#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)->Domains):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]"

   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/categories.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.
   The )library system command was not called after compilation.