Edit detail for SandBox Aldor Category Theory 4 revision 5 of 5

(1) -> <aldor>
#include "axiom"
#pile
#library    lBasics  "basics.ao"
import from lBasics

define AutomorphismCategory(Obj:Category,A:Obj):Category == Groups with
    aut:     (A->A,A->A) -> %  -- create an automorphism from a morphism and it's inverse
    aut: % ->(A->A,A->A)       -- create a morphism and it's inverse from an automorphism

+++
+++  If X is an object in any category, Aut X given below is the group
+++  of automorphisms.  If the category has Set and CountablyInfinite, 
+++  autmorphisms are said to be equal if they have equal values at each
+++  point in their domain.
+++
define Automorphism(Obj:Category):Category == with
    Aut: (A:Obj) -> AutomorphismCategory (Obj,A)
    default
        Aut(A:Obj):AutomorphismCategory(Obj,A) == 
            WW0:AutomorphismCategory(Obj,A) == add
                Rep == Record(iso:A->A,isi:A->A); import from Rep
                1:% == per [(a:A):A +-> a, (a:A):A +-> a]
                (x:%)=(y:%):Boolean == 
                     A has CountablyFinite with Set => 
                         import from A
                         forall? ( ((rep x).iso)(a) = ((rep y).iso)(a) for a in (elements$A)() )
                     error "Equality is not available for these automorphisms."
                import from o(Obj,A,A,A)
                (g:%)*(f:%):% == per [ ((rep g).iso) ** ((rep f).iso) , ((rep f).isi) ** ((rep g).isi) ]
                inv(f:%):% == per [ (rep f).isi, (rep f).iso ]
                aut(isomorphism:A->A,isomorphismInverse:A->A):% == per [isomorphism,isomorphismInverse]
                aut(f:%):(A->A,A->A) == explode rep f 
                coerce(f:%):OutputForm == message "[Automorphism]"
            WW0 add

define EndomorphismCategory(Obj:Category,A:Obj):Category == Monoids with
    end:        (A->A) -> %  -- create an endomorphisms from a morphism
    end:   % -> (A->A)       -- create a morphism from an endomorphism

+++
+++  If X is an object in any category, End X given below is the monoid
+++  of endomorphisms.  If the category has Set and CountablyInfinite,
+++  endomorphisms are computed to be equal if they have equal values at
+++  each point in their domain.
+++
define Endomorphism(Obj:Category):Category == with
    End: (A:Obj) -> EndomorphismCategory(Obj,A) 
    default
        End(A:Obj):EndomorphismCategory(Obj,A) == 
            WW1:EndomorphismCategory(Obj,A) == add
                Rep ==> A->A
                1:% == per ( (a:A):A +-> a )
                import from o(Obj,A,A,A)
                (x:%)=(y:%):Boolean == 
                    A has CountablyFinite with Set => 
                        import from A
                        forall? ( (rep x) a = (rep y) a for a in (elements$A)() )
                    error "Equality is not available for endomorphisms."
                (g:%)*(f:%):% == per ( (rep g)**(rep f) )
                end(f:A->A):% == per f
                end(f:%):(A->A) == rep f
                coerce(f:%):OutputForm == message "[Endomorphism]"
            WW1 add

define Morphisms(Obj:Category):Category == Automorphism Obj with Endomorphism Obj</aldor>

Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/morphisms.as 
      using Aldor compiler and options 
-O -Fasy -Fao -Flsp -lfricas -Mno-ALDOR_W_WillObsolete -DFriCAS -Y $FRICAS/algebra -I $FRICAS/algebra
      Use the system command )set compiler args to change these 
      options.
"/var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/morphisms.as", line 1: 
#include "axiom"
^
[L1 C1] #1 (Error) Could not open file `axiom'.

   The )library system command was not called after compilation.

)library basics.ao

   )library cannot find the file basics.ao.

)show Domains

   The )show system command is used to display information about types 
      or partial types. For example, )show Integer will show 
      information about Integer .

      Domains is not the name of a known type constructor. If you want 
      to see information about any operations named Domains , issue
                         )display operations Domains