(.60)*\dir{>};
"2'";"0'" **\dir{-} ?(.53)*\dir{>};
"1";"A" **\dir{-}; ?(.57)*\dir{>};
\endxy

\xy
(-12,-0)*{};
(12,0)*{};
\endxy
\qquad = \qquad \xy
(-12,12)*{}="0'";
(-12,-5)*{}="2'";
(12,0)*{}="";
"2'";"0'" **\dir{-} ?(.53)*\dir{>};
\endxy

\xy (-15,0)*{};(15,0)*{}; (-5,12)*{}; (-5,-12)*{};
**\dir{-}?(.5)*\dir{<}+(-4,0)*{\scriptstyle F(x)};
(5,0)*\xycircle(2.75,2.75){-}="f"; (5,12)**\dir{-}
?(.75)*\dir{>}+(5,-1)*{\scriptstyle \overline{F(x)}};
"f";(5,-12)**\dir{-} ?(.5)*\dir{<}+(5,-1)*{\scriptstyle F(\xb)};
(5,0)*{\scriptstyle F_{-1}};
\endxy
\qquad \qquad = \qquad \qquad \xy
(6,15)*{};
(-6,15)*{};
**\crv{(0,-3)}
?(.22)*\dir{>}+(4,-5.5)*{\scriptstyle i_{F(x)}^{-1}}
?(.87)*\dir{>};
(-6,-15)*{};
(6,-15)*{};
**\crv{(0,3)} ?(.20)*\dir{>} ?(.87)*\dir{>};+(-5,10.5)*{\scriptstyle \fix}
\endxy

\xy (0,0)*{\scriptstyle F_{-1}}; (-10,0)*{};(10,0)*{};
(0,0)*\xycircle(2.75,2.75){-}="f"; (0,14)**\dir{-}
?(.75)*\dir{>}+(5,-1)*{\scriptstyle \overline{F(x)}};
"f";(0,-14)**\dir{-} ?(.5)*\dir{<}+(5,-1)*{\scriptstyle F(\xb)};
\endxy
\qquad = \qquad \xy (-15,0)*{};(15,0)*{}; (-8,10)*{}="TL";
(0,10)*{}="TR"; (-8,-10)*{}="BL"; (0,-10)*{}="BR";
"TL";"BL" **\dir{-};
?(.5)*\dir{<};
"TR";"BR" **\dir{-};
?(.5)*\dir{>};
"TL";"TR" **\crv{(-8,18)& (0,18)};
"BL";"BR" **\crv{(-8,-18)& (0,-18)};
(8,0)*{\scriptstyle F_{-1}}; (8,0)*\xycircle(2.75,2.75){-}="f";
(8,14)**\dir{-} ?(.75)*\dir{>}+(5,-1)*{\scriptstyle
\overline{F(x)}}; "f";(8,-14)**\dir{-}
?(.5)*\dir{<}+(5,-1)*{\scriptstyle F(\xb)};
(3.5,5)*{\scriptstyle F(x)};
(-12,5)*{\scriptstyle \overline{F(x)}};
\endxy

\xy (-10,0)*{};(10,0)*{};
\endxy
\qquad = \qquad \xy (-15,0)*{};(15,0)*{}; (-8,10)*{}="TL";
(0,10)*{}="TR"; (-8,-10)*{}="BL"; (0,-10)*{}="BR"; (8,10)*{}="RU";
(8,-10)*{}="RB";
"TL";"BL" **\dir{-};
?(.5)*\dir{<};
"TR";"BR" **\dir{-};
?(.5)*\dir{>};
"TL";"TR" **\crv{(-8,18)& (0,18)};
"BL";"BR" **\crv{(-8,-18)& (0,-18)};
"RU";"TR" **\crv{~*=<2pt>{.}(8,4)& (0,4)};
"RB";"BR" **\crv{~*=<2pt>{.}(8,-4)& (0,-4)};
(8,0)*{\scriptstyle F_{-1}}; (8,0)*\xycircle(2.75,2.75){-}="f";
(8,14)**\dir{-} ?(.75)*\dir{>}+(5,-1)*{\scriptstyle
\overline{F(x)}}; "f";(8,-14)**\dir{-}
?(.5)*\dir{<}+(5,-1)*{\scriptstyle F(\xb)};
(-12,5)*{\scriptstyle \overline{F(x)}};
\endxy

\xy (-10,0)*{};(10,0)*{};
\endxy
\qquad = \qquad \xy (-15,0)*{};(15,0)*{}; (-8,10)*{}="TL";
(0,10)*{}="TR"; (-8,-10)*{}="BL"; (0,-10)*{}="BR"; (8,10)*{}="RU";
(8,16)*{}="RUEND"; (8,-16)*{}="RBEND"; (8,-10)*{}="RB";
"TL";"BL" **\dir{-};
?(.5)*\dir{<};
"RU";"RUEND" **\dir{-};
?(.15)*\dir{>}+(5,2)*{\scriptstyle \overline{F(x)}};
"RB";"RBEND" **\dir{-};
?(0)*\dir{<}+(5,-2)*{\scriptstyle F(\xb)};
"TL";"TR" **\crv{(-8,18)& (0,18)};
?(1)*\dir{>};
"BL";"BR" **\crv{(-8,-18)& (0,-18)};
?(.95)*\dir{<}+(4,8)*{\scriptstyle \fix} ;;
"RU";"TR" **\crv{(8,4)& (0,4)};
"RB";"BR" **\crv{(8,-4)& (0,-4)};
(-12,5)*{\scriptstyle \overline{F(x)}};
\endxy

\xy (-10,0)*{};(10,0)*{};
\endxy
\qquad = \qquad \xy (-15,0)*{};(15,0)*{}; (-8,10)*{}="TL";
(-8,-10)*{}="BL"; (0,-10)*{}="BR"; (8,-16)*{}="RBEND";
(8,-10)*{}="RB";
"TL";"BL" **\dir{-};
?(.5)*\dir{<}+(5,8)*{\scriptstyle \overline{F(x)}};
"RB";"RBEND" **\dir{-};
?(0)*\dir{<}+(5,-2)*{\scriptstyle F(\xb)};
"BL";"BR" **\crv{(-8,-18)& (0,-18)};
?(.95)*\dir{<}+(4,8)*{\scriptstyle \fix} ;;
"RB";"BR" **\crv{(8,-4)& (0,-4)};
\endxy

\xy (-15,0)*{};(15,0)*{}; (-5,12)*{}; (-5,-12)*{};
**\dir{-}?(.5)*\dir{<}+(-4,0)*{\scriptstyle F(x)};
(5,0)*{\scriptstyle F_{-1}}; (5,0)*\xycircle(2.75,2.75){-}="f";
(5,12)**\dir{-} ?(.75)*\dir{>}+(5,-1)*{\scriptstyle
\overline{F(x)}}; "f";(5,-12)**\dir{-}
?(.5)*\dir{<}+(5,-1)*{\scriptstyle F(\xb)};
\endxy
\qquad = \qquad \xy
(-10,12)*{}; (-10,-12)*{}; **\dir{-}?(.5)*\dir{<}+(-4,0)*{\scriptstyle F(x)};
(-2,12)*{}="1";
(6,-7) *{}="2";
(14,-7) *{}="3";
(14,-13)*{}="3'";
(18,0) *={}="";
"1";"2" **\crv{(-4,-21)&(5,-15)};
?(.15)*\dir{<}+(5,10)*{\scriptstyle \overline{F(x)}};
"2";"3" **\crv{(6,1)&(14,1)};
?(0)*\dir{<}+(4,8)*{\scriptstyle \fix} ;
?(.95)*\dir{<}+(5,-2)*{\scriptstyle F(\xb)};
"3";"3'" **\dir{-}

\endxy

\xy (-15,0)*{};(15,0)*{};
\endxy
\qquad = \qquad \xy
(-10,12)*{}; (-10,-12)*{}; **\dir{-}?(.5)*\dir{<}+(-4,0)*{\scriptstyle F(x)};
(-2,12)*{}="1";
(6,-7) *{}="2";
(14,-7) *{}="3";
(14,-13)*{}="3'";
(18,0) *={}="";
"1";"2" **\crv{(-4,-21)&(5,-15)};
?(.15)*\dir{<}+(5,10)*{\scriptstyle \overline{F(x)}};
"2";"3" **\crv{(6,1)&(14,1)};
?(0)*\dir{<}+(4,8)*{\scriptstyle \fix} ;
?(.95)*\dir{<}+(5,-2)*{\scriptstyle F(\xb)};
"3";"3'" **\dir{-};
(-2,6)*{}="X1";
(-10,6)*{}="X2";
"X1";"X2" **\crv{~*=<2pt>{.}(-2,-1)&(-10,-1)};
(-2,-6)*{}="XA";
(-10,-6)*{}="XB";
"XA";"XB" **\crv{~*=<2pt>{.}(-2,1)&(-10,1)};
\endxy

\xy (-15,0)*{};(15,0)*{};
\endxy
\qquad = \qquad \xy
(-10,12)*{}="0";
(-2,12)*{}="1";
(6,-7) *{}="2";
(14,-7) *{}="3";
(14,-13)*{}="3'";
(18,0) *={}="";
(-2,6)*{}="X1";
(-10,6)*{}="X2";
(-2,-6)*{}="XA";
(-10,-6)*{}="XB";
(-10,-12)*{}="XB'";
"XB";"XB'" **\dir{-}; ?(.4)*\dir{>};
"X1";"1" **\dir{-}; ?(.3)*\dir{>};
"X2";"0" **\dir{-}; ?(0)*\dir{<};
"XA";"2" **\crv{(-1,-15)&(6,-15)};
?(0)*\dir{<};
"2";"3" **\crv{(6,1)&(14,1)};
?(0)*\dir{<}+(4,8)*{\scriptstyle \fix} ;
?(.95)*\dir{<}+(5,-2)*{\scriptstyle F(\xb)};
"3";"3'" **\dir{-};
"X1";"X2" **\crv{(-2,-1)&(-10,-1)};
"XA";"XB" **\crv{(-2,1)&(-10,1)};
\endxy

\xy (-15,0)*{};(15,0)*{};
\endxy
\qquad = \qquad \xy
(-10,12)*{}="0";
(-2,12)*{}="1";
(6,-7) *{}="2";
(6,-13) *{}="2'";
(14,-7) *{}="3";
(14,-13)*{}="3'";
(18,0) *={}="";
(-2,6)*{}="X1";
(-10,6)*{}="X2";
"X1";"1" **\dir{-}; ?(.3)*\dir{>};
"X2";"0" **\dir{-}; ?(0)*\dir{<};
"2";"3" **\crv{(6,1)&(14,1)};
?(0)*\dir{<}+(4,8)*{\scriptstyle \fix} ;
?(.95)*\dir{<}+(5,-2)*{\scriptstyle F(\xb)};
"3";"3'" **\dir{-};
"2";"2'" **\dir{-};
"X1";"X2" **\crv{(-2,-1)&(-10,-1)};
\endxy

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