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last edited 17 years ago by Bill Page |
Edit detail for SandBox Aldor Category Theory 4 revision 4 of 5
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Editor: Bill Page
Time: 2007/11/20 20:34:10 GMT-8 |
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changed: -\begin{aldor} \begin{aldor}[morphisms]
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/zope2/var/LatexWiki/morphisms.as using AXIOM-XL compiler and
options
-O -Fasy -Fao -Flsp -laxiom -Mno-AXL_W_WillObsolete -DAxiom -Y $AXIOM/algebra
Use the system command )set compiler args to change these
options.
#1 (Warning) Deprecated message prefix: use `ALDOR_' instead of `_AXL'
Compiling Lisp source code from file ./morphisms.lsp
Issuing )library command for morphisms
Reading /var/zope2/var/LatexWiki/morphisms.asy
Morphisms is now explicitly exposed in frame initial
Morphisms will be automatically loaded when needed from
/var/zope2/var/LatexWiki/morphisms
Endomorphism is now explicitly exposed in frame initial
Endomorphism will be automatically loaded when needed from
/var/zope2/var/LatexWiki/morphisms
AutomorphismCategory is now explicitly exposed in frame initial
AutomorphismCategory will be automatically loaded when needed from
/var/zope2/var/LatexWiki/morphisms
Automorphism is now explicitly exposed in frame initial
Automorphism will be automatically loaded when needed from
/var/zope2/var/LatexWiki/morphisms
EndomorphismCategory is now explicitly exposed in frame initial
EndomorphismCategory will be automatically loaded when needed from
/var/zope2/var/LatexWiki/morphisms... --Bill Page, Tue, 20 Nov 2007 20:29:07 -0800 reply
[SandBox Aldor Category Theory 5]?