1 Generalizable All Variables.
2 Require Import Preamble.
3 Require Import Categories_ch1_3.
4 Require Import Functors_ch1_4.
6 (******************************************************************************)
7 (* Chapter 1.5: Isomorphisms *)
8 (******************************************************************************)
10 Class Isomorphism `{C:Category}{a b:C}(f:a~>b)(g:b~>a) : Prop :=
11 { iso_cmp1 : f >>> g ~~ id a
12 ; iso_cmp2 : g >>> f ~~ id b
14 (* TO DO: show isos are unique when they exist *)
21 ; iso_comp1 : iso_forward >>> iso_backward ~~ id a
22 ; iso_comp2 : iso_backward >>> iso_forward ~~ id b
23 (* TO DO: merge this with Isomorphism *)
25 Implicit Arguments iso_forward [ C a b Ob Hom ].
26 Implicit Arguments iso_backward [ C a b Ob Hom ].
27 Implicit Arguments iso_comp1 [ C a b Ob Hom ].
28 Implicit Arguments iso_comp2 [ C a b Ob Hom ].
29 Notation "a ≅ b" := (Isomorphic a b) : isomorphism_scope.
30 (* the sharp symbol "casts" an isomorphism to the morphism in the forward direction *)
31 Notation "# f" := (@iso_forward _ _ _ _ _ f) : isomorphism_scope.
32 Open Scope isomorphism_scope.
34 (* the inverse of an isomorphism is an isomorphism *)
35 Definition iso_inv `{C:Category}(a b:C)(is:Isomorphic a b) : Isomorphic b a.
36 intros; apply (Build_Isomorphic _ _ _ _ _ (iso_backward _) (iso_forward _));
37 [ apply iso_comp2 | apply iso_comp1 ].
39 Notation "f '⁻¹'" := (@iso_inv _ _ _ _ _ f) : isomorphism_scope.
41 (* identity maps are isomorphisms *)
42 Definition iso_id `{C:Category}(A:C) : Isomorphic A A.
43 intros; apply (Build_Isomorphic _ _ C A A (id A) (id A)); auto.
46 (* the composition of two isomorphisms is an isomorphism *)
47 Definition iso_comp `{C:Category}{a b c:C}(i1:Isomorphic a b)(i2:Isomorphic b c) : Isomorphic a c.
48 intros; apply (@Build_Isomorphic _ _ C a c (#i1 >>> #i2) (#i2⁻¹ >>> #i1⁻¹));
49 setoid_rewrite juggle3;
50 [ setoid_rewrite (iso_comp1 i2) | setoid_rewrite (iso_comp2 i1) ];
51 setoid_rewrite right_identity;
52 [ setoid_rewrite (iso_comp1 i1) | setoid_rewrite (iso_comp2 i2) ];
56 Definition functors_preserve_isos `{C1:Category}`{C2:Category}{Fo}(F:Functor C1 C2 Fo){a b:C1}(i:Isomorphic a b)
57 : Isomorphic (F a) (F b).
58 refine {| iso_forward := F \ (iso_forward i)
59 ; iso_backward := F \ (iso_backward i)
61 setoid_rewrite fmor_preserves_comp.
62 setoid_rewrite iso_comp1.
63 apply fmor_preserves_id.
64 setoid_rewrite fmor_preserves_comp.
65 setoid_rewrite iso_comp2.
66 apply fmor_preserves_id.
69 Lemma iso_shift_right `{C:Category} : forall {a b c:C}(f:b~>c)(g:a~>c)(i:Isomorphic b a), #i⁻¹ >>> f ~~ g -> f ~~ #i >>> g.
72 setoid_rewrite <- associativity.
73 setoid_rewrite iso_comp1.
78 Lemma iso_shift_right' `{C:Category} : forall {a b c:C}(f:b~>c)(g:a~>c)(i:Isomorphic a b), #i >>> f ~~ g -> f ~~ #i⁻¹ >>> g.
81 setoid_rewrite <- associativity.
82 setoid_rewrite iso_comp1.
87 Lemma iso_shift_left `{C:Category} : forall {a b c:C}(f:a~>b)(g:a~>c)(i:Isomorphic c b), f >>> #i⁻¹ ~~ g -> f ~~ g >>> #i.
90 setoid_rewrite associativity.
91 setoid_rewrite iso_comp1.
96 Lemma iso_shift_left' `{C:Category} : forall {a b c:C}(f:a~>b)(g:a~>c)(i:Isomorphic b c), f >>> #i ~~ g -> f ~~ g >>> #i⁻¹.
99 setoid_rewrite associativity.
100 setoid_rewrite iso_comp1.
102 apply right_identity.
105 Lemma isos_forward_equal_then_backward_equal `{C:Category}{a}{b}(i1 i2:a ≅ b) : #i1 ~~ #i2 -> #i1⁻¹ ~~ #i2⁻¹.
107 setoid_rewrite <- left_identity at 1.
108 setoid_rewrite <- (iso_comp2 i2).
109 setoid_rewrite associativity.
111 setoid_rewrite iso_comp1.
112 setoid_rewrite right_identity.
116 Lemma iso_inv_inv `{C:Category}{a}{b}(i:a ≅ b) : #(i⁻¹)⁻¹ ~~ #i.
117 unfold iso_inv; simpl.