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In a domain and in categories referenced in a domain the notation %
represents "this domain" (or self in some programming languages).  So
we commonly write for example:
  with
    f: (%,%) -> %

to indicate a function f which takes a pair of values in this domain and returns a value in this same domain - whatever domain we happen to be talking about in this context.

But what if we are interested in the domain as a functor? Suppose I was writing an "endofunctor" domain constructor like Set and I wanted to treat constructions like Set Set R, i.e. sets of sets as something special. E.g. \begin{spad} )abbrev domain MYSET MySet? MySet?(T:SetCategory?): SetAggregate?(T) with finiteAggregate join: MySet? MySet? T -> MySet? T == add Rep == List T --rep(x:%):Rep == x pretend Rep --per(x:Rep):% == x pretend %

Rep2 := List List T rep2(x:MySet? MySet? T):Rep2 == x pretend Rep2 per2(x:Rep2):MySet? MySet? T == x pretend MySet? MySet? T

coerce(x:%):OutputForm? == brace [i::OutputForm for i in rep x]? x = y == rep(x) = rep(y) construct(x) == per removeDuplicates(x)$Rep parts(x:%):List T == rep x join x == construct concat rep2 x \end{spad}

\begin{axiom} m1:MySet?(Integer) := construct([1,2,3]?)$MySet?(Integer) m2:MySet?(Integer) := construct([4,5,6]?)$MySet?(Integer) ml:=[m1,m2]? mm:MySet? MySet? Integer := construct([m1,m2]?)$MySet?(MySet?(Integer)) join mm \end{axiom}


Some or all expressions may not have rendered properly, because Axiom returned the following error:
Error: export FRICAS=/usr/local/lib/open-axiom/x86_64-unknown-linux-gnu/1.4.0-2011-02-20; export ALDORROOT=/usr/local/aldor/linux/1.1.0; export PATH=$ALDORROOT/bin:$PATH; export HOME=/var/zope2/var/LatexWiki; ulimit -t 600; export LD_LIBRARY_PATH=/usr/local/lib/open-axiom/x86_64-unknown-linux-gnu/1.4.0-2011-02-20/lib; LANG=en_US.UTF-8 $FRICAS/bin/FRICASsys < /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/7590631627605034976-25px.axm
/bin/sh: /usr/local/lib/open-axiom/x86_64-unknown-linux-gnu/1.4.0-2011-02-20/bin/FRICASsys: not found


Some or all expressions may not have rendered properly, because Latex returned the following error:
This is pdfTeXk, Version 3.141592-1.40.3 (Web2C 7.5.6)
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entering extended mode
(./2371516621003776631-16.0px.tex
LaTeX2e <2005/12/01>
Babel <v3.8h> and hyphenation patterns for english, usenglishmax, dumylang, noh
yphenation, arabic, farsi, croatian, ukrainian, russian, bulgarian, czech, slov
ak, danish, dutch, finnish, basque, french, german, ngerman, ibycus, greek, mon
ogreek, ancientgreek, hungarian, italian, latin, mongolian, norsk, icelandic, i
nterlingua, turkish, coptic, romanian, welsh, serbian, slovenian, estonian, esp
eranto, uppersorbian, indonesian, polish, portuguese, spanish, catalan, galicia
n, swedish, ukenglish, pinyin, loaded.
(/usr/share/texmf-texlive/tex/latex/base/article.cls
Document Class: article 2005/09/16 v1.4f Standard LaTeX document class
(/usr/share/texmf-texlive/tex/latex/base/size12.clo))
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(/usr/share/texmf-texlive/tex/latex/base/inputenc.sty
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(/usr/share/texmf-texlive/tex/latex/bbm/bbm.sty)
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(/usr/share/texmf-texlive/tex/latex/base/fontenc.sty
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(/usr/share/texmf-texlive/tex/latex/pstricks/pstricks.sty
(/usr/share/texmf-texlive/tex/generic/pstricks/pstricks.tex
`PSTricks' v1.15  <2006/12/22> (tvz)
(/usr/share/texmf-texlive/tex/generic/pstricks/pstricks.con))
(/usr/share/texmf/tex/latex/xcolor/xcolor.sty
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(/etc/texmf/tex/latex/config/graphics.cfg))))
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(/usr/share/texmf-texlive/tex/latex/xkeyval/pst-xkey.tex
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(/usr/share/texmf-texlive/tex/latex/xkeyval/xkeyval.tex)))
`pst-plot' v1.05, 2006/11/04 (tvz,dg,hv)))
(/usr/share/texmf-texlive/tex/latex/pstricks/pst-plot.sty
(/usr/share/texmf-texlive/tex/generic/pstricks/pst-plot.tex
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(/usr/share/texmf-texlive/tex/generic/multido/multido.tex
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(/usr/share/texmf-texlive/tex/latex/geometry/geometry.sty
(/usr/share/texmf-texlive/tex/xelatex/xetexconfig/geometry.cfg)

Package geometry Warning: `lmargin' and `rmargin' result in NEGATIVE (-108.405p t). `width' should be shortened in length.

) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `? option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/setspace/setspace.sty Package: `setspace 6.7 <2000/12/01> ) (/usr/share/texmf-texlive/tex/generic/xypic/xy.sty (/usr/share/texmf-texlive/tex/generic/xypic/xy.tex Bootstrap'ing: catcodes, docmode, (/usr/share/texmf-texlive/tex/generic/xypic/xyrecat.tex) (/usr/share/texmf-texlive/tex/generic/xypic/xyidioms.tex)

Xy-pic version 3.7 <1999/02/16> Copyright (c) 1991-1998 by Kristoffer H. Rose <krisrose@ens-lyon.fr> Xy-pic is free software: see the User's Guide for details.

Loading kernel: messages; fonts; allocations: state, direction, utility macros; pictures: \xy, positions, objects, decorations; kernel objects: directionals, circles, text; options; algorithms: directions, edges, connections; Xy-pic loaded)) (/usr/share/texmf-texlive/tex/generic/xypic/xyall.tex Xy-pic option: All features v.3.3 (/usr/share/texmf-texlive/tex/generic/xypic/xycurve.tex Xy-pic option: Curve and Spline extension v.3.7 curve, circles, loaded) (/usr/share/texmf-texlive/tex/generic/xypic/xyframe.tex Xy-pic option: Frame and Bracket extension v.3.7 loaded) (/usr/share/texmf-texlive/tex/generic/xypic/xycmtip.tex Xy-pic option: Computer Modern tip extension v.3.3 (/usr/share/texmf-texlive/tex/generic/xypic/xytips.tex Xy-pic option: More Tips extension v.3.3 loaded) loaded) (/usr/share/texmf-texlive/tex/generic/xypic/xyline.tex Xy-pic option: Line styles extension v.3.6 loaded) (/usr/share/texmf-texlive/tex/generic/xypic/xyrotate.tex Xy-pic option: Rotate and Scale extension v.3.3 loaded) (/usr/share/texmf-texlive/tex/generic/xypic/xycolor.tex Xy-pic option: Colour extension v.3.3 loaded) (/usr/share/texmf-texlive/tex/generic/xypic/xymatrix.tex Xy-pic option: Matrix feature v.3.4 loaded) (/usr/share/texmf-texlive/tex/generic/xypic/xyarrow.tex Xy-pic option: Arrow and Path feature v.3.5 path, \ar, loaded) (/usr/share/texmf-texlive/tex/generic/xypic/xygraph.tex Xy-pic option: Graph feature v.3.7 loaded) loaded) (/usr/share/texmf-texlive/tex/latex/tools/verbatim.sty) (/usr/share/texmf/tex/latex/graphviz/graphviz.sty (/usr/share/texmf-texlive/tex/latex/psfrag/psfrag.sty)) (/usr/share/texmf/tex/latex/sagetex.sty Writing sage input file 2371516621003776631-16.0px.sage ) (/usr/share/texmf-texlive/tex/latex/gnuplottex/gnuplottex.sty (/usr/share/texmf-texlive/tex/latex/base/latexsym.sty) (/usr/share/texmf-texlive/tex/latex/moreverb/moreverb.sty) (/usr/share/texmf-texlive/tex/latex/base/ifthen.sty)) (./2371516621003776631-16.0px.aux) (/usr/share/texmf-texlive/tex/latex/ucs/ucsencs.def) (/usr/share/texmf-texlive/tex/latex/jknapltx/ursfs.fd) (/usr/share/texmf-texlive/tex/latex/amsfonts/umsa.fd) (/usr/share/texmf-texlive/tex/latex/amsfonts/umsb.fd) (/usr/share/texmf-texlive/tex/latex/base/ulasy.fd) Missing $ inserted. <inserted text> $ l.140 \end{spad} \newpage [1] (/usr/share/texmf-texlive/tex/latex/base/t1cmtt.fd)

LaTeX Warning: Characters dropped after `\end{axiom}' on input line 147.

[2] (./2371516621003776631-16.0px.aux) ) (see the transcript file for additional information) Output written on 2371516621003776631-16.0px.dvi (2 pages, 1504 bytes). Transcript written on 2371516621003776631-16.0px.log.




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