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 Topics FrontPage Aldor FriCAS language SandBox SandBox Aldor Category Theory SandBox Aldor Category Theory 6 <-- You are here. 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 #library lCategories "categories.ao" import from lCategories +++ +++ Cartesian Closed Categories +++ define CartesianClosedCategory(Obj:Category):Category == MathCategory Obj with _ Product Obj with _ Exponential Obj +++ +++ 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 aldor Compiling FriCAS source code from file /var/zope2/var/LatexWiki/1722447487155098788-25px001.as using AXIOM-XL compiler and options -O -Fasy -Fao -Flsp -laxiom -Mno-ALDOR_W_WillObsolete -DAxiom -Y $AXIOM/algebra -I$AXIOM/algebra Use the system command )set compiler args to change these options. #1 (Fatal Error) The file BOOLEAN.ao' is newer than basics.ao'. The )library system command was not called after compilation. ... --Bill Page, Tue, 20 Nov 2007 22:50:22 -0800 replySandBox Aldor Category Theory 7

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