add inverse form of ni_commutes

[coq-categories.git] / src / NaturalIsomorphisms_ch7_5.v

diff --git a/src/NaturalIsomorphisms_ch7_5.v b/src/NaturalIsomorphisms_ch7_5.v
index 415d13c..ce97ce6 100644 (file)
--- a/src/NaturalIsomorphisms_ch7_5.v
+++ b/src/NaturalIsomorphisms_ch7_5.v
@@ -1,5 +1,5 @@
 Generalizable All Variables.
-Require Import Preamble.
+Require Import Notations.
 Require Import Categories_ch1_3.
 Require Import Functors_ch1_4.
 Require Import Isomorphisms_ch1_5.
@@ -19,6 +19,17 @@ Implicit Arguments ni_commutes [Ob Hom Ob0 Hom0 C1 C2 Fobj1 Fobj2 F1 F2 A B].
 Coercion ni_iso : NaturalIsomorphism >-> Funclass.
 Notation "F <~~~> G" := (@NaturalIsomorphism _ _ _ _ _ _ _ _ F G) : category_scope.
 
+(* same as ni_commutes, but phrased in terms of inverses *)
+Lemma ni_commutes' `(ni:NaturalIsomorphism) : forall `(f:A~>B), F2 \ f >>> #(ni_iso ni B)⁻¹ ~~ #(ni_iso ni A)⁻¹ >>> F1 \ f.
+  intros.
+  apply iso_shift_right'.
+  setoid_rewrite <- associativity.
+  symmetry.
+  apply iso_shift_left'.
+  symmetry.
+  apply ni_commutes.
+  Qed.
+
 (* FIXME: Lemma 7.11: natural isos are natural transformations in which every morphism is an iso *)
 
 (* every natural iso is invertible, and that inverse is also a natural iso *)
@@ -69,7 +80,7 @@ Definition ni_comp `{C:Category}`{D:Category}
    intros.
      destruct N1 as [ ni_iso1 ni_commutes1 ].
      destruct N2 as [ ni_iso2 ni_commutes2 ].
-   exists (fun A => id_comp (ni_iso1 A) (ni_iso2 A)).
+   exists (fun A => iso_comp (ni_iso1 A) (ni_iso2 A)).
    abstract (intros; simpl;
              setoid_rewrite <- associativity;
              setoid_rewrite <- ni_commutes1;
@@ -78,19 +89,49 @@ Definition ni_comp `{C:Category}`{D:Category}
              reflexivity).
    Defined.
 
+Definition ni_respects1
+  `{A:Category}`{B:Category}
+  {Fobj}(F:Functor A B Fobj)
+  `{C:Category}
+  {G0obj}(G0:Functor B C G0obj)
+  {G1obj}(G1:Functor B C G1obj)
+  : (G0 <~~~> G1) -> ((F >>>> G0) <~~~> (F >>>> G1)).
+  intro phi.
+  destruct phi as [ phi_niso phi_comm ].
+  refine {| ni_iso := (fun a => phi_niso (Fobj a)) |}.
+  intros; simpl; apply phi_comm.
+  Defined.
+
+Definition ni_respects2
+  `{A:Category}`{B:Category}
+  {F0obj}(F0:Functor A B F0obj)
+  {F1obj}(F1:Functor A B F1obj)
+  `{C:Category}
+  {Gobj}(G:Functor B C Gobj)
+  : (F0 <~~~> F1) -> ((F0 >>>> G) <~~~> (F1 >>>> G)).
+  intro phi.
+  destruct phi as [ phi_niso phi_comm ].
+  refine {| ni_iso := fun a => (@functors_preserve_isos _ _ _ _ _ _ _ G) _ _ (phi_niso a) |}.
+  intros; simpl.
+  setoid_rewrite fmor_preserves_comp.
+  apply fmor_respects.
+  apply phi_comm.
+  Defined.
+
 Definition ni_respects
-  `{A:Category}`{B:Category}`{C:Category}
-  {F0obj}{F0:Functor A B F0obj}
-  {F1obj}{F1:Functor A B F1obj}
-  {G0obj}{G0:Functor B C G0obj}
-  {G1obj}{G1:Functor B C G1obj}
+  `{A:Category}`{B:Category}
+  {F0obj}(F0:Functor A B F0obj)
+  {F1obj}(F1:Functor A B F1obj)
+  `{C:Category}
+  {G0obj}(G0:Functor B C G0obj)
+  {G1obj}(G1:Functor B C G1obj)
   : (F0 <~~~> F1) -> (G0 <~~~> G1) -> ((F0 >>>> G0) <~~~> (F1 >>>> G1)).
   intro phi.
   intro psi.
   destruct psi as [ psi_niso psi_comm ].
   destruct phi as [ phi_niso phi_comm ].
   refine {| ni_iso :=
-      (fun a => id_comp ((@functors_preserve_isos _ _ _ _ _ _ _ G0) _ _ (phi_niso a)) (psi_niso (F1obj a))) |}.
+      (fun a => iso_comp ((@functors_preserve_isos _ _ _ _ _ _ _ G0) _ _ (phi_niso a)) (psi_niso (F1obj a))) |}.
   abstract (intros; simpl;
             setoid_rewrite <- associativity;
             setoid_rewrite fmor_preserves_comp;
@@ -159,7 +200,7 @@ Definition if_comp `{C:Category}`{D:Category}
    intros.
      destruct N1 as [ ni_iso1 ni_commutes1 ].
      destruct N2 as [ ni_iso2 ni_commutes2 ].
-   exists (fun A => id_comp (ni_iso1 A) (ni_iso2 A)).
+   exists (fun A => iso_comp (ni_iso1 A) (ni_iso2 A)).
    abstract (intros; simpl;
              setoid_rewrite <- associativity;
              setoid_rewrite <- ni_commutes1;
@@ -201,17 +242,18 @@ Definition if_right_identity `{C1:Category}`{C2:Category} {Fobj1}(F1:Functor C1
   Defined.
 
 Definition if_respects
-  `{A:Category}`{B:Category}`{C:Category}
-  {F0obj}{F0:Functor A B F0obj}
-  {F1obj}{F1:Functor A B F1obj}
-  {G0obj}{G0:Functor B C G0obj}
-  {G1obj}{G1:Functor B C G1obj}
+  `{A:Category}`{B:Category}
+  {F0obj}(F0:Functor A B F0obj)
+  {F1obj}(F1:Functor A B F1obj)
+  `{C:Category}
+  {G0obj}(G0:Functor B C G0obj)
+  {G1obj}(G1:Functor B C G1obj)
   : (F0 ≃ F1) -> (G0 ≃ G1) -> ((F0 >>>> G0) ≃ (F1 >>>> G1)).
   intro phi.
   intro psi.
   destruct psi as [ psi_niso psi_comm ].
   destruct phi as [ phi_niso phi_comm ].
-  exists (fun a => id_comp ((@functors_preserve_isos _ _ _ _ _ _ _ G0) _ _ (phi_niso a)) (psi_niso (F1obj a))).
+  exists (fun a => iso_comp ((@functors_preserve_isos _ _ _ _ _ _ _ G0) _ _ (phi_niso a)) (psi_niso (F1obj a))).
   abstract (intros; simpl;
             setoid_rewrite <- associativity;
             setoid_rewrite fmor_preserves_comp;
@@ -227,11 +269,11 @@ Require Import ProductCategories_ch1_6_1.
   Context
   `{C1:Category}`{C2:Category}
   `{D1:Category}`{D2:Category}
-   {F1Obj}{F1:@Functor _ _ C1 _ _ D1 F1Obj}
-   {F2Obj}{F2:@Functor _ _ C2 _ _ D2 F2Obj}
+   {F1Obj}(F1:@Functor _ _ C1 _ _ D1 F1Obj)
+   {F2Obj}(F2:@Functor _ _ C2 _ _ D2 F2Obj)
   `{E1:Category}`{E2:Category}
-   {G1Obj}{G1:@Functor _ _ D1 _ _ E1 G1Obj}
-   {G2Obj}{G2:@Functor _ _ D2 _ _ E2 G2Obj}.
+   {G1Obj}(G1:@Functor _ _ D1 _ _ E1 G1Obj)
+   {G2Obj}(G2:@Functor _ _ D2 _ _ E2 G2Obj).
 
   Definition ni_prod_comp_iso A : (((F1 >>>> G1) **** (F2 >>>> G2)) A) ≅ (((F1 **** F2) >>>> (G1 **** G2)) A).
     unfold functor_fobj.