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(* 02110-1301 USA                                                     *)
(*****************************************************************************)
(*          Projet Coq - Calculus of Inductive Constructions V5.10           *)
(*****************************************************************************)
(*                                                                           *)
(*          Category Theory : Functors                                       *)
(*                                                                           *)
(*          Amokrane Saibi May 1994                                          *)
(*                                                                           *)
(*****************************************************************************)
Require Export Setoid_dup1.
Require Export Hom_Equality.
Set Implicit Arguments.
Unset Strict Implicit.
(* Functors *)
Section funct_def.
Variable C D : Category.
Section funct_laws.
Variables (FOb : C -> D)
(FMap : forall a b : C, Map (a --> b) (FOb a --> FOb b)).
Definition Fcomp_law :=
forall (a b c : C) (f : a --> b) (g : b --> c),
FMap a c (f o g) =_S FMap a b f o FMap b c g.
Definition Fid_law := forall a : C, FMap a a (Id a) =_S Id (FOb a).
End funct_laws.
Structure Functor : Type :=
{FOb :> C -> D;
FMap : forall a b : C, Map (a --> b) (FOb a --> FOb b);
Prf_Fcomp_law : Fcomp_law FMap;
Prf_Fid_law : Fid_law FMap}.
Definition FMor (F : Functor) (a b : C) (f : a --> b) := FMap F a b f.
(*** rewrite rules ***)

(* Specifications: *)
Lemma FPres :
forall (F : Functor) (a b : C) (f g : a --> b),
f =_S g -> FMor F f =_S FMor F g.
Lemma FComp :
forall (F : Functor) (a b c : C) (f : a --> b) (g : b --> c),
FMor F (f o g) =_S FMor F f o FMor F g.
Lemma FComp1 :
forall (F : Functor) (a b c : C) (f : a --> b) (g : b --> c),
FMor F f o FMor F g =_S FMor F (f o g).
Lemma FId : forall (F : Functor) (a : C), FMor F (Id a) =_S Id (FOb F a).
Lemma FId1 : forall (F : Functor) (a : C), Id (FOb F a) =_S FMor F (Id a).
End funct_def.