1 Generalizable All Variables.
2 Require Import Notations.
3 Require Import Categories_ch1_3.
4 Require Import Functors_ch1_4.
5 Require Import Isomorphisms_ch1_5.
7 (*******************************************************************************)
8 (* Chapter 7.5: Natural Isomorphisms *)
9 (*******************************************************************************)
12 Class NaturalIsomorphism `{C1:Category}`{C2:Category}{Fobj1 Fobj2:C1->C2}(F1:Functor C1 C2 Fobj1)(F2:Functor C1 C2 Fobj2) :=
13 { ni_iso : forall A, Fobj1 A ≅ Fobj2 A
14 ; ni_commutes : forall `(f:A~>B), #(ni_iso A) >>> F2 \ f ~~ F1 \ f >>> #(ni_iso B)
16 Implicit Arguments ni_iso [Ob Hom Ob0 Hom0 C1 C2 Fobj1 Fobj2 F1 F2].
17 Implicit Arguments ni_commutes [Ob Hom Ob0 Hom0 C1 C2 Fobj1 Fobj2 F1 F2 A B].
18 (* FIXME: coerce to NaturalTransformation instead *)
19 Coercion ni_iso : NaturalIsomorphism >-> Funclass.
20 Notation "F <~~~> G" := (@NaturalIsomorphism _ _ _ _ _ _ _ _ F G) : category_scope.
22 (* FIXME: Lemma 7.11: natural isos are natural transformations in which every morphism is an iso *)
24 (* every natural iso is invertible, and that inverse is also a natural iso *)
25 Definition ni_inv `(N:NaturalIsomorphism(F1:=F1)(F2:=F2)) : NaturalIsomorphism F2 F1.
26 intros; apply (Build_NaturalIsomorphism _ _ _ _ _ _ _ _ F2 F1 (fun A => iso_inv _ _ (ni_iso N A))).
27 abstract (intros; simpl;
28 set (ni_commutes N f) as qqq;
30 apply iso_shift_left' in qqq;
32 repeat setoid_rewrite <- associativity;
33 setoid_rewrite iso_comp2;
34 setoid_rewrite left_identity;
39 `{C1:Category}`{C2:Category}
40 {Fobj}(F:Functor C1 C2 Fobj)
41 : NaturalIsomorphism F F.
42 intros; apply (Build_NaturalIsomorphism _ _ _ _ _ _ _ _ F F (fun A => iso_id (F A))).
43 abstract (intros; simpl; setoid_rewrite left_identity; setoid_rewrite right_identity; reflexivity).
46 (* two different choices of composition order are naturally isomorphic (strictly, in fact) *)
47 Instance ni_associativity
48 `{C1:Category}`{C2:Category}`{C3:Category}`{C4:Category}
49 {Fobj1}(F1:Functor C1 C2 Fobj1)
50 {Fobj2}(F2:Functor C2 C3 Fobj2)
51 {Fobj3}(F3:Functor C3 C4 Fobj3)
53 ((F1 >>>> F2) >>>> F3) <~~~> (F1 >>>> (F2 >>>> F3)) :=
54 { ni_iso := fun A => iso_id (F3 (F2 (F1 A))) }.
57 setoid_rewrite left_identity;
58 setoid_rewrite right_identity;
62 Definition ni_comp `{C:Category}`{D:Category}
63 {F1Obj}{F1:@Functor _ _ C _ _ D F1Obj}
64 {F2Obj}{F2:@Functor _ _ C _ _ D F2Obj}
65 {F3Obj}{F3:@Functor _ _ C _ _ D F3Obj}
66 (N1:NaturalIsomorphism F1 F2)
67 (N2:NaturalIsomorphism F2 F3)
68 : NaturalIsomorphism F1 F3.
70 destruct N1 as [ ni_iso1 ni_commutes1 ].
71 destruct N2 as [ ni_iso2 ni_commutes2 ].
72 exists (fun A => iso_comp (ni_iso1 A) (ni_iso2 A)).
73 abstract (intros; simpl;
74 setoid_rewrite <- associativity;
75 setoid_rewrite <- ni_commutes1;
76 setoid_rewrite associativity;
77 setoid_rewrite <- ni_commutes2;
81 Definition ni_respects1
82 `{A:Category}`{B:Category}
83 {Fobj}(F:Functor A B Fobj)
85 {G0obj}(G0:Functor B C G0obj)
86 {G1obj}(G1:Functor B C G1obj)
87 : (G0 <~~~> G1) -> ((F >>>> G0) <~~~> (F >>>> G1)).
89 destruct phi as [ phi_niso phi_comm ].
90 refine {| ni_iso := (fun a => phi_niso (Fobj a)) |}.
91 intros; simpl; apply phi_comm.
94 Definition ni_respects2
95 `{A:Category}`{B:Category}
96 {F0obj}(F0:Functor A B F0obj)
97 {F1obj}(F1:Functor A B F1obj)
99 {Gobj}(G:Functor B C Gobj)
100 : (F0 <~~~> F1) -> ((F0 >>>> G) <~~~> (F1 >>>> G)).
102 destruct phi as [ phi_niso phi_comm ].
103 refine {| ni_iso := fun a => (@functors_preserve_isos _ _ _ _ _ _ _ G) _ _ (phi_niso a) |}.
105 setoid_rewrite fmor_preserves_comp.
110 Definition ni_respects
111 `{A:Category}`{B:Category}
112 {F0obj}(F0:Functor A B F0obj)
113 {F1obj}(F1:Functor A B F1obj)
115 {G0obj}(G0:Functor B C G0obj)
116 {G1obj}(G1:Functor B C G1obj)
117 : (F0 <~~~> F1) -> (G0 <~~~> G1) -> ((F0 >>>> G0) <~~~> (F1 >>>> G1)).
120 destruct psi as [ psi_niso psi_comm ].
121 destruct phi as [ phi_niso phi_comm ].
123 (fun a => iso_comp ((@functors_preserve_isos _ _ _ _ _ _ _ G0) _ _ (phi_niso a)) (psi_niso (F1obj a))) |}.
124 abstract (intros; simpl;
125 setoid_rewrite <- associativity;
126 setoid_rewrite fmor_preserves_comp;
127 setoid_rewrite <- phi_comm;
128 setoid_rewrite <- fmor_preserves_comp;
129 setoid_rewrite associativity;
130 apply comp_respects; try reflexivity;
135 * Some structures (like monoidal and premonoidal functors) use the isomorphism
136 * component of a natural isomorphism in an "informative" way; these structures
137 * should use NaturalIsomorphism.
139 * However, in other situations the actual iso used is irrelevant; all
140 * that matters is the fact that a natural family of them exists. In
141 * these cases we can speed up Coq (and the extracted program)
142 * considerably by making the family of isos belong to [Prop] rather
143 * than [Type]. IsomorphicFunctors does this -- it's essentially a
144 * copy of NaturalIsomorphism which lives in [Prop].
147 (* Definition 7.10 *)
148 Definition IsomorphicFunctors `{C1:Category}`{C2:Category}{Fobj1 Fobj2:C1->C2}(F1:Functor C1 C2 Fobj1)(F2:Functor C1 C2 Fobj2) :=
149 exists ni_iso : (forall A, Fobj1 A ≅ Fobj2 A),
150 forall `(f:A~>B), #(ni_iso A) >>> F2 \ f ~~ F1 \ f >>> #(ni_iso B).
151 Notation "F ≃ G" := (@IsomorphicFunctors _ _ _ _ _ _ _ _ F G) : category_scope.
153 Definition if_id `{C:Category}`{D:Category}{Fobj}(F:Functor C D Fobj) : IsomorphicFunctors F F.
154 exists (fun A => iso_id (F A)).
158 [ apply left_identity |
160 apply right_identity]).
163 (* every natural iso is invertible, and that inverse is also a natural iso *)
165 `{C1:Category}`{C2:Category}{Fobj1 Fobj2:C1->C2}{F1:Functor C1 C2 Fobj1}{F2:Functor C1 C2 Fobj2}
166 (N:IsomorphicFunctors F1 F2) : IsomorphicFunctors F2 F1.
168 destruct N as [ ni_iso ni_commutes ].
169 exists (fun A => iso_inv _ _ (ni_iso A)).
172 set (ni_commutes _ _ f) as qq.
174 apply iso_shift_left' in qq.
176 repeat setoid_rewrite <- associativity.
177 setoid_rewrite iso_comp2.
178 setoid_rewrite left_identity.
182 Definition if_comp `{C:Category}`{D:Category}
183 {F1Obj}{F1:@Functor _ _ C _ _ D F1Obj}
184 {F2Obj}{F2:@Functor _ _ C _ _ D F2Obj}
185 {F3Obj}{F3:@Functor _ _ C _ _ D F3Obj}
186 (N1:IsomorphicFunctors F1 F2)
187 (N2:IsomorphicFunctors F2 F3)
188 : IsomorphicFunctors F1 F3.
190 destruct N1 as [ ni_iso1 ni_commutes1 ].
191 destruct N2 as [ ni_iso2 ni_commutes2 ].
192 exists (fun A => iso_comp (ni_iso1 A) (ni_iso2 A)).
193 abstract (intros; simpl;
194 setoid_rewrite <- associativity;
195 setoid_rewrite <- ni_commutes1;
196 setoid_rewrite associativity;
197 setoid_rewrite <- ni_commutes2;
201 (* two different choices of composition order are naturally isomorphic (strictly, in fact) *)
202 Definition if_associativity
203 `{C1:Category}`{C2:Category}`{C3:Category}`{C4:Category}
204 {Fobj1}(F1:Functor C1 C2 Fobj1)
205 {Fobj2}(F2:Functor C2 C3 Fobj2)
206 {Fobj3}(F3:Functor C3 C4 Fobj3)
208 ((F1 >>>> F2) >>>> F3) ≃ (F1 >>>> (F2 >>>> F3)).
209 exists (fun A => iso_id (F3 (F2 (F1 A)))).
212 setoid_rewrite left_identity;
213 setoid_rewrite right_identity;
217 Definition if_left_identity `{C1:Category}`{C2:Category} {Fobj1}(F1:Functor C1 C2 Fobj1) : (functor_id _ >>>> F1) ≃ F1.
218 exists (fun a => iso_id (F1 a)).
219 abstract (intros; unfold functor_comp; simpl;
220 setoid_rewrite left_identity;
221 setoid_rewrite right_identity;
225 Definition if_right_identity `{C1:Category}`{C2:Category} {Fobj1}(F1:Functor C1 C2 Fobj1) : (F1 >>>> functor_id _) ≃ F1.
226 exists (fun a => iso_id (F1 a)).
227 abstract (intros; unfold functor_comp; simpl;
228 setoid_rewrite left_identity;
229 setoid_rewrite right_identity;
233 Definition if_respects
234 `{A:Category}`{B:Category}
235 {F0obj}(F0:Functor A B F0obj)
236 {F1obj}(F1:Functor A B F1obj)
238 {G0obj}(G0:Functor B C G0obj)
239 {G1obj}(G1:Functor B C G1obj)
240 : (F0 ≃ F1) -> (G0 ≃ G1) -> ((F0 >>>> G0) ≃ (F1 >>>> G1)).
243 destruct psi as [ psi_niso psi_comm ].
244 destruct phi as [ phi_niso phi_comm ].
245 exists (fun a => iso_comp ((@functors_preserve_isos _ _ _ _ _ _ _ G0) _ _ (phi_niso a)) (psi_niso (F1obj a))).
246 abstract (intros; simpl;
247 setoid_rewrite <- associativity;
248 setoid_rewrite fmor_preserves_comp;
249 setoid_rewrite <- phi_comm;
250 setoid_rewrite <- fmor_preserves_comp;
251 setoid_rewrite associativity;
252 apply comp_respects; try reflexivity;
256 Section ni_prod_comp.
257 Require Import ProductCategories_ch1_6_1.
259 `{C1:Category}`{C2:Category}
260 `{D1:Category}`{D2:Category}
261 {F1Obj}(F1:@Functor _ _ C1 _ _ D1 F1Obj)
262 {F2Obj}(F2:@Functor _ _ C2 _ _ D2 F2Obj)
263 `{E1:Category}`{E2:Category}
264 {G1Obj}(G1:@Functor _ _ D1 _ _ E1 G1Obj)
265 {G2Obj}(G2:@Functor _ _ D2 _ _ E2 G2Obj).
267 Definition ni_prod_comp_iso A : (((F1 >>>> G1) **** (F2 >>>> G2)) A) ≅ (((F1 **** F2) >>>> (G1 **** G2)) A).
269 unfold functor_product_fobj.
274 Lemma ni_prod_comp : (F1 >>>> G1) **** (F2 >>>> G2) <~~~> (F1 **** F2) >>>> (G1 **** G2).
275 refine {| ni_iso := ni_prod_comp_iso |}.
280 setoid_rewrite left_identity.
281 setoid_rewrite right_identity.