
(* -*- mode: coq; coq-prog-args: ("-emacs" "-nois" "-R" "." "Top" "-top" "bug_hom_anom_10") -*- *)
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   coqtop version trunk (June 2016) *)

Reserved Notation "x -> y" (at level 99, right associativity, y at level 200).
Reserved Notation "x * y" (at level 40, left associativity).
Delimit Scope type_scope with type.
Open Scope type_scope.
Global Set Universe Polymorphism.
Notation "A -> B" := (forall (_ : A), B) : type_scope.
Set Implicit Arguments.
Global Set Nonrecursive Elimination Schemes.
Record prod (A B : Type) := pair { fst : A ; snd : B }.
Notation "x * y" := (prod x y) : type_scope.
Axiom admit : forall {T}, T.
Delimit Scope function_scope with function.
Notation compose := (fun g f x => g (f x)).
Notation "g 'o' f" := (compose g%function f%function) (at level 40, left associativity) : function_scope.
Record PreCategory :=
  Build_PreCategory {
      object :> Type;
      morphism : object -> object -> Type;
      identity : forall x, morphism x x }.
Bind Scope category_scope with PreCategory.
Record Functor (C D : PreCategory) := { object_of :> C -> D }.
Bind Scope functor_scope with Functor.
Class Isomorphic {C : PreCategory} (s d : C) := {}.
Definition oppositeC (C : PreCategory) : PreCategory
  := @Build_PreCategory C (fun s d => morphism C d s) admit.
Notation "C ^op" := (oppositeC C) (at level 3, format "C '^op'") : category_scope.
Definition oppositeF C D (F : Functor C D) : Functor C^op D^op
  := Build_Functor (C^op) (D^op) (object_of F).
Definition set_cat : PreCategory := @Build_PreCategory Type (fun x y => x -> y) admit.
Definition prodC (C D : PreCategory) : PreCategory
  := @Build_PreCategory
       (C * D)%type
       (fun s d => (morphism C (fst s) (fst d)
                    * morphism D (snd s) (snd d))%type)
       admit.
Infix "*" := prodC : category_scope.
Section composition.
  Variables B C D E : PreCategory.
  Definition composeF (G : Functor D E) (F : Functor C D) : Functor C E := Build_Functor C E (fun c => G (F c)).
End composition.
Infix "o" := composeF : functor_scope.
Definition fstF {C D} : Functor (C * D) C := admit.
Definition sndF {C D} : Functor (C * D) D := admit.
Definition prodF C D D' (F : Functor C D) (F' : Functor C D') : Functor C (D * D') := admit.
Local Infix "*" := prodF : functor_scope.
Definition pairF C D C' D' (F : Functor C D) (F' : Functor C' D') : Functor (C * C') (D * D')
  := (F o fstF) * (F' o sndF).
Section hom_functor.
  Variable C : PreCategory.
  Local Notation obj_of c'c :=
    ((morphism
        C
        (fst (c'c : object (C^op * C)))
        (snd (c'c : object (C^op * C))))).
  Definition hom_functor : Functor (C^op * C) set_cat
    := Build_Functor (C^op * C) set_cat (fun c'c => obj_of c'c).
End hom_functor.
Definition identityF C : Functor C C := admit.
Definition functor_category (C D : PreCategory) : PreCategory
  := @Build_PreCategory (Functor C D) admit admit.
Local Notation "C -> D" := (functor_category C D) : category_scope.
Definition NaturalIsomorphism (C D : PreCategory) F G := @Isomorphic (C -> D) F G.

Section Adjunction.
  Variables C D : PreCategory.
  Variable F : Functor C D.
  Variable G : Functor D C.
  
  Record AdjunctionHom :=
    {
      mate_of : @NaturalIsomorphism
                  (prodC (oppositeC C) D)
                  (@set_cat)
                  (@composeF
                     (prodC (oppositeC C) D)
                     (prodC (oppositeC D) D)
                     (@set_cat) (@hom_functor D)
                     (@pairF (oppositeC C)
                             (oppositeC D) D D
                             (@oppositeF C D F) (identityF D)))
                  (@composeF
                     (prodC (oppositeC C) D)
                     (prodC (oppositeC C) C)
                     (@set_cat) (@hom_functor C)
                     (@pairF (oppositeC C)
                             (oppositeC C) D C
                             (identityF (oppositeC C)) G))
    }.
End Adjunction.