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Appended is the latest version.  It is, I hope, very close to the final
version.  I have gotten almost no feedback from the first version.  I
encourage you to try it and let me know any problems or questions.  The
next email will be the revised version of the documentation.


%%%%%%%%%%%%%%%%%%%%%%%%%% diagxy.tex %%%%%%%%%%%%%%%
 % This is a package of commutative diagram macros built on top of Xy-pic
 % by Michael Barr (email: barr@barrs.org).  This may be freely
 % distributed, unchanged, for non-commercial use.  If changed, it must be
 % renamed.  Inclusion in a commercial software package is also
 % permitted, provided I receive one free copy of the software package for
 % my personal use.  There are no guarantees that this package is good for
 % anything.  I have tested it with LaTeX 2e, LaTeX 2.09 and Plain TeX.
 % Although I know of no reason it will not work with AMSTeX, I have not
 % tested it.

\input xy
\xyoption{arrow}

\newdir{ >}{{ }*!/-9pt/@{>}}
\newdir{ (}{{ }*!/-5pt/@{(}}
\newdir^{ (}{{ }*!/-5pt/@^{(}}
\newdir{< }{!/9pt/@{<}*{ }}

\newbox\Label%
\newdimen\high%
\newdimen\deep%
\newdimen\ul%
\newcount\deltax%
\newcount\deltay%
\newcount\deltaX%
\newcount\deltaY%

\newdimen\wdth
\newcount\xend%
\newcount\yend%
\newcount\Xend
\newcount\Yend
\newcount\xpos%
\newcount\ypos%
\newcount\default \default=500%
\newcount\defaultmargin \defaultmargin=200%
\newif\ifat%
\newcount\topw%
\newcount\botw%
\newcount\Xpos%
\newcount\Ypos%
\def\ratchet#1#2{\ifnum#1<#2\global #1=#2\fi}%

\newcount\atcode \atcode=\catcode`\@%
\catcode`\@=11
\expandafter\ifx\csname @ifnextchar\endcsname\relax
\def\ifnextchar#1#2#3{\let\@tempe
#1\def\@tempa{#2}\def\@tempb{#3}\futurelet
    \@tempc\@ifnch}
\def\@ifnch{\ifx \@tempc \@sptoken \let\@tempd\@xifnch
      \else \ifx \@tempc \@tempe\let\@tempd\@tempa\else\let\@tempd\@tempb\fi
      \fi \@tempd}
\def\:{\let\@sptoken= } \:  % this makes \@sptoken a space token
\def\:{\@xifnch} \expandafter\def\: {\futurelet\@tempc\@ifnch}
\else
\let\ifnextchar\@ifnextchar
\fi
\ul=.01em%
\X@xbase =.01em%
\Y@ybase =.01em%
\def\scalefactor#1{\ul=#1\ul \X@xbase=#1\X@xbase \Y@ybase=#1\Y@ybase}%
\catcode`\@=12%

\let\bfig\xy%
\let\efig\endxy%

\def\car#1#2\nil{#1}%

\def\morphism(#1,#2)|#3|/#4/<#5,#6>[#7`#8`#9]{%
\xend#1\advance \xend by #5%
\yend#2\advance \yend by #6%
\domorphism(#1,#2)|#3|/#4/<#5,#6>[#7`#8`#9]}

\def\domorphism(#1,#2)|#3|/#4/<#5,#6>[#7`#8`#9]{%
\edef\next{#4}%
\ifx\next\empty \atfalse
\else \def\next{\car#4\nil}%
   \if\next@\relax\attrue \else \atfalse \fi
\fi%
\ifat%
 \edef\next{#4}%
 \ifx\next\empty%
  \POS(#1,#2)*+{#7}\ar#4 (\xend,\yend)*+{#8}%
 \else \if#3l%
  \ifnum #6>0%
   \POS(#1,#2)*+{#7}\ar#4^-{#9} (\xend,\yend)*+{#8}%
  \else%
   \POS(#1,#2)*+{#7}\ar#4_-{#9} (\xend,\yend)*+{#8}%
  \fi%
 \else \if#3m%
   \edef\next{#9}%
   \ifx\next\empty
     \POS(#1,#2)*+{#7}\ar#4 (\xend,\yend)*+{#8}%
   \else
     \setbox\Label=\hbox{\kern .5pt $\labelstyle #9$\kern .5pt}%
     \high=\ht\Label \advance\high by 1pt \ht\Label=\high%
     \deep=\dp\Label \advance\deep by 1pt \dp\Label=\deep%
     \POS(#1,#2)*+{#7}\ar#4|-{\box\Label} (\xend,\yend)*+{#8}%
   \fi
 \else \if#3r%
  \ifnum #6<0%
   \POS(#1,#2)*+{#7}\ar#4^-{#9} (\xend,\yend)*+{#8}%
  \else%
   \POS(#1,#2)*+{#7}\ar#4_-{#9} (\xend,\yend)*+{#8}%
  \fi%
 \else \if#3a%
  \ifnum #5>0%
   \POS(#1,#2)*+{#7}\ar#4^-{#9} (\xend,\yend)*+{#8}%
  \else%
   \POS(#1,#2)*+{#7}\ar#4_-{#9} (\xend,\yend)*+{#8}%
  \fi%
 \else \if#3b%
  \ifnum #5<0%
   \POS(#1,#2)*+{#7}\ar#4^-{#9} (\xend,\yend)*+{#8}%
  \else%
   \POS(#1,#2)*+{#7}\ar#4_-{#9} (\xend,\yend)*+{#8}%
  \fi%
 \else
   \POS(#1,#2)*+{#7}\ar#4 (\xend,\yend)*+{#8}%
 \fi\fi\fi\fi\fi\fi%
\else%
\edef\next{#4}%
 \ifx\next\empty%
  \POS(#1,#2)*+{#7}; (\xend,\yend)*+{#8}%
 \else \edef\next{#4}%
  \ifx\next\empty%
   \POS(#1,#2)*+{#7}\ar (\xend,\yend)*+{#8}%
 \else \if#3l%
  \ifnum #6>0%
   \POS(#1,#2)*+{#7}\ar@{#4}^-{#9} (\xend,\yend)*+{#8}%
  \else%
   \POS(#1,#2)*+{#7}\ar@{#4}_-{#9} (\xend,\yend)*+{#8}%
  \fi%
 \else \if#3m%
   \edef\next{#9}%
   \ifx\next\empty
     \POS(#1,#2)*+{#7}\ar@{#4} (\xend,\yend)*+{#8}%
   \else
     \setbox\Label=\hbox{\kern .5pt $\labelstyle #9$\kern .5pt}%
     \high=\ht\Label \advance\high by 1pt \ht\Label=\high%
     \deep=\dp\Label \advance\deep by 1pt \dp\Label=\deep%
     \POS(#1,#2)*+{#7}\ar@{#4}|-{\box\Label} (\xend,\yend)*+{#8}%
   \fi
 \else \if#3r%
  \ifnum #6<0%
   \POS(#1,#2)*+{#7}\ar@{#4}^-{#9} (\xend,\yend)*+{#8}%
  \else%
   \POS(#1,#2)*+{#7}\ar@{#4}_-{#9} (\xend,\yend)*+{#8}%
  \fi%
 \else \if#3a%
  \ifnum #5>0%
   \POS(#1,#2)*+{#7}\ar@{#4}^-{#9} (\xend,\yend)*+{#8}%
  \else%
   \POS(#1,#2)*+{#7}\ar@{#4}_-{#9} (\xend,\yend)*+{#8}%
  \fi%
 \else \if#3b%
  \ifnum #5<0%
   \POS(#1,#2)*+{#7}\ar@{#4}^-{#9} (\xend,\yend)*+{#8}%
  \else%
   \POS(#1,#2)*+{#7}\ar@{#4}_-{#9} (\xend,\yend)*+{#8}%
  \fi%
 \else
   \POS(#1,#2)*+{#7}\ar@{#4} (\xend,\yend)*+{#8}%
 \fi\fi\fi\fi\fi\fi\fi%
\fi\ignorespaces}%

\def\location(#1,#2){\xpos=#1\ypos=#2\ignorespaces}%
\def\squareplacements|#1`#2`#3`#4|{%
    \def\xa{#1}\def\xb{#2}\def\xc{#3}\def\xd{#4}\ignorespaces}%
\def\sizes<#1,#2>{\deltax=#1\deltay=#2\ignorespaces}%
\def\squaredata[#1`#2`#3`#4;#5`#6`#7`#8]{%
    \def\nodea{#1}\def\nodeb{#2}\def\nodec{#3}\def\noded{#4}%
    \def\labela{#5}\def\labelb{#6}\def\labelc{#7}\def\labeld{#8}\ignorespaces}%
\def\squaredata[#1`#2`#3`#4;#5]{%
 \def\nodea{#1}\def\nodeb{#2}\def\nodec{#3}\def\noded{#4}%
 \edef\next{#5}%
 \ifx\next\empty
  \def\labela{}\def\labelb{}\def\labelc{}\def\labeld{}%
 \else\gdef\next[##1`##2`##3`##4]{%
  \def\labela{##1}\def\labelb{##2}\def\labelc{##3}\def\labeld{##4}}%
 \next[#5]%
 \fi \ignorespaces}%

\def\squarepppp(#1)|#2|/#3`#4`#5`#6/<#7>[#8]{%
\location(#1)%
\squareplacements|#2|%
\sizes<#7>%
\squaredata[#8]%
\morphism(\xpos,\ypos)|\xd|/#6/<\deltax,0>[\nodec`\noded`\labeld]%
\advance \ypos by \deltay%
\morphism(\xpos,\ypos)|\xb|/#4/<0,-\deltay>[\nodea`\nodec`\labelb]%
\morphism(\xpos,\ypos)|\xa|/#3/<\deltax,0>[\nodea`\nodeb`\labela]%
 \advance \xpos by \deltax%
\morphism(\xpos,\ypos)|\xc|/#5/<0,-\deltay>[\nodeb`\noded`\labelc]%
\ignorespaces}%

\def\square{\ifnextchar({\squarep}{\squarep(0,0)}}%
\def\squarep(#1){\ifnextchar|{\squarepp(#1)}{\squarepp(#1)|a`l`r`b|}}%
\def\squarepp(#1)|#2|{\ifnextchar/{\squareppp(#1)|#2|}%
    {\squareppp(#1)|#2|/>`>`>`>/}}%
\def\squareppp(#1)|#2|/#3`#4`#5`#6/{%
    \ifnextchar<{\squarepppp(#1)|#2|/#3`#4`#5`#6/}%
    {\squarepppp(#1)|#2|/#3`#4`#5`#6/<\default,\default>}}%

\def\triplacements|#1`#2`#3|{\def\xa{#1}\def\xb{#2}\def\xc{#3}}%
\def\tridata[#1`#2`#3;#4`#5`#6]{%
    \def\nodea{#1}\def\nodeb{#2}\def\nodec{#3}%
    \def\labela{#4}\def\labelb{#5}\def\labelc{#6}}%

\def\ptrianglepppp(#1)|#2|/#3`#4`#5/<#6>[#7]{%
\location(#1)%
\triplacements|#2|%
\sizes<#6>%
\tridata[#7]%
\advance\ypos by \deltay%
\morphism(\xpos,\ypos)|\xa|/#3/<\deltax,0>[\nodea`\nodeb`\labela]%
\morphism(\xpos,\ypos)|\xb|/#4/<0,-\deltay>[\nodea`\nodec`\labelb]%
\advance\xpos by \deltax%
\morphism(\xpos,\ypos)|\xc|/#5/<-\deltax,-\deltay>[\nodeb`\nodec`\labelc]%
\ignorespaces}%

\def\ptriangle{\ifnextchar({\ptrianglep}{\ptrianglep(0,0)}}%
\def\ptrianglep(#1){\ifnextchar|{\ptrianglepp(#1)}{\ptrianglepp(#1)|a`l`r|}}%
\def\ptrianglepp(#1)|#2|{\ifnextchar/{\ptriangleppp(#1)|#2|}%
    {\ptriangleppp(#1)|#2|/>`>`>/}}%
\def\ptriangleppp(#1)|#2|/#3`#4`#5/{%
    \ifnextchar<{\ptrianglepppp(#1)|#2|/#3`#4`#5/}%
    {\ptrianglepppp(#1)|#2|/#3`#4`#5/<\default,\default>}}%

\def\qtrianglepppp(#1)|#2|/#3`#4`#5/<#6>[#7]{%
\location(#1)%
\triplacements|#2|%
\sizes<#6>%
\tridata[#7]%
\advance\ypos by \deltay%
\morphism(\xpos,\ypos)|\xa|/#3/<\deltax,0>[\nodea`\nodeb`\labela]%
\morphism(\xpos,\ypos)|\xb|/#4/<\deltax,-\deltay>[\nodea`\nodec`\labelb]%
\advance\xpos by \deltax%
\morphism(\xpos,\ypos)|\xc|/#5/<0,-\deltay>[\nodeb`\nodec`\labelc]%
\ignorespaces}%

\def\qtriangle{\ifnextchar({\qtrianglep}{\qtrianglep(0,0)}}%
\def\qtrianglep(#1){\ifnextchar|{\qtrianglepp(#1)}{\qtrianglepp(#1)|a`l`r|}}%
\def\qtrianglepp(#1)|#2|{\ifnextchar/{\qtriangleppp(#1)|#2|}%
    {\qtriangleppp(#1)|#2|/>`>`>/}}%
\def\qtriangleppp(#1)|#2|/#3`#4`#5/{%
    \ifnextchar<{\qtrianglepppp(#1)|#2|/#3`#4`#5/}%
    {\qtrianglepppp(#1)|#2|/#3`#4`#5/<\default,\default>}}%

\def\dtrianglepppp(#1)|#2|/#3`#4`#5/<#6>[#7]{%
\location(#1)%
\triplacements|#2|%
\sizes<#6>%
\tridata[#7]%
\morphism(\xpos,\ypos)|\xc|/#5/<\deltax,0>[\nodeb`\nodec`\labelc]%
\advance\ypos by \deltay\advance \xpos by \deltax%
\morphism(\xpos,\ypos)|\xa|/#3/<-\deltax,-\deltay>[\nodea`\nodeb`\labela]%
\morphism(\xpos,\ypos)|\xb|/#4/<0,-\deltay>[\nodea`\nodec`\labelb]%
\ignorespaces}%

\def\dtriangle{\ifnextchar({\dtrianglep}{\dtrianglep(0,0)}}%
\def\dtrianglep(#1){\ifnextchar|{\dtrianglepp(#1)}{\dtrianglepp(#1)|l`r`b|}}%
\def\dtrianglepp(#1)|#2|{\ifnextchar/{\dtriangleppp(#1)|#2|}%
    {\dtriangleppp(#1)|#2|/>`>`>/}}%
\def\dtriangleppp(#1)|#2|/#3`#4`#5/{%
    \ifnextchar<{\dtrianglepppp(#1)|#2|/#3`#4`#5/}%
    {\dtrianglepppp(#1)|#2|/#3`#4`#5/<\default,\default>}}%

\def\btrianglepppp(#1)|#2|/#3`#4`#5/<#6>[#7]{%
\location(#1)%
\triplacements|#2|%
\sizes<#6>%
\tridata[#7]%
\morphism(\xpos,\ypos)|\xc|/#5/<\deltax,0>[\nodeb`\nodec`\labelc]%
\advance\ypos by \deltay%
\morphism(\xpos,\ypos)|\xa|/#3/<0,-\deltay>[\nodea`\nodeb`\labela]%
\morphism(\xpos,\ypos)|\xb|/#4/<\deltax,-\deltay>[\nodea`\nodec`\labelb]%
\ignorespaces}%

\def\btriangle{\ifnextchar({\btrianglep}{\btrianglep(0,0)}}%
\def\btrianglep(#1){\ifnextchar|{\btrianglepp(#1)}{\btrianglepp(#1)|l`r`b|}}%
\def\btrianglepp(#1)|#2|{\ifnextchar/{\btriangleppp(#1)|#2|}%
    {\btriangleppp(#1)|#2|/>`>`>/}}%
\def\btriangleppp(#1)|#2|/#3`#4`#5/{%
    \ifnextchar<{\btrianglepppp(#1)|#2|/#3`#4`#5/}%
    {\btrianglepppp(#1)|#2|/#3`#4`#5/<\default,\default>}}%

\def\Atrianglepppp(#1)|#2|/#3`#4`#5/<#6>[#7]{%
\location(#1)%
\triplacements|#2|%
\sizes<#6>%
\tridata[#7]%
\multiply\deltax by 2%
\morphism(\xpos,\ypos)|\xc|/#5/<\deltax,0>[\nodeb`\nodec`\labelc]%
\divide\deltax by 2
\advance\ypos by \deltay\advance\xpos by \deltax%
\morphism(\xpos,\ypos)|\xa|/#3/<-\deltax,-\deltay>[\nodea`\nodeb`\labela]%
\morphism(\xpos,\ypos)|\xb|/#4/<\deltax,-\deltay>[\nodea`\nodec`\labelb]%
\ignorespaces}%

\def\Atriangle{\ifnextchar({\Atrianglep}{\Atrianglep(0,0)}}%
\def\Atrianglep(#1){\ifnextchar|{\Atrianglepp(#1)}{\Atrianglepp(#1)|l`r`b|}}%
\def\Atrianglepp(#1)|#2|{\ifnextchar/{\Atriangleppp(#1)|#2|}%
    {\Atriangleppp(#1)|#2|/>`>`>/}}%
\def\Atriangleppp(#1)|#2|/#3`#4`#5/{%
    \ifnextchar<{\Atrianglepppp(#1)|#2|/#3`#4`#5/}%
    {\Atrianglepppp(#1)|#2|/#3`#4`#5/<\default,\default>}}%

\def\Vtrianglepppp(#1)|#2|/#3`#4`#5/<#6>[#7]{%
\location(#1)%
\triplacements|#2|%
\sizes<#6>%
\tridata[#7]%
\advance\ypos by \deltay%
\morphism(\xpos,\ypos)|\xb|/#4/<\deltax,-\deltay>[\nodea`\nodec`\labelb]%
\multiply\deltax by 2%
\morphism(\xpos,\ypos)|\xa|/#3/<\deltax,0>[\nodea`\nodeb`\labela]%
\advance\xpos by \deltax \divide \deltax by 2
\morphism(\xpos,\ypos)|\xc|/#5/<-\deltax,-\deltay>[\nodeb`\nodec`\labelc]%
\ignorespaces}%

\def\Vtriangle{\ifnextchar({\Vtrianglep}{\Vtrianglep(0,0)}}%
\def\Vtrianglep(#1){\ifnextchar|{\Vtrianglepp(#1)}{\Vtrianglepp(#1)|a`l`b|}}%
\def\Vtrianglepp(#1)|#2|{\ifnextchar/{\Vtriangleppp(#1)|#2|}%
    {\Vtriangleppp(#1)|#2|/>`>`>/}}%
\def\Vtriangleppp(#1)|#2|/#3`#4`#5/{%
    \ifnextchar<{\Vtrianglepppp(#1)|#2|/#3`#4`#5/}%
    {\Vtrianglepppp(#1)|#2|/#3`#4`#5/<\default,\default>}}%

\def\Ctrianglepppp(#1)|#2|/#3`#4`#5/<#6>[#7]{%
\location(#1)%
\triplacements|#2|%
\sizes<#6>%
\tridata[#7]%
\advance \ypos by \deltay%
\morphism(\xpos,\ypos)|\xc|/#5/<\deltax,-\deltay>[\nodeb`\nodec`\labelc]%
\advance\ypos by \deltay \advance \xpos by \deltax%
\morphism(\xpos,\ypos)|\xa|/#3/<-\deltax,-\deltay>[\nodea`\nodeb`\labela]%
\multiply\deltay by 2%
\morphism(\xpos,\ypos)|\xb|/#4/<0,-\deltay>[\nodea`\nodec`\labelb]%
\ignorespaces}%

\def\Ctriangle{\ifnextchar({\Ctrianglep}{\Ctrianglep(0,0)}}%
\def\Ctrianglep(#1){\ifnextchar|{\Ctrianglepp(#1)}{\Ctrianglepp(#1)|a`r`b|}}%
\def\Ctrianglepp(#1)|#2|{\ifnextchar/{\Ctriangleppp(#1)|#2|}%
    {\Ctriangleppp(#1)|#2|/>`>`>/}}%
\def\Ctriangleppp(#1)|#2|/#3`#4`#5/{%
    \ifnextchar<{\Ctrianglepppp(#1)|#2|/#3`#4`#5/}%
    {\Ctrianglepppp(#1)|#2|/#3`#4`#5/<\default,\default>}}%

\def\Dtrianglepppp(#1)|#2|/#3`#4`#5/<#6>[#7]{%
\location(#1)%
\triplacements|#2|%
\sizes<#6>%
\tridata[#7]%
\advance\xpos by \deltax \advance\ypos by \deltay%
\morphism(\xpos,\ypos)|\xc|/#5/<-\deltax,-\deltay>[\nodeb`\nodec`\labelc]%
\advance\xpos by -\deltax \advance\ypos by \deltay%
\morphism(\xpos,\ypos)|\xb|/#4/<\deltax,-\deltay>[\nodea`\nodeb`\labelb]%
\multiply \deltay by 2%
\morphism(\xpos,\ypos)|\xa|/#3/<0,-\deltay>[\nodea`\nodec`\labela]%
\ignorespaces}%

\def\Dtriangle{\ifnextchar({\Dtrianglep}{\Dtrianglep(0,0)}}%
\def\Dtrianglep(#1){\ifnextchar|{\Dtrianglepp(#1)}{\Dtrianglepp(#1)|a`l`b|}}%
\def\Dtrianglepp(#1)|#2|{\ifnextchar/{\Dtriangleppp(#1)|#2|}%
    {\Dtriangleppp(#1)|#2|/>`>`>/}}%
\def\Dtriangleppp(#1)|#2|/#3`#4`#5/{%
    \ifnextchar<{\Dtrianglepppp(#1)|#2|/#3`#4`#5/}%
    {\Dtrianglepppp(#1)|#2|/#3`#4`#5/<\default,\default>}}%

\def\tripairplacements|#1`#2`#3`#4`#5|{\def\xa{#1}\def\xb{#2}\def\xc{#3}%
\def\xd{#4}\def\xe{#5}}%
\def\tripairdata[#1`#2`#3`#4;#5`#6`#7`#8`#9]{%
 \def\nodea{#1}\def\nodeb{#2}\def\nodec{#3}\def\noded{#4}%
 \def\labela{#5}\def\labelb{#6}\def\labelc{#7}\def\labeld{#8}\def\labele{#9}}%


\def\Atrianglepairpppp(#1)|#2|/#3`#4`#5`#6`#7/<#8>[#9]{%
\location(#1)%
\tripairplacements|#2|%
\sizes<#8>%
\tripairdata[#9]%
\morphism(\xpos,\ypos)|\xd|/#6/<\deltax,0>[\nodeb`\nodec`\labeld]%
\advance\xpos by \deltax%
\morphism(\xpos,\ypos)|\xe|/#7/<\deltax,0>[\nodec`\noded`\labele]%
\advance\ypos by \deltay%
\morphism(\xpos,\ypos)|\xa|/#3/<-\deltax,-\deltay>[\nodea`\nodeb`\labela]%
\morphism(\xpos,\ypos)|\xb|/#4/<0,-\deltay>[\nodea`\nodec`\labelb]%
\morphism(\xpos,\ypos)|\xc|/#5/<\deltax,-\deltay>[\nodea`\noded`\labelc]%
\ignorespaces}%

\def\Atrianglepair{\ifnextchar({\Atrianglepairp}{\Atrianglepairp(0,0)}}%
\def\Atrianglepairp(#1){\ifnextchar|{\Atrianglepairpp(#1)}%
{\Atrianglepairpp(#1)|l`m`r`b`b|}}%
\def\Atrianglepairpp(#1)|#2|{\ifnextchar/{\Atrianglepairppp(#1)|#2|}%
    {\Atrianglepairppp(#1)|#2|/>`>`>`>`>/}}%
\def\Atrianglepairppp(#1)|#2|/#3`#4`#5`#6`#7/{%
    \ifnextchar<{\Atrianglepairpppp(#1)|#2|/#3`#4`#5`#6`#7/}%
    {\Atrianglepairpppp(#1)|#2|/#3`#4`#5`#6`#7/<\default,\default>}}%

\def\Vtrianglepairpppp(#1)|#2|/#3`#4`#5`#6`#7/<#8>[#9]{%
\location(#1)%
\tripairplacements|#2|%
\sizes<#8>%
\tripairdata[#9]%
\advance\ypos by \deltay%
\morphism(\xpos,\ypos)|\xa|/#3/<\deltax,0>[\nodea`\nodeb`\labela]%
\morphism(\xpos,\ypos)|\xc|/#5/<\deltax,-\deltay>[\nodea`\noded`\labelc]%
\advance\xpos by \deltax%
\morphism(\xpos,\ypos)|\xb|/#4/<\deltax,0>[\nodeb`\nodec`\labelb]%
\morphism(\xpos,\ypos)|\xd|/#6/<0,-\deltay>[\nodeb`\noded`\labeld]%
\advance\xpos by \deltax%
\morphism(\xpos,\ypos)|\xe|/#7/<-\deltax,-\deltay>[\nodec`\noded`\labele]%
\ignorespaces}%

\def\Vtrianglepair{\ifnextchar({\Vtrianglepairp}{\Vtrianglepairp(0,0)}}%
\def\Vtrianglepairp(#1){\ifnextchar|{\Vtrianglepairpp(#1)}%
{\Vtrianglepairpp(#1)|a`a`l`m`r|}}%
\def\Vtrianglepairpp(#1)|#2|{\ifnextchar/{\Vtrianglepairppp(#1)|#2|}%
    {\Vtrianglepairppp(#1)|#2|/>`>`>`>`>/}}%
\def\Vtrianglepairppp(#1)|#2|/#3`#4`#5`#6`#7/{%
    \ifnextchar<{\Vtrianglepairpppp(#1)|#2|/#3`#4`#5`#6`#7/}%
    {\Vtrianglepairpppp(#1)|#2|/#3`#4`#5`#6`#7/<\default,\default>}}%

\def\Ctrianglepairpppp(#1)|#2|/#3`#4`#5`#6`#7/<#8>[#9]{%
\location(#1)%
\tripairplacements|#2|%
\sizes<#8>%
\tripairdata[#9]%
\advance\ypos by \deltay%
\morphism(\xpos,\ypos)|\xe|/#7/<0,-\deltay>[\nodec`\noded`\labele]%
\advance\xpos by -\deltax%
\morphism(\xpos,\ypos)|\xc|/#5/<\deltax,0>[\nodeb`\nodec`\labelc]%
\morphism(\xpos,\ypos)|\xd|/#6/<\deltax,-\deltay>[\nodeb`\noded`\labeld]%
\advance\ypos by \deltay%
\advance\xpos by \deltax%
\morphism(\xpos,\ypos)|\xa|/#3/<-\deltax,-\deltay>[\nodea`\nodeb`\labela]%
\morphism(\xpos,\ypos)|\xb|/#4/<0,-\deltay>[\nodea`\nodec`\labelb]%
\ignorespaces}%

\def\Ctrianglepair{\ifnextchar({\Ctrianglepairp}{\Ctrianglepairp(0,0)}}%
\def\Ctrianglepairp(#1){\ifnextchar|{\Ctrianglepairpp(#1)}%
{\Ctrianglepairpp(#1)|l`r`m`l`r|}}%
\def\Ctrianglepairpp(#1)|#2|{\ifnextchar/{\Ctrianglepairppp(#1)|#2|}%
    {\Ctrianglepairppp(#1)|#2|/>`>`>`>`>/}}%
\def\Ctrianglepairppp(#1)|#2|/#3`#4`#5`#6`#7/{%
    \ifnextchar<{\Ctrianglepairpppp(#1)|#2|/#3`#4`#5`#6`#7/}%
    {\Ctrianglepairpppp(#1)|#2|/#3`#4`#5`#6`#7/<\default,\default>}}%

\def\Dtrianglepairpppp(#1)|#2|/#3`#4`#5`#6`#7/<#8>[#9]{%
\location(#1)%
\tripairplacements|#2|%
\sizes<#8>%
\tripairdata[#9]%
\advance\ypos by \deltay%
\morphism(\xpos,\ypos)|\xc|/#5/<\deltax,0>[\nodeb`\nodec`\labelc]%
\morphism(\xpos,\ypos)|\xd|/#6/<0,-\deltay>[\nodeb`\noded`\labeld]%
\advance\ypos by \deltay%
\morphism(\xpos,\ypos)|\xa|/#3/<0,-\deltay>[\nodea`\nodeb`\labela]%
\morphism(\xpos,\ypos)|\xb|/#4/<\deltax,-\deltay>[\nodea`\nodec`\labelb]%
\advance\ypos by -\deltay%
\advance\xpos by \deltax%
\morphism(\xpos,\ypos)|\xe|/#7/<-\deltax,-\deltay>[\nodec`\noded`\labele]%
\ignorespaces}%

\def\Dtrianglepair{\ifnextchar({\Dtrianglepairp}{\Dtrianglepairp(0,0)}}%
\def\Dtrianglepairp(#1){\ifnextchar|{\Dtrianglepairpp(#1)}%
{\Dtrianglepairpp(#1)|l`r`m`l`r|}}%
\def\Dtrianglepairpp(#1)|#2|{\ifnextchar/{\Dtrianglepairppp(#1)|#2|}%
    {\Dtrianglepairppp(#1)|#2|/>`>`>`>`>/}}%
\def\Dtrianglepairppp(#1)|#2|/#3`#4`#5`#6`#7/{%
    \ifnextchar<{\Dtrianglepairpppp(#1)|#2|/#3`#4`#5`#6`#7/}%
    {\Dtrianglepairpppp(#1)|#2|/#3`#4`#5`#6`#7/<\default,\default>}}%

\def\place(#1,#2)[#3]{\POS(#1,#2)*+{#3}}%

\def\pullback#1]#2]{\square#1]\trident#2]}%
\def\tridentdata[#1;#2`#3`#4]{\def\nodee{#1}\def\labele{#2}%
   \def\labelf{#3}\def\labelg{#4}}%
\def\trident{\ifnextchar|{\tridentp}{\tridentp|a`m`b|}}%

\def\tridentp|#1|{\ifnextchar/{\tridentpp|#1|}{\tridentpp|#1|/{>}`{>}`{>}/}}%

\def\tridentpp|#1|/#2/{\ifnextchar<{\tridentppp|#1|/#2/}%
  {\tridentppp|#1|/#2/<500,500>}}%

\def\tridentppp|#1`#2`#3|/#4`#5`#6/<#7,#8>[#9]{%
\tridentdata[#9]%
\advance \xpos by -\deltax%
\advance \xpos by -#7\advance \ypos by #8%
\advance\deltax by #7%
\morphism(\xpos,\ypos)|#1|/#4/<\deltax,-#8>[\nodee`\nodeb`\labele]%
\advance\deltax by -#7%
\morphism(\xpos,\ypos)|#2|/#5/<#7,-#8>[\nodee`\nodea`\labelf]%
\advance\deltay by #8%
\morphism(\xpos,\ypos)|#3|/#6/<#7,-\deltay>[\nodee`\nodec`\labelg]%
\ignorespaces}%


\def\setmorphismwidth#1#2#3#4{%
 \setbox0=\hbox{$#1#3#3#2$}#4=\wd0%
 \divide #4 by 2 \divide #4 by \ul%
 \advance #4 by \defaultmargin \ratchet{#4}{500}%
}%


\def\setSquarewidth[#1`#2`#3`#4;#5`#6`#7`#8]{%
 \setmorphismwidth{#1}{#2}{#5}{\topw}%
 \setmorphismwidth{#3}{#4}{#8}{\botw}%
\ratchet{\topw}{\botw}}%


\def\Squarepppp(#1)|#2|/#3/<#4>[#5]{%
 \setSquarewidth[#5]%
 \squarepppp(#1)|#2|/#3/<\topw,#4>[#5]%
\ignorespaces}%

\def\Square{\ifnextchar({\Squarep}{\Squarep(0,0)}}%
\def\Squarep(#1){\ifnextchar|{\Squarepp(#1)}{\Squarepp(#1)|a`l`r`b|}}%
\def\Squarepp(#1)|#2|{\ifnextchar/{\Squareppp(#1)|#2|}%
    {\Squareppp(#1)|#2|/>`>`>`>/}}%
\def\Squareppp(#1)|#2|/#3`#4`#5`#6/{%
    \ifnextchar<{\Squarepppp(#1)|#2|/#3`#4`#5`#6/}%
    {\Squarepppp(#1)|#2|/#3`#4`#5`#6/<\default>}}%

\def\Placements|#1`#2`#3`#4`#5`#6`#7|{%
 \def\Xa{#1}%
 \def\Xb{#2}%
 \def\Xc{#3}%
 \def\Xd{#4}%
 \def\Xe{#5}%
 \def\Xf{#6}%
 \def\Xg{#7}%
}%

\def\Nodes[#1`#2`#3`#4`#5`#6]{%
 \def\Nodea{#1}%
 \def\Nodeb{#2}%
 \def\Nodec{#3}%
 \def\Noded{#4}%
 \def\Nodee{#5}%
 \def\Nodef{#6}%
}%

\def\Labels[#1`#2`#3`#4`#5`#6`#7]{%
 \def\Labela{#1}%
 \def\Labelb{#2}%
 \def\Labelc{#3}%
 \def\Labeld{#4}%
 \def\Labele{#5}%
 \def\Labelf{#6}%
 \def\Labelg{#7}%
}%


\def\hSquarespppp(#1,#2)|#3|/#4/<#5>[#6;#7]{%
\Xpos=#1\Ypos=#2%
\Placements|#3|%
\deltaY=#5%
\Nodes[#6]%
\Labels[#7]%
\dohSquares/#4/}%

\def\dohSquares/#1`#2`#3`#4`#5`#6`#7/{%
\Squarepppp(\Xpos,\Ypos)|\Xa`\Xc`\Xd`\Xf|/#1`#3`#4`#6/<\deltaY>%
 [\Nodea`\Nodeb`\Noded`\Nodee;\Labela`\Labelc`\Labeld`\Labelf]%
 \advance \Xpos by \topw
\Squarepppp(\Xpos,\Ypos)|\Xb`\Xd`\Xe`\Xg|/#2``#5`#7/<\deltaY>%
[\Nodeb`\Nodec`\Nodee`\Nodef;\Labelb``\Labele`\Labelg]%
\ignorespaces}%

\def\hSquares{\ifnextchar({\hSquaresp}{\hSquaresp(0,0)}}%
\def\hSquaresp(#1){\ifnextchar|{\hSquarespp(#1)}{\hSquarespp%
(#1)|a`a`l`m`r`b`b|}}%
\def\hSquarespp(#1)|#2|{\ifnextchar/{\hSquaresppp(#1)|#2|}%
    {\hSquaresppp(#1)|#2|/>`>`>`>`>`>`>/}}%
\def\hSquaresppp(#1)|#2|/#3/{%
    \ifnextchar<{\hSquarespppp(#1)|#2|/#3/}%
    {\hSquarespppp(#1)|#2|/#3/<\default>}}%




\def\vSquarespppp(#1,#2)|#3|/#4/<#5,#6>[#7;#8]{%
\Xpos=#1\Ypos=#2%
\Placements|#3|%
\deltaX=#5%
\deltaY=#6%
\Nodes[#7]%
\Labels[#8]%
\dovSquares/#4/\ignorespaces}%

\def\dovSquares/#1`#2`#3`#4`#5`#6`#7/{%
\setmorphismwidth{\Nodea}{\Nodeb}{\Labela}{\topw}%
\setmorphismwidth{\Nodec}{\Noded}{\Labeld}{\botw}%
\ratchet{\topw}{\botw}%
\setmorphismwidth{\Nodee}{\Nodef}{\Labelg}{\botw}%
\ratchet{\topw}{\botw}%
\square(\Xpos,\Ypos)|\Xd`\Xe`\Xf`\Xg|/`#5`#6`#7/<\topw,\deltaX>%
 [\Nodec`\Noded`\Nodee`\Nodef;`\Labele`\Labelf`\Labelg]%
\advance \Ypos by \deltaX%
\square(\Xpos,\Ypos)|\Xa`\Xb`\Xc`\Xd|/#1`#2`#3`#4/<\topw,\deltaY>%
 [\Nodea`\Nodeb`\Nodec`\Noded;\Labela`\Labelb`\Labelc`\Labeld]%
}%

\def\vSquares{\ifnextchar({\vSquaresp}{\vSquaresp(0,0)}}%
\def\vSquaresp(#1){\ifnextchar|{\vSquarespp(#1)}{\vSquarespp%
(#1)|a`l`r`m`l`r`b|}}%
\def\vSquarespp(#1)|#2|{\ifnextchar/{\vSquaresppp(#1)|#2|}%
    {\vSquaresppp(#1)|#2|/>`>`>`>`>`>`>/}}%
\def\vSquaresppp(#1)|#2|/#3/{%
    \ifnextchar<{\vSquarespppp(#1)|#2|/#3/}%
    {\vSquarespppp(#1)|#2|/#3/<\default>}}%

\def\osquarepppp(#1)|#2|/#3`#4`#5`#6/<#7>[#8]{\squarepppp%
 (#1)|#2|/#3`#4`#5`#6/<#7>[#8]%
 \let\Nodea\nodea\let\Nodeb\nodeb%
\let\Nodec\nodec\let\Noded\noded\Xpos=\xpos\Ypos=\ypos%
\deltaX=\deltax \deltaY=\deltay \isquare}

\def\cube{\ifnextchar({\osquarep}{\osquarep(0,0)}}%
\def\osquarep(#1){\ifnextchar|{\osquarepp(#1)}{\osquarepp(#1)|a`l`r`b|}}%
\def\osquarepp(#1)|#2|{\ifnextchar/{\osquareppp(#1)|#2|}%
    {\osquareppp(#1)|#2|/>`>`>`>/}}%
\def\osquareppp(#1)|#2|/#3`#4`#5`#6/{%
    \ifnextchar<{\osquarepppp(#1)|#2|/#3`#4`#5`#6/}%
    {\osquarepppp(#1)|#2|/#3`#4`#5`#6/<1500,1500>}}%

\def\isquarepppp(#1)|#2|/#3`#4`#5`#6/<#7>[#8]{%
 \squarepppp(#1)|#2|/#3`#4`#5`#6/<#7>[#8]%
\ifnextchar|{\cubep}{\cubep|m`m`m`m|}}%
\def\cubep|#1|{\ifnextchar/{\cubepp|#1|}{\cubepp|#1|/>`>`>`>/}}%

\def\isquare{\ifnextchar({\isquarep}{\isquarep(\default,\default)}}%
\def\isquarep(#1){\ifnextchar|{\isquarepp(#1)}{\isquarepp(#1)|a`l`r`b|}}
\def\isquarepp(#1)|#2|{\ifnextchar/{\isquareppp(#1)|#2|}%
    {\isquareppp(#1)|#2|/>`>`>`>/}}%
\def\isquareppp(#1)|#2|/#3`#4`#5`#6/{%
    \ifnextchar<{\isquarepppp(#1)|#2|/#3`#4`#5`#6/}%
    {\isquarepppp(#1)|#2|/#3`#4`#5`#6/<500,500>}}%

\def\cubepp|#1`#2`#3`#4|/#5`#6`#7`#8/[#9]{%
\def\Labels[##1`##2`##3`##4]{\gdef\Labela{##1}%
\gdef\Labelb{##2}\gdef\Labelc{##3}\gdef\Labeld{##4}}\Labels[#9]%
\xend\xpos \yend\ypos
\Xend\xend\advance\Xend by -\Xpos
\Yend\yend\advance\Yend by -\Ypos
\domorphism(\Xpos,\Ypos)|#2|/#6/<\Xend,\Yend>[\Nodeb`\nodeb`\Labelb]%
\advance\Xpos by-\deltaX
\advance\xend by-\deltax
\Xend\xend\advance\Xend by -\Xpos
\domorphism(\Xpos,\Ypos)|#1|/#5/<\Xend,\Yend>[\Nodea`\nodea`\Labela]%
\advance\Ypos by-\deltaY
\advance\yend by-\deltay
\Yend\yend\advance\Yend by -\Ypos
\domorphism(\Xpos,\Ypos)|#3|/#7/<\Xend,\Yend>[\Nodec`\nodec`\Labelc]%
\advance\Xpos by\deltaX
\advance\xend by\deltax
\Xend\xend\advance\Xend by -\Xpos
\domorphism(\Xpos,\Ypos)|#4|/#8/<\Xend,\Yend>[\Noded`\noded`\Labeld]%
\ignorespaces}

\def\setwdth#1#2{\setbox0\hbox{$#1$}\wdth=\wd0
\setbox0\hbox{$#2$}\ifnum\wdth<\wd0 \wdth=\wd0 \fi}

\def\tx^#1_#2{\allowbreak\edef\next{#1}\edef\nextt{#2}
\setwdth{#1}{#2}\deltax=\wdth \divide \deltax by \ul
\advance \deltax by 100 \ratchet{\deltax}{200}
\ifx\next\empty
  \ifx\nextt\empty
    \>\xy \ar(\deltax,0)\endxy\>%
  \else
    \>\xy \ar_{#2}(\deltax,0)\endxy\>%
  \fi
\else
  \ifx\nextt\empty
    \>\xy \ar^{#1}(\deltax,0)\endxy\>%
  \else
    \>\xy \ar^{#1}_{#2}(\deltax,0)\endxy\>%
  \fi
\fi}

\def\txx^#1{\ifnextchar_{\tx^{#1}}{\tx^{#1}_{}}}
\def\to{\ifnextchar^{\txx}{\txx^{}}}

\def\txleft^#1_#2{\allowbreak\edef\next{#1}\edef\nextt{#2}
\setwdth{#1}{#2}\deltax=\wdth \divide \deltax by \ul
\advance \deltax by 100 \ratchet{\deltax}{200}
\ifx\next\empty
  \ifx\nextt\empty
    \>\xy \ar@{<-}(\deltax,0)\endxy\>%
  \else
    \>\xy \ar@{<-}_{#2}(\deltax,0)\endxy\>%
  \fi
\else
  \ifx\nextt\empty
    \>\xy \ar@{<-}^{#1}(\deltax,0)\endxy\>%
  \else
    \>\xy \ar@{<-}^{#1}_{#2}(\deltax,0)\endxy\>%
  \fi
\fi}

\def\txxleft^#1{\ifnextchar_{\txleft^{#1}}{\txleft^{#1}_{}}}
\def\toleft{\ifnextchar^{\txxleft}{\txxleft^{}}}

\def\monx^#1_#2{\allowbreak\edef\next{#1}\edef\nextt{#2}
\setwdth{#1}{#2}\deltax=\wdth \divide \deltax by \ul
\advance \deltax by 125 \ratchet{\deltax}{225}
\ifx\next\empty
  \ifx\nextt\empty
    \>\xy \ar@{ >->}(\deltax,0)\endxy\>%
  \else
    \>\xy \ar@{ >->}_{#2}(\deltax,0)\endxy\>%
  \fi
\else
  \ifx\nextt\empty
    \>\xy \ar@{ >->}^{#1}(\deltax,0)\endxy\>%
  \else
    \>\xy \ar@{ >->}^{#1}_{#2}(\deltax,0)\endxy\>%
  \fi
\fi}

\def\monxx^#1{\ifnextchar_{\monx^{#1}}{\monx^{#1}_{}}}
\def\mon{\ifnextchar^{\monxx}{\monxx^{}}}

\def\monleftx^#1_#2{\allowbreak\edef\next{#1}\edef\nextt{#2}
\setwdth{#1}{#2}\deltax=\wdth \divide \deltax by \ul
\advance \deltax by 125 \ratchet{\deltax}{225}
\ifx\next\empty
  \ifx\nextt\empty
    \>\xy \ar@{<-< }(\deltax,0)\endxy\>%
  \else
    \>\xy \ar@{<-< }_{#2}(\deltax,0)\endxy\>%
  \fi
\else
  \ifx\nextt\empty
    \>\xy \ar@{<-< }^{#1}(\deltax,0)\endxy\>%
  \else
    \>\xy \ar@{<-< }^{#1}_{#2}(\deltax,0)\endxy\>%
  \fi
\fi}

\def\monleftxx^#1{\ifnextchar_{\monleftx^{#1}}{\monleftx^{#1}_{}}}
\def\monleft{\ifnextchar^{\monleftxx}{\monleftxx^{}}}

\def\\epileftx^#1_#2{\allowbreak\edef\next{#1}\edef\nextt{#2}
\setwdth{#1}{#2}\deltax=\wdth \divide \deltax by \ul
\advance \deltax by 125 \ratchet{\deltax}{225}
\ifx\next\empty
  \ifx\nextt\empty
    \>\xy \ar@{<<-}(\deltax,0)\endxy\>%
  \else
    \>\xy \ar@{<<-}_{#2}(\deltax,0)\endxy\>%
  \fi
\else
  \ifx\nextt\empty
    \>\xy \ar@{<<-}^{#1}(\deltax,0)\endxy\>%
  \else
    \>\xy \ar@{<<-}^{#1}_{#2}(\deltax,0)\endxy\>%
  \fi
\fi}

\def\\epileftxx^#1{\ifnextchar_{\\epileftx^{#1}}{\\epileftx^{#1}_{}}}
\def\\epileft{\ifnextchar^{\\epileftxx}{\\epileftxx^{}}}

\def\epix^#1_#2{\allowbreak\edef\next{#1}\edef\nextt{#2}
\setwdth{#1}{#2}\deltax=\wdth \divide \deltax by \ul
\advance \deltax by 125 \ratchet{\deltax}{225}
\ifx\next\empty
  \ifx\nextt\empty
    \>\xy \ar@{->>}(\deltax,0)\endxy\>%
  \else
    \>\xy \ar@{->>}_{#2}(\deltax,0)\endxy\>%
  \fi
\else
  \ifx\nextt\empty
    \>\xy \ar@{->>}^{#1}(\deltax,0)\endxy\>%
  \else
    \>\xy \ar@{->>}^{#1}_{#2}(\deltax,0)\endxy\>%
  \fi
\fi}

\def\epixx^#1{\ifnextchar_{\epix^{#1}}{\epix^{#1}_{}}}
\def\epi{\ifnextchar^{\epixx}{\epixx^{}}}

\def\twx^#1_#2{\allowbreak\edef\next{#1}\edef\nextt{#2}
\setwdth{#1}{#2}\deltax=\wdth \divide \deltax by \ul
\advance \deltax by 100 \ratchet{\deltax}{200}
\ifx\next\empty
  \ifx\nextt\empty
    \>\xy
    \ar@<2.5pt>(\deltax,0)
    \ar@<-2.5pt>(\deltax,0)
    \endxy\>%
  \else
    \>\xy
    \ar@<2.5pt>(\deltax,0)
    \ar@<-2.5pt>_{#2}(\deltax,0)
    \endxy\>%
  \fi
\else
  \ifx\nextt\empty
    \>\xy
    \ar@<2.5pt>^{#1}(\deltax,0)
    \ar@<-2.5pt>(\deltax,0)
    \endxy\>%
  \else
    \>\xy
    \ar@<2.5pt>^{#1}(\deltax,0)
    \ar@<-2.5pt>_{#2}(\deltax,0)
    \endxy\>%
  \fi
\fi}

\def\twxx^#1{\ifnextchar_{\twx^{#1}}{\twx^{#1}_{}}}
\def\two{\ifnextchar^{\twxx}{\twxx^{}}}

\def\twleftx^#1_#2{\allowbreak\edef\next{#1}\edef\nextt{#2}
\setwdth{#1}{#2}\deltax=\wdth \divide \deltax by \ul
\advance \deltax by 100 \ratchet{\deltax}{200}
\ifx\next\empty
  \ifx\nextt\empty
    \>\xy
    \ar@{<-}@<2.5pt>(\deltax,0)
    \ar@{<-}@<-2.5pt>(\deltax,0)
    \endxy\>%
  \else
    \>\xy
    \ar@{<-}@<2.5pt>(\deltax,0)
    \ar@{<-}@<-2.5pt>_{#2}(\deltax,0)
    \endxy\>%
  \fi
\else
  \ifx\nextt\empty
    \>\xy
    \ar@{<-}@<2.5pt>^{#1}(\deltax,0)
    \ar@{<-}@<-2.5pt>(\deltax,0)
    \endxy\>%
  \else
    \>\xy
    \ar@{<-}@<2.5pt>^{#1}(\deltax,0)
    \ar@{<-}@<-2.5pt>_{#2}(\deltax,0)
    \endxy\>%
  \fi
\fi}

\def\twleftxx^#1{\ifnextchar_{\twleftx^{#1}}{\twleftx^{#1}_{}}}
\def\twoleft{\ifnextchar^{\twleftxx}{\twleftxx^{}}}

\def\twoar(#1,#2){%
 \deltax#1\deltay#2%
 \deltaX=\ifnum\deltax<0-\fi\deltax
 \deltaY=\ifnum\deltay<0-\fi\deltay
 \multiply\deltax by 300
 \multiply\deltay by 300
 \ifnum \deltaX > \deltaY
    \multiply \deltaX by 3 \advance \deltaX by \deltaY
 \else
    \multiply \deltaY by 3 \advance \deltaX by \deltaY
 \fi
 \divide \deltax by \deltaX
 \divide \deltay by \deltaX
 \xy \ar@{=>}(\deltax,\deltay) \endxy
}

\catcode`\@=\atcode%