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Using OpenAxiom
\begin{axiom}
)version
\end{axiom}

\begin{spad} )abbrev domain MORPH Morphism T ==> SetCategory Morphism(source:T, target:T): with domain:%->T codomain:%->T coerce:(source->target) -> % coerce:%->(source->target) coerce:%->OutputForm elt:(%,source)->target == (source->target) add -- Rep == (source->target) Rep == SExpression domain(p:%):T == source codomain(p:%):T == target coerce(r:(source->target)):% == per(r pretend SExpression) coerce(p:%):(source->target) == (rep(p) pretend (source->target)) coerce(p:%):OutputForm == p pretend OutputForm elt(f,x) == (rep(f) pretend (source->target))(x) \end{spad}

The naive representation of Rep == (source->target) gives the following error:

  Error: Caught fatal error [memory may be damaged]
  ...
  Error signalled by EVALHOOK.
  Backtrace: system:universal-error-handler > evalhook > lambda > lambda-closure > block > apply > APPLY

But using SExpression seems to work. \begin{axiom} )show Morphism(Float,Integer) f:Morphism(Float,Integer) f:=(x:Float):Integer +->wholePart(x) f(1.1) (f::(Float->Integer))(1.1) domain f codomain f (f::(domain(f)->codomain(f)))(1.1) g:Morphism(Integer,String):=(i:Integer):String +-> i::String g(1985) \end{axiom}

Next let's study how this sort of thing is accomplished in a similar domain. \begin{spad} )abbrev domain HOM Hom ++ Based on Automorphism by: Manuel Bronstein ++ Description: ++ Hom(R,S) is the domain of homomorphisms - Mapping(S,R). Hom(R:SetCategory?,S:SetCategory?): Join(SetCategory?, Eltable(R, S)) with morphism: (R -> S) -> % ++ morphism(f) returns the non-invertible morphism given by f. == add -- Why doesn't the obvious representation just (R->S) work? Rep == SExpression? morphism(f) == per(f pretend SExpression?) elt(f, r) == (rep(f) pretend (R->S))(r)

--Rep == List(R -> S) --morphism(f) == per([f #1]?) --elt(f, r) == (rep(f)(1))(r)

--Rep := ((R, Integer) -> S) --morphism(f) == (#2; f(#1)) --elt(f, r) == (g := f pretend ((R, Integer) -> S); g(r, 1))

f = g == EQ(f, g)$Lisp coerce(f:%):OutputForm? == message("theMap ...")

\end{spad}

\begin{axiom} p:=morphism((x:Integer):Float +-> x+1.0)$Hom(Integer,Float) p(2) p(-1) \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/1.4.0-2009-10-13; 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/1.4.0-2009-10-13/lib; LANG=en_US.UTF-8 $FRICAS/bin/FRICASsys < /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/8560717210793468152-25px.axm
/bin/sh: /usr/local/lib/open-axiom/x86_64-unknown-linux/1.4.0-2009-10-13/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
(./7813582976431298218-16.0px.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))
(/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty
(/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def))
(/usr/share/texmf-texlive/tex/latex/base/inputenc.sty
(/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def))
(/usr/share/texmf-texlive/tex/latex/bbm/bbm.sty)
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(/usr/share/texmf-texlive/tex/latex/base/t1enc.def))
(/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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`pst-plot' v1.05, 2006/11/04 (tvz,dg,hv)))
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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 7813582976431298218-16.0px.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)) (./7813582976431298218-16.0px.aux) (/usr/share/texmf-texlive/tex/latex/ucs/ucsencs.def) (/usr/share/texmf-texlive/tex/latex/base/t1cmtt.fd)

LaTeX Warning: Characters dropped after `\end{axiom}' on input line 123.

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LaTeX Warning: Characters dropped after `\end{axiom}' on input line 155.

You can't use `macro parameter character #' in horizontal mode. l.171 --morphism(f) == per([f # 1]) You can't use `macro parameter character #' in horizontal mode. l.175 ... --morphism(f) == (# 2; f(#1)) You can't use `macro parameter character #' in horizontal mode. l.175 ...rphism(f) == (#2; f(# 1)) (/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.180

[2]

LaTeX Warning: Characters dropped after `\end{axiom}' on input line 186.

[3] (./7813582976431298218-16.0px.aux) ) (see the transcript file for additional information) Output written on 7813582976431298218-16.0px.dvi (3 pages, 2696 bytes). Transcript written on 7813582976431298218-16.0px.log.




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