login  home  contents  what's new  discussion  bug reports     help  links  subscribe  changes  refresh  edit

Edit detail for SandBox Aldor Category Theory 5 revision 3 of 7

1 2 3 4 5 6 7
Editor: Bill Page
Time: 2007/11/20 20:47:22 GMT-8
Note:

changed:
-+++
-+++  Cartesian Closed Categories
-+++
-define CartesianClosedCategory(Obj:Category):Category == MathCategory Obj with _
-                                                              Product Obj with _
-                                                          Exponential Obj 


changed:
-+++
-+++  Direct Product of objects and morphisms
-+++	
-define Product(Obj:Category):Category == with
-    Product: (A:Obj,B:Obj) -> ( AB:Obj, AB->A, AB->B, (X:Obj)->(X->A,X->B)->(X->AB) )
-    Product: (A1:Obj,B1:Obj,  A2:Obj,B2:Obj) -> (AB1:Obj,AB2:Obj,(A1->A2,B1->B2)->(AB1->AB2))
-    *:(Obj,Obj)-> with Obj
-    default
-        Product(A1:Obj,B1:Obj,A2:Obj,B2:Obj):(AB1:Obj,AB2:Obj,(A1->A2,B1->B2)->(AB1->AB2)) ==
-	    (ab1:Obj,pa1:ab1->A1,pb1:ab1->B1, product1: (X:Obj) -> (X->A1,X->B1) -> (X->ab1)) == Product(A1,B1)
-	    (ab2:Obj,pa2:ab2->A2,pb2:ab2->B2, product2: (X:Obj) -> (X->A2,X->B2) -> (X->ab2)) == Product(A2,B2)
-	    (f:A1->A2)*(g:B1->B2):(ab1->ab2) == product2 ( ab1 )( (x:ab1):A2 +-> f pa1 x, (x:ab1):B2 +-> g pb1 x )
-	    (ab1,ab2,*)
-        (A:Obj)*(B:Obj): with Obj == 
-            (AB:Obj,pa:AB->A,pb:AB->B,product:(X:Obj)->(X->A,X->B)->(X->AB)) == Product(A,B)
-            AB add
-        
-+++
-+++  Direct Sum
-+++
-define CoProduct(Obj:Category):Category == with
-    CoProduct: (A:Obj,B:Obj) -> ( AB:Obj, A->AB, B->AB, (X:Obj)->(A->X,B->X)->(AB->X) )
-    CoProduct: (A1:Obj,B1:Obj, A2:Obj,B2:Obj) -> (AB1:Obj,AB2:Obj,(A2->A1,B2->B1)->(AB2->AB1))
-    +:(Obj,Obj)-> with Obj
-    default
-        CoProduct(A1:Obj,B1:Obj,A2:Obj,B2:Obj):(AB1:Obj,AB2:Obj,(A2->A1,B2->B1)->(AB2->AB1)) == 
-	    (ab1:Obj,ia1:A1->ab1,ib1:B1->ab1, sum1: (X:Obj) -> (A1->X,B1->X) -> (ab1->X)) == CoProduct(A1,B1)
-	    (ab2:Obj,ia2:A2->ab2,ib2:B2->ab2, sum2: (X:Obj) -> (A2->X,B2->X) -> (ab2->X)) == CoProduct(A2,B2)
-	    (f:A2->A1)+(g:B2->B1):(ab2->ab1) == sum2 ( ab1 ) ( (x:A2):ab1 +-> ia1 f x, (x:B2):ab1 +-> ib1 g x )
-	    (ab1,ab2,+)
-        (A:Obj)+(B:Obj): with Obj == 
-            (AB:Obj,ia:A->AB,ib:B->AB,product:(X:Obj)->(A->X,B->X)->(AB->X)) == CoProduct(A,B)
-            AB add
-            
-+++
-+++  Multiple Direct Product of a Single Object
-+++    
-define MultiProduct(Obj:Category):Category == with
-    Product:(A:Obj,n:Integer)   -> (Prod:Obj,Integer->(Prod->A),(X:Obj)->(Tuple (X->A))->(X->Prod))
-    ^:(Obj,Integer) -> with Obj
-    default
-        (A:Obj)^(n:Integer): with Obj ==
-	    (Prod:Obj,project:Integer->(Prod->A),product:(X:Obj)->(Tuple (X->A))->(X->Prod)) == Product(A,n)
-	    Prod add
-
-+++
-+++  Multiple Direct Sum of a Single Object
-+++	    
-define CoMultiProduct(Obj:Category):Category == with
-    CoProduct:(A:Obj,n:Integer) -> ( Sum:Obj,Integer->(A->Sum),(X:Obj)->(Tuple (A->X))->(Sum->X))
-    ..:(Obj,Integer) -> with Obj
-    default
-        (A:Obj)..(n:Integer): with Obj == 
-	    (Sum:Obj,insert:Integer->(A->Sum),sum:(X:Obj)->(Tuple (A->X))->(Sum->X)) == CoProduct(A,n)
-	    Sum add
-            
-+++
-+++  Quotients
-+++ 
-define Quotient(Obj:Category):Category == with
-    Quotient:  (A:Obj,B:Obj) -> (A->B) -> (Quo:Obj,A->Quo,(X:Obj)->(A->X)->(Quo->X))
-    /:(A:Obj,B:Obj) -> ((A->B)->Obj)
-    default
-        (A:Obj)/(B:Obj):((A->B)->Obj) ==
-	    (f:A->B):Obj +-> 
-	        (Q:Obj,insert:A->Q,quo:(X:Obj)->(A->X)->(Q->X)) == Quotient(A,B)(f)
-		Q add
-                
-+++
-+++   Subobjects (the dual of Quotients)
-+++    
-define Subobject(Obj:Category):Category == with
-    Subobject: (A:Obj,B:Obj) -> (B->A) -> (Sub:Obj,Sub->A,(X:Obj)->(X->A)->(X->Sub))
-    \:(A:Obj,B:Obj)->((B->A)->Obj)
-    default
-        (A:Obj)\(B:Obj):(B->A)->Obj == 
-	    (f:(B->A)):Obj +->
-	        (S:Obj,include:S->A,sub:(X:Obj)->(X->A)->(X->S)) == Subobject(A,B)(f)
-		S add 
               

aldor
#include "axiom" #pile #library lBasics "basics.ao" import from lBasics #library lMorphisms "morphisms.ao" import from lMorphisms #library lAdjoints "adjoints.ao" import from lAdjoints +++ +++ The Aldor category of mathematical categories +++ define MathCategory(Obj:Category):Category == Id Obj with Compose Obj with Morphisms Obj +++ +++ One sometimes needs the Hom-style categories where morphisms from A to B +++ are objects in Hom(A,B) rather than A->B. +++ define MathCategory(Obj:Category,Hom:(Obj,Obj)->Domain):Category == with id: (A:Obj) -> Hom(A,A) compose: (A:Obj,B:Obj,C:Obj) -> (Hom(A,B),Hom(B,C)) -> Hom(A,C) +++ +++ Identities +++ define Id(Obj:Category):Category == with id: (A:Obj) -> (A->A) default id(A:Obj):(A->A) == (a:A):A +-> a +++ +++ Composition of Morphisms +++ define Compose(Obj:Category):Category == with compose: (A:Obj,B:Obj,C:Obj) -> (A->B,B->C) -> (A->C) default compose(A:Obj,B:Obj,C:Obj)(f:A->B,g:B->C):(A->C) == (a:A):C +-> g f a +++ +++ Initial Objects +++ define Initial(Obj:Category):Category == with Zero: () -> Obj zero: (A:Obj) -> (Zero()->A) -- 0: Obj -- 0: (A:Obj)->(0->A) +++ +++ Final Objects +++ define Final(Obj:Category):Category == with One: () -> Obj one: (A:Obj) -> (A->One()) -- 1: Obj -- 1: (A:Obj)->(A->1) +++ +++ Equalizer +++ define Equalizer(Obj:Category):Category == with Equalizer: (A:Obj,B:Obj,A->B,A->B) -> (E:Obj,E->A) +++ +++ CoEqualizer +++ define CoEqualizer(Obj:Category):Category == with CoEqualizer: (A:Obj,B:Obj,B->A,B->A) -> (E:Obj,A->E) +++ +++ Pullback Square +++ define Pullback(Obj:Category):Category == with Pullback: (A:Obj,C:Obj,B:Obj) -> (A->C,B->C) -> ( Pullback:Obj, Pullback->A,Pullback->B,(X:Obj,X->A,X->B) -> (X->Pullback)) +++ +++ Pushout Square, the dual of a Pullback Square +++ define Pushout(Obj:Category):Category == with Pushout: (A:Obj,C:Obj,B:Obj) -> (C->A,C->B) -> ( Pushout:Obj, A->Pushout,B->Pushout, (X:Obj,A->X,A->X) -> ( Pushout->X)) +++ +++ Exponential object +++ define Exp(Obj:Category,E:Obj):Category == with rightProductFunctor: Obj -> Obj expFunctor: Obj -> Obj Adjoint(Obj,Obj,rightProductFunctor,expFunctor) define Exponential(Obj:Category):Category == with Exp: (E:Obj) -> Exp(Obj,E) define Hom(Obj:Category):Category == with Hom: (Obj,Obj) -> Obj define Hom?(Obj:Category):Category == with hom?: (Obj,Obj) -> Boolean -- hom?(A,B) answers "Are there any Homs from A to B?" +++ +++ Decategorification +++ define Isomorphic(Obj:Category):Category == with isomorphic?: (A:Obj,B:Obj) -> Boolean Decategorify: Set with { object: Obj -> % } default Decategorify: Set with { object: Obj -> % } == add Rep == Obj object(A:Obj):% == per A (A:%)=(B:%):Boolean == isomorphic? ( rep A, rep B) coerce(A:%):OutputForm == message "[Object]"
aldor
   Compiling FriCAS source code from file 
      /var/zope2/var/LatexWiki/2878892845306364262-25px001.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 
      ./2878892845306364262-25px001.lsp
   Issuing )library command for 2878892845306364262-25px001
   Reading /var/zope2/var/LatexWiki/2878892845306364262-25px001.asy
   Isomorphic is now explicitly exposed in frame initial 
   Isomorphic will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/2878892845306364262-25px001
   Hom? is now explicitly exposed in frame initial 
   Hom? will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/2878892845306364262-25px001
   Hom is now explicitly exposed in frame initial 
   Hom will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/2878892845306364262-25px001
   Exponential is now explicitly exposed in frame initial 
   Exponential will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/2878892845306364262-25px001
   Pushout is now explicitly exposed in frame initial 
   Pushout will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/2878892845306364262-25px001
   Pullback is now explicitly exposed in frame initial 
   Pullback will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/2878892845306364262-25px001
   CoEqualizer is now explicitly exposed in frame initial 
   CoEqualizer will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/2878892845306364262-25px001
   Equalizer is now explicitly exposed in frame initial 
   Equalizer will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/2878892845306364262-25px001
   Final is now explicitly exposed in frame initial 
   Final will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/2878892845306364262-25px001
   Initial is now explicitly exposed in frame initial 
   Initial will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/2878892845306364262-25px001
   Compose is now explicitly exposed in frame initial 
   Compose will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/2878892845306364262-25px001
   Id is now explicitly exposed in frame initial 
   Id will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/2878892845306364262-25px001
   MathCategory is now explicitly exposed in frame initial 
   MathCategory will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/2878892845306364262-25px001
   Exp is now explicitly exposed in frame initial 
   Exp will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/2878892845306364262-25px001