Library HoTT.Pointed.Loops
Require Import HoTT.Basics HoTT.Types.
Require Import HFiber Factorization Truncations.Core Truncations.Connectedness HProp.
Require Import Pointed.Core Pointed.pEquiv.
Require Import WildCat.
Require Import Spaces.Nat.Core.
Local Open Scope pointed_scope.
Local Open Scope path_scope.
Require Import HFiber Factorization Truncations.Core Truncations.Connectedness HProp.
Require Import Pointed.Core Pointed.pEquiv.
Require Import WildCat.
Require Import Spaces.Nat.Core.
Local Open Scope pointed_scope.
Local Open Scope path_scope.
Global Instance ispointed_loops A (a : A) : IsPointed (a = a) := 1.
Definition loops (A : pType) : pType
:= [point A = point A, 1].
Definition iterated_loops (n : nat) (A : pType) : pType
:= nat_iter n loops A.
(* Inner unfolding for iterated loops *)
Definition unfold_iterated_loops (n : nat) (X : pType)
: iterated_loops n.+1 X = iterated_loops n (loops X)
:= nat_iter_succ_r _ _ _.
Definition loops (A : pType) : pType
:= [point A = point A, 1].
Definition iterated_loops (n : nat) (A : pType) : pType
:= nat_iter n loops A.
(* Inner unfolding for iterated loops *)
Definition unfold_iterated_loops (n : nat) (X : pType)
: iterated_loops n.+1 X = iterated_loops n (loops X)
:= nat_iter_succ_r _ _ _.
The loop space decreases the truncation level by one. We don't bother making this an instance because it is automatically found by typeclass search, but we record it here in case anyone is looking for it.
Similarly for connectedness.
Definition isconnected_loops `{Univalence} {n} (A : pType)
`{IsConnected n.+1 A} : IsConnected n (loops A) := _.
Lemma pequiv_loops_punit : loops pUnit <~>* pUnit.
Proof.
snrapply Build_pEquiv'.
{ srapply (equiv_adjointify (fun _ ⇒ tt) (fun _ ⇒ idpath)).
1: by intros [].
rapply path_contr. }
reflexivity.
Defined.
`{IsConnected n.+1 A} : IsConnected n (loops A) := _.
Lemma pequiv_loops_punit : loops pUnit <~>* pUnit.
Proof.
snrapply Build_pEquiv'.
{ srapply (equiv_adjointify (fun _ ⇒ tt) (fun _ ⇒ idpath)).
1: by intros [].
rapply path_contr. }
reflexivity.
Defined.
Global Instance is0functor_loops : Is0Functor loops.
Proof.
apply Build_Is0Functor.
intros A B f.
refine (Build_pMap (loops A) (loops B)
(fun p ⇒ (point_eq f)^ @ (ap f p @ point_eq f)) _).
refine (_ @ concat_Vp (point_eq f)).
apply whiskerL. apply concat_1p.
Defined.
Global Instance is1functor_loops : Is1Functor loops.
Proof.
apply Build_Is1Functor.
Proof.
apply Build_Is0Functor.
intros A B f.
refine (Build_pMap (loops A) (loops B)
(fun p ⇒ (point_eq f)^ @ (ap f p @ point_eq f)) _).
refine (_ @ concat_Vp (point_eq f)).
apply whiskerL. apply concat_1p.
Defined.
Global Instance is1functor_loops : Is1Functor loops.
Proof.
apply Build_Is1Functor.
Action on 2-cells
- intros A B f g p.
pointed_reduce.
srapply Build_pHomotopy; cbn.
{ intro q.
refine (_ @ (concat_p1 _)^ @ (concat_1p _)^).
apply moveR_Vp.
apply (concat_Ap (fun x ⇒ p x @ 1)). }
simpl. generalize (p point0). generalize (g point0).
intros _ []. reflexivity.
pointed_reduce.
srapply Build_pHomotopy; cbn.
{ intro q.
refine (_ @ (concat_p1 _)^ @ (concat_1p _)^).
apply moveR_Vp.
apply (concat_Ap (fun x ⇒ p x @ 1)). }
simpl. generalize (p point0). generalize (g point0).
intros _ []. reflexivity.
Preservation of identity.
- intros A.
srapply Build_pHomotopy.
{ intro p.
refine (concat_1p _ @ concat_p1 _ @ ap_idmap _). }
reflexivity.
srapply Build_pHomotopy.
{ intro p.
refine (concat_1p _ @ concat_p1 _ @ ap_idmap _). }
reflexivity.
Preservation of compositon.
- intros A B c g f.
srapply Build_pHomotopy.
{ intros p. cbn.
refine ((inv_pp _ _ @@ 1) @ concat_pp_p _ _ _ @ _).
apply whiskerL.
refine (((ap_V _ _)^ @@ 1) @ _ @ concat_p_pp _ _ _ @ ((ap_pp _ _ _)^ @@ 1)).
apply whiskerL.
refine (_ @ concat_p_pp _ _ _ @ ((ap_pp _ _ _)^ @@ 1)).
apply whiskerR.
apply ap_compose. }
by pointed_reduce.
Defined.
srapply Build_pHomotopy.
{ intros p. cbn.
refine ((inv_pp _ _ @@ 1) @ concat_pp_p _ _ _ @ _).
apply whiskerL.
refine (((ap_V _ _)^ @@ 1) @ _ @ concat_p_pp _ _ _ @ ((ap_pp _ _ _)^ @@ 1)).
apply whiskerL.
refine (_ @ concat_p_pp _ _ _ @ ((ap_pp _ _ _)^ @@ 1)).
apply whiskerR.
apply ap_compose. }
by pointed_reduce.
Defined.
Lemma fmap_loops_pp {X Y : pType} (f : X ->* Y) (x y : loops X)
: fmap loops f (x @ y) = fmap loops f x @ fmap loops f y.
Proof.
pointed_reduce_rewrite.
apply ap_pp.
Defined.
: fmap loops f (x @ y) = fmap loops f x @ fmap loops f y.
Proof.
pointed_reduce_rewrite.
apply ap_pp.
Defined.
Loops is a pointed functor
Global Instance ispointedfunctor_loops : IsPointedFunctor loops.
Proof.
snrapply Build_IsPointedFunctor'.
1-4: exact _.
exact pequiv_loops_punit.
Defined.
Lemma fmap_loops_pconst {A B : pType} : fmap loops (@pconst A B) ==* pconst.
Proof.
rapply fmap_zero_morphism.
Defined.
Proof.
snrapply Build_IsPointedFunctor'.
1-4: exact _.
exact pequiv_loops_punit.
Defined.
Lemma fmap_loops_pconst {A B : pType} : fmap loops (@pconst A B) ==* pconst.
Proof.
rapply fmap_zero_morphism.
Defined.
Global Instance is0functor_iterated_loops n : Is0Functor (iterated_loops n).
Proof.
induction n.
1: exact _.
nrapply is0functor_compose; exact _.
Defined.
Global Instance is1functor_iterated_loops n : Is1Functor (iterated_loops n).
Proof.
induction n.
1: exact _.
nrapply is1functor_compose; exact _.
Defined.
Lemma fmap_iterated_loops_pp {X Y : pType} (f : X ->* Y) n
(x y : iterated_loops n.+1 X)
: fmap (iterated_loops n.+1) f (x @ y)
= fmap (iterated_loops n.+1) f x @ fmap (iterated_loops n.+1) f y.
Proof.
apply fmap_loops_pp.
Defined.
Proof.
induction n.
1: exact _.
nrapply is0functor_compose; exact _.
Defined.
Global Instance is1functor_iterated_loops n : Is1Functor (iterated_loops n).
Proof.
induction n.
1: exact _.
nrapply is1functor_compose; exact _.
Defined.
Lemma fmap_iterated_loops_pp {X Y : pType} (f : X ->* Y) n
(x y : iterated_loops n.+1 X)
: fmap (iterated_loops n.+1) f (x @ y)
= fmap (iterated_loops n.+1) f x @ fmap (iterated_loops n.+1) f y.
Proof.
apply fmap_loops_pp.
Defined.
Definition hfiber_fmap_loops {A B : pType} (f : A ->* B) (p : loops B)
: {q : loops A & ap f q = (point_eq f @ p) @ (point_eq f)^}
<~> hfiber (fmap loops f) p.
Proof.
apply equiv_functor_sigma_id; intros q.
refine (equiv_moveR_Vp _ _ _ oE _).
apply equiv_moveR_pM.
Defined.
: {q : loops A & ap f q = (point_eq f @ p) @ (point_eq f)^}
<~> hfiber (fmap loops f) p.
Proof.
apply equiv_functor_sigma_id; intros q.
refine (equiv_moveR_Vp _ _ _ oE _).
apply equiv_moveR_pM.
Defined.
The loop space functor decreases the truncation level by one.
Global Instance istrunc_fmap_loops {n} (A B : pType) (f : A ->* B)
`{IsTruncMap n.+1 _ _ f} : IsTruncMap n (fmap loops f).
Proof.
intro p. apply (istrunc_equiv_istrunc _ (hfiber_fmap_loops f p)).
Defined.
`{IsTruncMap n.+1 _ _ f} : IsTruncMap n (fmap loops f).
Proof.
intro p. apply (istrunc_equiv_istrunc _ (hfiber_fmap_loops f p)).
Defined.
And likewise the connectedness.
Global Instance isconnected_fmap_loops `{Univalence} {n : trunc_index}
(A B : pType) (f : A ->* B) `{IsConnMap n.+1 _ _ f}
: IsConnMap n (fmap loops f).
Proof.
intros p; eapply isconnected_equiv'.
- refine (hfiber_fmap_loops f p oE _).
symmetry; apply hfiber_ap.
- exact _.
Defined.
Definition isconnected_iterated_fmap_loops `{Univalence}
(k : nat) (A B : pType) (f : A ->* B)
: ∀ n : trunc_index, IsConnMap (trunc_index_inc' n k) f
→ IsConnMap n (fmap (iterated_loops k) f).
Proof.
induction k; intros n C.
- exact C.
- apply isconnected_fmap_loops.
apply IHk.
exact C.
Defined.
(A B : pType) (f : A ->* B) `{IsConnMap n.+1 _ _ f}
: IsConnMap n (fmap loops f).
Proof.
intros p; eapply isconnected_equiv'.
- refine (hfiber_fmap_loops f p oE _).
symmetry; apply hfiber_ap.
- exact _.
Defined.
Definition isconnected_iterated_fmap_loops `{Univalence}
(k : nat) (A B : pType) (f : A ->* B)
: ∀ n : trunc_index, IsConnMap (trunc_index_inc' n k) f
→ IsConnMap n (fmap (iterated_loops k) f).
Proof.
induction k; intros n C.
- exact C.
- apply isconnected_fmap_loops.
apply IHk.
exact C.
Defined.
It follows that loop spaces "commute with images".
Definition equiv_loops_image `{Univalence} n {A B : pType} (f : A ->* B)
: loops ([image n.+1 f, factor1 (image n.+1 f) (point A)])
<~> image n (fmap loops f).
Proof.
set (C := [image n.+1 f, factor1 (image n.+1 f) (point A)]).
pose (g := Build_pMap A C (factor1 (image n.+1 f)) 1).
pose (h := Build_pMap C B (factor2 (image n.+1 f)) (point_eq f)).
transparent assert (I : (Factorization
(@IsConnMap n) (@MapIn n) (fmap loops f))).
{ refine (@Build_Factorization (@IsConnMap n) (@MapIn n)
(loops A) (loops B) (fmap loops f) (loops C)
(fmap loops g) (fmap loops h) _ _ _).
intros x; symmetry.
refine (_ @ fmap_comp loops g h x).
simpl.
abstract (rewrite !concat_1p; reflexivity). }
exact (path_intermediate (path_factor (O_factsys n) (fmap loops f) I
(image n (fmap loops f)))).
Defined.
: loops ([image n.+1 f, factor1 (image n.+1 f) (point A)])
<~> image n (fmap loops f).
Proof.
set (C := [image n.+1 f, factor1 (image n.+1 f) (point A)]).
pose (g := Build_pMap A C (factor1 (image n.+1 f)) 1).
pose (h := Build_pMap C B (factor2 (image n.+1 f)) (point_eq f)).
transparent assert (I : (Factorization
(@IsConnMap n) (@MapIn n) (fmap loops f))).
{ refine (@Build_Factorization (@IsConnMap n) (@MapIn n)
(loops A) (loops B) (fmap loops f) (loops C)
(fmap loops g) (fmap loops h) _ _ _).
intros x; symmetry.
refine (_ @ fmap_comp loops g h x).
simpl.
abstract (rewrite !concat_1p; reflexivity). }
exact (path_intermediate (path_factor (O_factsys n) (fmap loops f) I
(image n (fmap loops f)))).
Defined.
Loop inversion is a pointed equivalence
Definition loops_inv (A : pType) : loops A <~>* loops A.
Proof.
srapply Build_pEquiv.
1: exact (Build_pMap (loops A) (loops A) inverse 1).
apply isequiv_path_inverse.
Defined.
Proof.
srapply Build_pEquiv.
1: exact (Build_pMap (loops A) (loops A) inverse 1).
apply isequiv_path_inverse.
Defined.
Loops functor preserves equivalences
A version of unfold_iterated_loops that's an equivalence rather than an equality. We could get this from the equality, but it's more useful to construct it explicitly since then we can reason about it.
Definition unfold_iterated_loops' (n : nat) (X : pType)
: iterated_loops n.+1 X <~>* iterated_loops n (loops X).
Proof.
induction n.
1: reflexivity.
change (iterated_loops n.+2 X)
with (loops (iterated_loops n.+1 X)).
apply pequiv_fmap_loops, IHn.
Defined.
: iterated_loops n.+1 X <~>* iterated_loops n (loops X).
Proof.
induction n.
1: reflexivity.
change (iterated_loops n.+2 X)
with (loops (iterated_loops n.+1 X)).
apply pequiv_fmap_loops, IHn.
Defined.
For instance, we can prove that it's natural.
Definition unfold_iterated_fmap_loops {A B : pType} (n : nat) (f : A ->* B)
: (unfold_iterated_loops' n B) o× (fmap (iterated_loops n.+1) f)
==* (fmap (iterated_loops n) (fmap loops f)) o× (unfold_iterated_loops' n A).
Proof.
induction n.
- srefine (Build_pHomotopy _ _).
+ reflexivity.
+ cbn. apply moveL_pV.
refine (concat_1p _ @ _).
refine (concat_1p _ @ _).
refine (_ @ (concat_p1 _)^).
exact ((ap_idmap _)^).
- refine ((fmap_comp loops _ _)^* @* _).
refine (_ @* (fmap_comp loops _ _)).
rapply (fmap2 loops).
apply IHn.
Defined.
: (unfold_iterated_loops' n B) o× (fmap (iterated_loops n.+1) f)
==* (fmap (iterated_loops n) (fmap loops f)) o× (unfold_iterated_loops' n A).
Proof.
induction n.
- srefine (Build_pHomotopy _ _).
+ reflexivity.
+ cbn. apply moveL_pV.
refine (concat_1p _ @ _).
refine (concat_1p _ @ _).
refine (_ @ (concat_p1 _)^).
exact ((ap_idmap _)^).
- refine ((fmap_comp loops _ _)^* @* _).
refine (_ @* (fmap_comp loops _ _)).
rapply (fmap2 loops).
apply IHn.
Defined.
Iterated loops preserves equivalences
Definition pequiv_fmap_iterated_loops {A B} n
: A <~>* B → iterated_loops n A <~>* iterated_loops n B
:= emap (iterated_loops n).
: A <~>* B → iterated_loops n A <~>* iterated_loops n B
:= emap (iterated_loops n).
Loops preserves products.
Lemma loops_prod (X Y : pType) : loops (X × Y) <~>* loops X × loops Y.
Proof.
snrapply Build_pEquiv'.
1: exact (equiv_path_prod (point (X × Y)) (point (X × Y)))^-1%equiv.
reflexivity.
Defined.
Proof.
snrapply Build_pEquiv'.
1: exact (equiv_path_prod (point (X × Y)) (point (X × Y)))^-1%equiv.
reflexivity.
Defined.
There is a natural map from loops (X × Y) to loops X × loops Y, and ideally it would definitionally underly the equivalence loops_prod. That's not the case, but we show that loops_prod is homotopic to the expected maps after projecting to each factor.
Definition pfst_loops_prod (X Y : pType)
: pfst o× loops_prod X Y ==* fmap loops pfst.
Proof.
snrapply Build_pHomotopy.
- intro p; simpl.
rhs nrapply concat_1p.
symmetry; apply concat_p1.
- reflexivity.
Defined.
Definition psnd_loops_prod (X Y : pType)
: psnd o× loops_prod X Y ==* fmap loops psnd.
Proof.
snrapply Build_pHomotopy.
- intro p; simpl.
rhs nrapply concat_1p.
symmetry; apply concat_p1.
- reflexivity.
Defined.
: pfst o× loops_prod X Y ==* fmap loops pfst.
Proof.
snrapply Build_pHomotopy.
- intro p; simpl.
rhs nrapply concat_1p.
symmetry; apply concat_p1.
- reflexivity.
Defined.
Definition psnd_loops_prod (X Y : pType)
: psnd o× loops_prod X Y ==* fmap loops psnd.
Proof.
snrapply Build_pHomotopy.
- intro p; simpl.
rhs nrapply concat_1p.
symmetry; apply concat_p1.
- reflexivity.
Defined.
Iterated loops of products are products of iterated loops.
Lemma iterated_loops_prod (X Y : pType) {n}
: iterated_loops n (X × Y) <~>* (iterated_loops n X) × (iterated_loops n Y).
Proof.
induction n as [|n IHn].
1: reflexivity.
exact (loops_prod _ _ o×E emap loops IHn).
Defined.
: iterated_loops n (X × Y) <~>* (iterated_loops n X) × (iterated_loops n Y).
Proof.
induction n as [|n IHn].
1: reflexivity.
exact (loops_prod _ _ o×E emap loops IHn).
Defined.
Similarly, we compute the projections here.
Definition pfst_iterated_loops_prod (X Y : pType) {n}
: pfst o× iterated_loops_prod X Y ==* fmap (iterated_loops n) pfst.
Proof.
induction n as [|n IHn].
- reflexivity.
- change (_ ==* ?R) with (pfst o× (loops_prod _ _ o× fmap loops (iterated_loops_prod _ _)) ==* R).
refine ((pmap_compose_assoc _ _ _)^* @* _).
refine (pmap_prewhisker _ (pfst_loops_prod _ _) @* _).
refine ((fmap_comp loops _ _)^* @* _).
change (?L ==* _) with (L ==* fmap loops (fmap (iterated_loops n) pfst)).
rapply (fmap2 loops); simpl.
exact IHn.
Defined.
Definition psnd_iterated_loops_prod (X Y : pType) {n}
: psnd o× iterated_loops_prod X Y ==* fmap (iterated_loops n) psnd.
Proof.
induction n as [|n IHn].
- reflexivity.
- change (_ ==* ?R) with (psnd o× (loops_prod _ _ o× fmap loops (iterated_loops_prod _ _)) ==* R).
refine ((pmap_compose_assoc _ _ _)^* @* _).
refine (pmap_prewhisker _ (psnd_loops_prod _ _) @* _).
refine ((fmap_comp loops _ _)^* @* _).
change (?L ==* _) with (L ==* fmap loops (fmap (iterated_loops n) psnd)).
rapply (fmap2 loops); simpl.
exact IHn.
Defined.
(* A dependent form of loops *)
Definition loopsD {A} : pFam A → pFam (loops A)
:= fun Pp ⇒ Build_pFam (fun q : loops A ⇒ transport Pp q (dpoint Pp) = (dpoint Pp)) 1.
Global Instance istrunc_pfam_loopsD {n} {A} (P : pFam A)
{H :IsTrunc_pFam n.+1 P}
: IsTrunc_pFam n (loopsD P).
Proof.
intros a. pose (H (point A)). exact _.
Defined.
(* psigma and loops 'commute' *)
Lemma loops_psigma_commute (A : pType) (P : pFam A)
: loops (psigma P) <~>* psigma (loopsD P).
Proof.
snrapply Build_pEquiv'.
1: exact (equiv_path_sigma _ _ _)^-1%equiv.
reflexivity.
Defined.
(* product and loops 'commute' *)
Lemma loops_pproduct_commute `{Funext} (A : Type) (F : A → pType)
: loops (pproduct F) <~>* pproduct (loops o F).
Proof.
snrapply Build_pEquiv'.
1: apply equiv_apD10.
reflexivity.
Defined.
(* product and iterated loops commute *)
Lemma iterated_loops_pproduct_commute `{Funext} (A : Type) (F : A → pType) (n : nat)
: iterated_loops n (pproduct F) <~>* pproduct (iterated_loops n o F).
Proof.
induction n.
1: reflexivity.
refine (loops_pproduct_commute _ _ o×E _).
rapply (emap loops).
exact IHn.
Defined.
(* Loops neutralise sigmas when truncated *)
Lemma loops_psigma_trunc (n : nat) : ∀ (Aa : pType)
(Pp : pFam Aa) (istrunc_Pp : IsTrunc_pFam (trunc_index_inc minus_two n) Pp),
iterated_loops n (psigma Pp)
<~>* iterated_loops n Aa.
Proof.
induction n.
{ intros A P p.
snrapply Build_pEquiv'.
1: refine (@equiv_sigma_contr _ _ p).
reflexivity. }
intros A P p.
refine (pequiv_inverse (unfold_iterated_loops' _ _) o×E _ o×E unfold_iterated_loops' _ _).
refine (IHn _ _ _ o×E _).
rapply (emap (iterated_loops _)).
apply loops_psigma_commute.
Defined.
(* We can convert between loops in a type and loops in Type at that type. *)
Definition loops_type `{Univalence} (A : Type@{i})
: loops [Type@{i}, A] <~>* [A <~> A, equiv_idmap].
Proof.
apply issig_pequiv'.
∃ (equiv_equiv_path A A).
reflexivity.
Defined.
(* An iterated version. Note that the statement with "-1" substituted for n is loops [Type, A] <~>* [A → A, idmap], which is not true in general. Compare the previous result. *)
Lemma local_global_looping `{Univalence} (A : Type@{i}) (n : nat)
: iterated_loops@{j} n.+2 [Type@{i}, A]
<~>* pproduct (fun a ⇒ iterated_loops@{j} n.+1 [A, a]).
Proof.
induction n.
{ refine (_ o×E emap loops (loops_type A)).
apply issig_pequiv'.
∃ (equiv_inverse (equiv_path_arrow 1%equiv 1%equiv)
oE equiv_inverse (equiv_path_equiv 1%equiv 1%equiv)).
reflexivity. }
exact (loops_pproduct_commute _ _ o×E emap loops IHn).
Defined.
(* 7.2.7 *)
Theorem equiv_istrunc_istrunc_loops `{Funext} n X
: IsTrunc n.+2 X <~> ∀ (x : X), IsTrunc n.+1 (loops [X, x]).
Proof.
srapply equiv_iff_hprop.
intro tr_loops.
apply istrunc_S; intros x y.
apply istrunc_S; intros p.
destruct p.
nrapply tr_loops.
Defined.
(* 7.2.9, with n here meaning the same as n-1 there. Note that n.-1 in the statement is short for trunc_index_pred (nat_to_trunc_index n) which is definitionally equal to (trunc_index_inc minus_two n).+1. *)
Theorem equiv_istrunc_contr_iterated_loops `{Funext} (n : nat)
: ∀ A, IsTrunc n.-1 A <~> ∀ a : A, Contr (iterated_loops n [A, a]).
Proof.
induction n; intro A.
{ cbn. exact equiv_hprop_inhabited_contr. }
refine (_ oE equiv_istrunc_istrunc_loops n.-2 _).
srapply equiv_functor_forall_id.
intro a.
cbn beta.
refine (_ oE IHn (loops [A, a])).
refine (equiv_inO_equiv (-2) (unfold_iterated_loops' n [A,a])^-1 oE _).
rapply equiv_iff_hprop.
intros X p.
refine (@contr_equiv' _ _ _ X).
rapply (emap (iterated_loops _)).
srapply Build_pEquiv'.
1: exact (equiv_concat_lr p 1).
cbn; unfold ispointed_loops.
exact (concat_p1 _ @ concat_p1 _).
Defined.
: pfst o× iterated_loops_prod X Y ==* fmap (iterated_loops n) pfst.
Proof.
induction n as [|n IHn].
- reflexivity.
- change (_ ==* ?R) with (pfst o× (loops_prod _ _ o× fmap loops (iterated_loops_prod _ _)) ==* R).
refine ((pmap_compose_assoc _ _ _)^* @* _).
refine (pmap_prewhisker _ (pfst_loops_prod _ _) @* _).
refine ((fmap_comp loops _ _)^* @* _).
change (?L ==* _) with (L ==* fmap loops (fmap (iterated_loops n) pfst)).
rapply (fmap2 loops); simpl.
exact IHn.
Defined.
Definition psnd_iterated_loops_prod (X Y : pType) {n}
: psnd o× iterated_loops_prod X Y ==* fmap (iterated_loops n) psnd.
Proof.
induction n as [|n IHn].
- reflexivity.
- change (_ ==* ?R) with (psnd o× (loops_prod _ _ o× fmap loops (iterated_loops_prod _ _)) ==* R).
refine ((pmap_compose_assoc _ _ _)^* @* _).
refine (pmap_prewhisker _ (psnd_loops_prod _ _) @* _).
refine ((fmap_comp loops _ _)^* @* _).
change (?L ==* _) with (L ==* fmap loops (fmap (iterated_loops n) psnd)).
rapply (fmap2 loops); simpl.
exact IHn.
Defined.
(* A dependent form of loops *)
Definition loopsD {A} : pFam A → pFam (loops A)
:= fun Pp ⇒ Build_pFam (fun q : loops A ⇒ transport Pp q (dpoint Pp) = (dpoint Pp)) 1.
Global Instance istrunc_pfam_loopsD {n} {A} (P : pFam A)
{H :IsTrunc_pFam n.+1 P}
: IsTrunc_pFam n (loopsD P).
Proof.
intros a. pose (H (point A)). exact _.
Defined.
(* psigma and loops 'commute' *)
Lemma loops_psigma_commute (A : pType) (P : pFam A)
: loops (psigma P) <~>* psigma (loopsD P).
Proof.
snrapply Build_pEquiv'.
1: exact (equiv_path_sigma _ _ _)^-1%equiv.
reflexivity.
Defined.
(* product and loops 'commute' *)
Lemma loops_pproduct_commute `{Funext} (A : Type) (F : A → pType)
: loops (pproduct F) <~>* pproduct (loops o F).
Proof.
snrapply Build_pEquiv'.
1: apply equiv_apD10.
reflexivity.
Defined.
(* product and iterated loops commute *)
Lemma iterated_loops_pproduct_commute `{Funext} (A : Type) (F : A → pType) (n : nat)
: iterated_loops n (pproduct F) <~>* pproduct (iterated_loops n o F).
Proof.
induction n.
1: reflexivity.
refine (loops_pproduct_commute _ _ o×E _).
rapply (emap loops).
exact IHn.
Defined.
(* Loops neutralise sigmas when truncated *)
Lemma loops_psigma_trunc (n : nat) : ∀ (Aa : pType)
(Pp : pFam Aa) (istrunc_Pp : IsTrunc_pFam (trunc_index_inc minus_two n) Pp),
iterated_loops n (psigma Pp)
<~>* iterated_loops n Aa.
Proof.
induction n.
{ intros A P p.
snrapply Build_pEquiv'.
1: refine (@equiv_sigma_contr _ _ p).
reflexivity. }
intros A P p.
refine (pequiv_inverse (unfold_iterated_loops' _ _) o×E _ o×E unfold_iterated_loops' _ _).
refine (IHn _ _ _ o×E _).
rapply (emap (iterated_loops _)).
apply loops_psigma_commute.
Defined.
(* We can convert between loops in a type and loops in Type at that type. *)
Definition loops_type `{Univalence} (A : Type@{i})
: loops [Type@{i}, A] <~>* [A <~> A, equiv_idmap].
Proof.
apply issig_pequiv'.
∃ (equiv_equiv_path A A).
reflexivity.
Defined.
(* An iterated version. Note that the statement with "-1" substituted for n is loops [Type, A] <~>* [A → A, idmap], which is not true in general. Compare the previous result. *)
Lemma local_global_looping `{Univalence} (A : Type@{i}) (n : nat)
: iterated_loops@{j} n.+2 [Type@{i}, A]
<~>* pproduct (fun a ⇒ iterated_loops@{j} n.+1 [A, a]).
Proof.
induction n.
{ refine (_ o×E emap loops (loops_type A)).
apply issig_pequiv'.
∃ (equiv_inverse (equiv_path_arrow 1%equiv 1%equiv)
oE equiv_inverse (equiv_path_equiv 1%equiv 1%equiv)).
reflexivity. }
exact (loops_pproduct_commute _ _ o×E emap loops IHn).
Defined.
(* 7.2.7 *)
Theorem equiv_istrunc_istrunc_loops `{Funext} n X
: IsTrunc n.+2 X <~> ∀ (x : X), IsTrunc n.+1 (loops [X, x]).
Proof.
srapply equiv_iff_hprop.
intro tr_loops.
apply istrunc_S; intros x y.
apply istrunc_S; intros p.
destruct p.
nrapply tr_loops.
Defined.
(* 7.2.9, with n here meaning the same as n-1 there. Note that n.-1 in the statement is short for trunc_index_pred (nat_to_trunc_index n) which is definitionally equal to (trunc_index_inc minus_two n).+1. *)
Theorem equiv_istrunc_contr_iterated_loops `{Funext} (n : nat)
: ∀ A, IsTrunc n.-1 A <~> ∀ a : A, Contr (iterated_loops n [A, a]).
Proof.
induction n; intro A.
{ cbn. exact equiv_hprop_inhabited_contr. }
refine (_ oE equiv_istrunc_istrunc_loops n.-2 _).
srapply equiv_functor_forall_id.
intro a.
cbn beta.
refine (_ oE IHn (loops [A, a])).
refine (equiv_inO_equiv (-2) (unfold_iterated_loops' n [A,a])^-1 oE _).
rapply equiv_iff_hprop.
intros X p.
refine (@contr_equiv' _ _ _ X).
rapply (emap (iterated_loops _)).
srapply Build_pEquiv'.
1: exact (equiv_concat_lr p 1).
cbn; unfold ispointed_loops.
exact (concat_p1 _ @ concat_p1 _).
Defined.
loops_inv is a natural transformation.
Global Instance is1natural_loops_inv : Is1Natural loops loops loops_inv.
Proof.
intros A B f.
srapply Build_pHomotopy.
+ intros p. refine (inv_Vp _ _ @ whiskerR _ (point_eq f) @ concat_pp_p _ _ _).
refine (inv_pp _ _ @ whiskerL (point_eq f)^ (ap_V f p)^).
+ pointed_reduce. reflexivity.
Defined.
Proof.
intros A B f.
srapply Build_pHomotopy.
+ intros p. refine (inv_Vp _ _ @ whiskerR _ (point_eq f) @ concat_pp_p _ _ _).
refine (inv_pp _ _ @ whiskerL (point_eq f)^ (ap_V f p)^).
+ pointed_reduce. reflexivity.
Defined.
Loops on the pointed type of dependent pointed maps correspond to pointed dependent maps into a family of loops. We define this in this direction, because the forward map is pointed by reflexivity.