Timings for Equiv.v
(* -*- mode: coq; mode: visual-line -*- *)
Require Import HoTT.Basics.
Require Import Types.Sigma Types.Forall Types.Arrow Types.Paths.
Local Open Scope path_scope.
Global Instance contr_map_isequiv {A B} (f : A -> B) `{IsEquiv _ _ f}
: IsTruncMap (-2) f.
intros b; refine (contr_equiv' {a : A & a = f^-1 b} _).
apply equiv_functor_sigma_id; intros a.
apply equiv_moveR_equiv_M.
Definition isequiv_contr_map {A B} (f : A -> B) `{IsTruncMap (-2) A B f}
: IsEquiv f.
intros b; exact (center {a : A & f a = b}).1.
exact (center {a : A & f a = b}).2.
exact (@contr {x : A & f x = f a} _ (a;1))..1.
rewrite <- transport_paths_l, <- transport_compose.
exact ((@contr {x : A & f x = f a} _ (a;1))..2).
(** As usual, we can't make both of these [Instances]. *)
#[local]
Hint Immediate isequiv_contr_map : typeclass_instances.
(** It follows that when proving a map is an equivalence, we may assume its codomain is inhabited. *)
Definition isequiv_inhab_codomain {A B} (f : A -> B) (feq : B -> IsEquiv f)
: IsEquiv f.
Global Instance contr_sect_equiv {A B} (f : A -> B) `{IsEquiv A B f}
: Contr {g : B -> A & f o g == idmap}.
refine (contr_change_center (f^-1 ; eisretr f)).
refine (contr_equiv' { g : B -> A & f o g = idmap } _).
(* Typeclass inference finds this contractible instance: it's the fiber over [idmap] of postcomposition with [f], and the latter is an equivalence since [f] is. *)
apply equiv_functor_sigma_id; intros g.
Global Instance contr_retr_equiv {A B} (f : A -> B) `{IsEquiv A B f}
: Contr {g : B -> A & g o f == idmap}.
refine (contr_change_center (f^-1 ; eissect f)).
refine (contr_equiv' { g : B -> A & g o f = idmap } _).
apply equiv_functor_sigma_id; intros g.
(** We begin by showing that, assuming function extensionality, [IsEquiv f] is an hprop. *)
Global Instance hprop_isequiv {A B} (f : A -> B)
: IsHProp (IsEquiv f).
(** We will show that assuming [f] is an equivalence, [IsEquiv f] decomposes into a sigma of two contractible types. *)
apply hprop_inhabited_contr; intros feq.
nrefine (contr_equiv' _ (issig_isequiv f oE (equiv_sigma_assoc' _ _)^-1)).
srefine (contr_equiv' _ (equiv_contr_sigma _)^-1).
(** Each of these types is equivalent to a based homotopy space. The first is exactly [contr_sect_equiv]. *)
1: rapply contr_sect_equiv.
(** The second requires a bit more work. *)
refine (contr_equiv' { s : f^-1 o f == idmap & eissect f == s } _).
apply equiv_functor_sigma_id; intros s; cbn.
apply equiv_functor_forall_id; intros a.
refine (equiv_concat_l (eisadj f a) _ oE _).
(** Now since [IsEquiv f] and the assertion that its fibers are contractible are both HProps, logical equivalence implies equivalence. *)
Definition equiv_contr_map_isequiv {A B} (f : A -> B)
: IsTruncMap (-2) f <~> IsEquiv f.
(** Both directions are found by typeclass inference! *)
(** Thus, paths of equivalences are equivalent to paths of functions. *)
Lemma equiv_path_equiv {A B : Type} (e1 e2 : A <~> B)
: (e1 = e2 :> (A -> B)) <~> (e1 = e2 :> (A <~> B)).
equiv_via ((issig_equiv A B) ^-1 e1 = (issig_equiv A B) ^-1 e2).
2: symmetry; apply equiv_ap; refine _.
exact (equiv_path_sigma_hprop ((issig_equiv A B)^-1 e1) ((issig_equiv A B)^-1 e2)).
Definition path_equiv {A B : Type} {e1 e2 : A <~> B}
: (e1 = e2 :> (A -> B)) -> (e1 = e2 :> (A <~> B))
:= equiv_path_equiv e1 e2.
Global Instance isequiv_path_equiv {A B : Type} {e1 e2 : A <~> B}
: IsEquiv (@path_equiv _ _ e1 e2)
(* Coq can find this instance by itself, but it's slow. *)
:= equiv_isequiv (equiv_path_equiv e1 e2).
(** The inverse equivalence is homotopic to [ap equiv_fun], so that is also an equivalence. *)
Global Instance isequiv_ap_equiv_fun {A B : Type} (e1 e2 : A <~> B)
: IsEquiv (ap (x:=e1) (y:=e2) (@equiv_fun A B)).
snrapply isequiv_homotopic.
exact (equiv_path_equiv e1 e2)^-1%equiv.
exact (ap_compose (fun v => (equiv_fun v; equiv_isequiv v)) pr1 p)^.
(** This implies that types of equivalences inherit truncation. Note that we only state the theorem for [n.+1]-truncatedness, since it is not true for contractibility: if [B] is contractible but [A] is not, then [A <~> B] is not contractible because it is not inhabited.
Don't confuse this lemma with [trunc_equiv], which says that if [A] is truncated and [A] is equivalent to [B], then [B] is truncated. It would be nice to find a better pair of names for them. *)
Global Instance istrunc_equiv {n : trunc_index} {A B : Type} `{IsTrunc n.+1 B}
: IsTrunc n.+1 (A <~> B).
apply (istrunc_equiv_istrunc _ (equiv_path_equiv e1 e2)).
(** In the contractible case, we have to assume that *both* types are contractible to get a contractible type of equivalences. *)
Global Instance contr_equiv_contr_contr {A B : Type} `{Contr A} `{Contr B}
: Contr (A <~> B).
apply (Build_Contr _ equiv_contr_contr).
apply path_equiv, path_forall.
(** The type of *automorphisms* of an hprop is always contractible *)
Global Instance contr_aut_hprop A `{IsHProp A}
: Contr (A <~> A).
apply (Build_Contr _ 1%equiv).
intros e; apply path_equiv, path_forall.
intros ?; apply path_ishprop.
(** Equivalences are functorial under equivalences. *)
Definition functor_equiv {A B C D} (h : A <~> C) (k : B <~> D)
: (A <~> B) -> (C <~> D)
:= fun f => ((k oE f) oE h^-1).
Global Instance isequiv_functor_equiv {A B C D} (h : A <~> C) (k : B <~> D)
: IsEquiv (functor_equiv h k).
refine (isequiv_adjointify _
(functor_equiv (equiv_inverse h) (equiv_inverse k)) _ _).
intros f; apply path_equiv, path_arrow; intros x; simpl.
exact (eisretr k _ @ ap f (eisretr h x)).
intros g; apply path_equiv, path_arrow; intros x; simpl.
exact (eissect k _ @ ap g (eissect h x)).
Definition equiv_functor_equiv {A B C D} (h : A <~> C) (k : B <~> D)
: (A <~> B) <~> (C <~> D)
:= Build_Equiv _ _ (functor_equiv h k) _.
(** Reversing equivalences is an equivalence *)
Global Instance isequiv_equiv_inverse {A B}
: IsEquiv (@equiv_inverse A B).
refine (isequiv_adjointify _ equiv_inverse _ _);
intros e; apply path_equiv; reflexivity.
Definition equiv_equiv_inverse A B
: (A <~> B) <~> (B <~> A)
:= Build_Equiv _ _ (@equiv_inverse A B) _.
(** If [functor_sigma idmap g] is an equivalence then so is g *)
Definition isequiv_from_functor_sigma {A} (P Q : A -> Type)
(g : forall a, P a -> Q a) `{!IsEquiv (functor_sigma idmap g)}
: forall (a : A), IsEquiv (g a).
intros a; apply isequiv_contr_map.
apply istruncmap_from_functor_sigma.
(** Theorem 4.7.7 *)
(** [functor_sigma idmap g] is an equivalence if and only if g is *)
Definition equiv_total_iff_equiv_fiberwise {A} (P Q : A -> Type)
(g : forall a, P a -> Q a)
: IsEquiv (functor_sigma idmap g) <-> forall a, IsEquiv (g a).
apply isequiv_from_functor_sigma.
apply isequiv_functor_sigma.
(** We make this a global hint outside of the section. *)
#[export] Hint Immediate isequiv_contr_map : typeclass_instances.
(** This is like [transport_arrow], but for a family of equivalence types. It just shows that the functions are homotopic. *)
Definition transport_equiv {A : Type} {B C : A -> Type}
{x1 x2 : A} (p : x1 = x2) (f : B x1 <~> C x1) (y : B x2)
: (transport (fun x => B x <~> C x) p f) y = p # (f (p^ # y)).
(** A version that shows that the underlying functions are equal. *)
Definition transport_equiv' {A : Type} {B C : A -> Type}
{x1 x2 : A} (p : x1 = x2) (f : B x1 <~> C x1)
: transport (fun x => B x <~> C x) p f = (equiv_transport _ p) oE f oE (equiv_transport _ p^) :> (B x2 -> C x2).
(** A version that shows that the equivalences are equal. Here we do need [Funext], for [path_equiv]. *)
Definition transport_equiv'' `{Funext} {A : Type} {B C : A -> Type}
{x1 x2 : A} (p : x1 = x2) (f : B x1 <~> C x1)
: transport (fun x => B x <~> C x) p f = (equiv_transport _ p) oE f oE (equiv_transport _ p^).