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(* File reduced by coq-bug-finder from original input, then from 13191 lines to 1315 lines, then from 1601 lines to 595 lines, then from 585 lines to 379 lines *)
(* coqc version 8.5beta1 (March 2015) compiled on Mar 3 2015 3:50:31 with OCaml 4.01.0
coqtop version cagnode15:/afs/csail.mit.edu/u/j/jgross/coq-8.5,v8.5 (ac62cda8a4f488b94033b108c37556877232137a) *)
Axiom admit : False.
Ltac admit := exfalso; exact admit.
Global Set Primitive Projections.
Notation projT1 := proj1_sig (only parsing).
Notation projT2 := proj2_sig (only parsing).
Definition relation (A : Type) := A -> A -> Type.
Class Reflexive {A} (R : relation A) :=
reflexivity : forall x : A, R x x.
Class Symmetric {A} (R : relation A) :=
symmetry : forall x y, R x y -> R y x.
Notation idmap := (fun x => x).
Delimit Scope function_scope with function.
Delimit Scope path_scope with path.
Delimit Scope fibration_scope with fibration.
Open Scope path_scope.
Open Scope fibration_scope.
Open Scope function_scope.
Notation pr1 := projT1.
Notation pr2 := projT2.
Notation "x .1" := (pr1 x) (at level 3, format "x '.1'") : fibration_scope.
Notation "x .2" := (pr2 x) (at level 3, format "x '.2'") : fibration_scope.
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.
Inductive paths {A : Type} (a : A) : A -> Type :=
idpath : paths a a.
Arguments idpath {A a} , [A] a.
Notation "x = y :> A" := (@paths A x y) : type_scope.
Notation "x = y" := (x = y :>_) : type_scope.
Definition inverse {A : Type} {x y : A} (p : x = y) : y = x
:= match p with idpath => idpath end.
Global Instance symmetric_paths {A} : Symmetric (@paths A) | 0 := @inverse A.
Definition concat {A : Type} {x y z : A} (p : x = y) (q : y = z) : x = z :=
match p, q with idpath, idpath => idpath end.
Notation "1" := idpath : path_scope.
Notation "p @ q" := (concat p%path q%path) (at level 20) : path_scope.
Notation "p ^" := (inverse p%path) (at level 3, format "p '^'") : path_scope.
Definition transport {A : Type} (P : A -> Type) {x y : A} (p : x = y) (u : P x) : P y :=
match p with idpath => u end.
Definition ap {A B:Type} (f:A -> B) {x y:A} (p:x = y) : f x = f y
:= match p with idpath => idpath end.
Definition pointwise_paths {A} {P:A->Type} (f g:forall x:A, P x)
:= forall x:A, f x = g x.
Notation "f == g" := (pointwise_paths f g) (at level 70, no associativity) : type_scope.
Definition apD10 {A} {B:A->Type} {f g : forall x, B x} (h:f=g)
: f == g
:= fun x => match h with idpath => 1 end.
Definition Sect {A B : Type} (s : A -> B) (r : B -> A) :=
forall x : A, r (s x) = x.
Class IsEquiv {A B : Type} (f : A -> B) := BuildIsEquiv {
equiv_inv : B -> A ;
eisretr : Sect equiv_inv f;
eissect : Sect f equiv_inv;
eisadj : forall x : A, eisretr (f x) = ap f (eissect x)
}.
Notation "f ^-1" := (@equiv_inv _ _ f _) (at level 3, format "f '^-1'") : function_scope.
Class Contr_internal (A : Type) := BuildContr {
center : A ;
contr : (forall y : A, center = y)
}.
Inductive trunc_index : Type :=
| minus_two : trunc_index
| trunc_S : trunc_index -> trunc_index.
Notation "n .+1" := (trunc_S n) (at level 2, left associativity, format "n .+1") : trunc_scope.
Local Open Scope trunc_scope.
Notation "-2" := minus_two (at level 0) : trunc_scope.
Notation "-1" := (-2.+1) (at level 0) : trunc_scope.
Notation "0" := (-1.+1) : trunc_scope.
Fixpoint IsTrunc_internal (n : trunc_index) (A : Type) : Type :=
match n with
| -2 => Contr_internal A
| n'.+1 => forall (x y : A), IsTrunc_internal n' (x = y)
end.
Class IsTrunc (n : trunc_index) (A : Type) : Type :=
Trunc_is_trunc : IsTrunc_internal n A.
Tactic Notation "transparent" "assert" "(" ident(name) ":" constr(type) ")" :=
refine (let __transparent_assert_hypothesis := (_ : type) in _);
[
| (
let H := match goal with H := _ |- _ => constr:(H) end in
rename H into name) ].
Definition transport_idmap_ap A (P : A -> Type) x y (p : x = y) (u : P x)
: transport P p u = transport idmap (ap P p) u
:= match p with idpath => idpath end.
Section Adjointify.
Context {A B : Type} (f : A -> B) (g : B -> A).
Context (isretr : Sect g f) (issect : Sect f g).
Let issect' := fun x =>
ap g (ap f (issect x)^) @ ap g (isretr (f x)) @ issect x.
Let is_adjoint' (a : A) : isretr (f a) = ap f (issect' a).
admit.
Defined.
Definition isequiv_adjointify : IsEquiv f
:= BuildIsEquiv A B f g isretr issect' is_adjoint'.
End Adjointify.
Record TruncType (n : trunc_index) := BuildTruncType {
trunctype_type : Type ;
istrunc_trunctype_type : IsTrunc n trunctype_type
}.
Arguments trunctype_type {_} _.
Coercion trunctype_type : TruncType >-> Sortclass.
Notation "n -Type" := (TruncType n) (at level 1) : type_scope.
Notation hSet := 0-Type.
Module Export Category.
Module Export Core.
Set Implicit Arguments.
Delimit Scope morphism_scope with morphism.
Delimit Scope category_scope with category.
Delimit Scope object_scope with object.
Record PreCategory :=
Build_PreCategory' {
object :> Type;
morphism : object -> object -> Type;
identity : forall x, morphism x x;
compose : forall s d d',
morphism d d'
-> morphism s d
-> morphism s d'
where "f 'o' g" := (compose f g);
associativity : forall x1 x2 x3 x4
(m1 : morphism x1 x2)
(m2 : morphism x2 x3)
(m3 : morphism x3 x4),
(m3 o m2) o m1 = m3 o (m2 o m1);
associativity_sym : forall x1 x2 x3 x4
(m1 : morphism x1 x2)
(m2 : morphism x2 x3)
(m3 : morphism x3 x4),
m3 o (m2 o m1) = (m3 o m2) o m1;
left_identity : forall a b (f : morphism a b), identity b o f = f;
right_identity : forall a b (f : morphism a b), f o identity a = f;
identity_identity : forall x, identity x o identity x = identity x
}.
Arguments identity {!C%category} / x%object : rename.
Arguments compose {!C%category} / {s d d'}%object (m1 m2)%morphism : rename.
Definition Build_PreCategory
object morphism compose identity
associativity left_identity right_identity
:= @Build_PreCategory'
object
morphism
compose
identity
associativity
(fun _ _ _ _ _ _ _ => symmetry _ _ (associativity _ _ _ _ _ _ _))
left_identity
right_identity
(fun _ => left_identity _ _ _).
Module Export CategoryCoreNotations.
Infix "o" := compose : morphism_scope.
Notation "1" := (identity _) : morphism_scope.
End CategoryCoreNotations.
End Core.
End Category.
Module Export Core.
Set Implicit Arguments.
Delimit Scope functor_scope with functor.
Local Open Scope morphism_scope.
Section Functor.
Variables C D : PreCategory.
Record Functor :=
{
object_of :> C -> D;
morphism_of : forall s d, morphism C s d
-> morphism D (object_of s) (object_of d);
composition_of : forall s d d'
(m1 : morphism C s d) (m2: morphism C d d'),
morphism_of _ _ (m2 o m1)
= (morphism_of _ _ m2) o (morphism_of _ _ m1);
identity_of : forall x, morphism_of _ _ (identity x)
= identity (object_of x)
}.
End Functor.
Arguments morphism_of [C%category] [D%category] F%functor [s%object d%object] m%morphism : rename, simpl nomatch.
End Core.
Module Export Morphisms.
Set Implicit Arguments.
Local Open Scope category_scope.
Local Open Scope morphism_scope.
Class IsIsomorphism {C : PreCategory} {s d} (m : morphism C s d) :=
{
morphism_inverse : morphism C d s;
left_inverse : morphism_inverse o m = identity _;
right_inverse : m o morphism_inverse = identity _
}.
Class Isomorphic {C : PreCategory} s d :=
{
morphism_isomorphic :> morphism C s d;
isisomorphism_isomorphic :> IsIsomorphism morphism_isomorphic
}.
Coercion morphism_isomorphic : Isomorphic >-> morphism.
Local Infix "<~=~>" := Isomorphic (at level 70, no associativity) : category_scope.
Section iso_equiv_relation.
Variable C : PreCategory.
Global Instance isisomorphism_identity (x : C) : IsIsomorphism (identity x)
:= {| morphism_inverse := identity x;
left_inverse := left_identity C x x (identity x);
right_inverse := right_identity C x x (identity x) |}.
Global Instance isomorphic_refl : Reflexive (@Isomorphic C)
:= fun x : C => {| morphism_isomorphic := identity x |}.
Definition idtoiso (x y : C) (H : x = y) : Isomorphic x y
:= match H in (_ = y0) return (x <~=~> y0) with
| 1%path => reflexivity x
end.
End iso_equiv_relation.
End Morphisms.
Notation IsCategory C := (forall s d : object C, IsEquiv (@idtoiso C s d)).
Notation isotoid C s d := (@equiv_inv _ _ (@idtoiso C s d) _).
Notation cat_of obj :=
(@Build_PreCategory obj
(fun x y => x -> y)
(fun _ x => x)
(fun _ _ _ f g => f o g)%core
(fun _ _ _ _ _ _ _ => idpath)
(fun _ _ _ => idpath)
(fun _ _ _ => idpath)
).
Definition set_cat : PreCategory := cat_of hSet.
Set Implicit Arguments.
Local Open Scope morphism_scope.
Section Grothendieck.
Variable C : PreCategory.
Variable F : Functor C set_cat.
Record Pair :=
{
c : C;
x : F c
}.
Local Notation Gmorphism s d :=
{ f : morphism C s.(c) d.(c)
| morphism_of F f s.(x) = d.(x) }.
Definition identity_H s
:= apD10 (identity_of F s.(c)) s.(x).
Definition Gidentity s : Gmorphism s s.
Proof.
exists 1.
apply identity_H.
Defined.
Definition Gcategory : PreCategory.
Proof.
refine (@Build_PreCategory
Pair
(fun s d => Gmorphism s d)
Gidentity
_
_
_
_); admit.
Defined.
End Grothendieck.
Lemma isotoid_1 {C} `{IsCategory C} {x : C} {H : IsIsomorphism (identity x)}
: isotoid C x x {| morphism_isomorphic := (identity x) ; isisomorphism_isomorphic := H |}
= idpath.
admit.
Defined.
Generalizable All Variables.
Section Grothendieck2.
Context `{IsCategory C}.
Variable F : Functor C set_cat.
Instance iscategory_grothendieck_toset : IsCategory (Gcategory F).
Proof.
intros s d.
refine (isequiv_adjointify _ _ _ _).
{
intro m.
transparent assert (H' : (s.(c) = d.(c))).
{
apply (idtoiso C (x := s.(c)) (y := d.(c)))^-1%function.
exists (m : morphism _ _ _).1.
admit.
}
{
transitivity {| x := transport (fun x => F x) H' s.(x) |}.
admit.
{
change d with {| c := d.(c) ; x := d.(x) |}; simpl.
apply ap.
subst H'.
simpl.
refine (transport_idmap_ap _ (fun x => F x : Type) _ _ _ _ @ _ @ (m : morphism _ _ _).2).
change (fun x => F x : Type) with (trunctype_type o object_of F)%function.
admit.
}
}
}
{
admit.
}
{
intro x.
hnf in s, d.
destruct x.
simpl.
erewrite @isotoid_1.
|
