Library HoTT.Homotopy.Smash
Require Import Basics.Overture Basics.PathGroupoids Basics.Tactics Basics.Equivalences.
Require Import Types.Sum Types.Bool Types.Paths Types.Forall.
Require Import WildCat.Core WildCat.Bifunctor WildCat.Equiv.
Require Import Colimits.Pushout.
Require Import Cubical.DPath.
Require Import Pointed.Core.
Local Open Scope pointed_scope.
Local Open Scope dpath_scope.
Local Open Scope path_scope.
(* Definition of smash product *)
Definition sum_to_prod (X Y : pType) : X + Y → X × Y
:= sum_ind _ (fun x ⇒ (x, point Y)) (fun y ⇒ (point X, y)).
Definition sum_to_bool X Y : X + Y → Bool
:= sum_ind _ (fun _ ⇒ false) (fun _ ⇒ true).
Definition Smash@{u v w | u ≤ w, v ≤ w} (X : pType@{u}) (Y : pType@{v}) : pType@{w}
:= [Pushout@{w w w w} (sum_to_prod@{w w w} X Y) (sum_to_bool@{u v w} X Y), pushl (point X, point Y)].
Section Smash.
Context {X Y : pType}.
Definition sm (x : X) (y : Y) : Smash X Y := pushl (x, y).
Definition auxl : Smash X Y := pushr false.
Definition auxr : Smash X Y := pushr true.
Definition gluel (x : X) : sm x pt = auxl
:= pglue (f:=sum_to_prod X Y) (g:=sum_to_bool X Y) (inl x).
Definition gluer (y : Y) : sm pt y = auxr
:= pglue (f:=sum_to_prod X Y) (g:=sum_to_bool X Y) (inr y).
Definition gluel' (x x' : X) : sm x pt = sm x' pt
:= gluel x @ (gluel x')^.
Definition gluer' (y y' : Y) : sm pt y = sm pt y'
:= gluer y @ (gluer y')^.
Definition glue (x : X) (y : Y) : sm x pt = sm pt y
:= gluel' x pt @ gluer' pt y.
Definition glue_pt_left (y : Y) : glue pt y = gluer' pt y.
Proof.
refine (_ @ concat_1p _).
apply whiskerR, concat_pV.
Defined.
Definition glue_pt_right (x : X) : glue x pt = gluel' x pt.
Proof.
refine (_ @ concat_p1 _).
apply whiskerL, concat_pV.
Defined.
Definition ap_sm_left {x x' : X} (p : x = x')
: ap (fun t ⇒ sm t pt) p = gluel' x x'.
Proof.
destruct p.
symmetry.
apply concat_pV.
Defined.
Definition ap_sm_right {y y' : Y} (p : y = y')
: ap (sm pt) p = gluer' y y'.
Proof.
destruct p.
symmetry.
apply concat_pV.
Defined.
Definition Smash_ind {P : Smash X Y → Type}
(Psm : ∀ a b, P (sm a b)) (Pl : P auxl) (Pr : P auxr)
(Pgl : ∀ a, DPath P (gluel a) (Psm a pt) Pl)
(Pgr : ∀ b, DPath P (gluer b) (Psm pt b) Pr)
: ∀ x : Smash X Y, P x.
Proof.
srapply Pushout_ind.
+ intros [a b].
apply Psm.
+ exact (Bool_ind _ Pr Pl).
+ srapply sum_ind.
- exact Pgl.
- exact Pgr.
Defined.
Definition Smash_ind_beta_gluel {P : Smash X Y → Type}
{Psm : ∀ a b, P (sm a b)} {Pl : P auxl} {Pr : P auxr}
(Pgl : ∀ a, DPath P (gluel a) (Psm a pt) Pl)
(Pgr : ∀ b, DPath P (gluer b) (Psm pt b) Pr) (a : X)
: apD (Smash_ind Psm Pl Pr Pgl Pgr) (gluel a) = Pgl a
:= Pushout_ind_beta_pglue P _ _ _ (inl a).
Definition Smash_ind_beta_gluer {P : Smash X Y → Type}
{Psm : ∀ a b, P (sm a b)} {Pl : P auxl} {Pr : P auxr}
(Pgl : ∀ a, DPath P (gluel a) (Psm a pt) Pl)
(Pgr : ∀ b, DPath P (gluer b) (Psm pt b) Pr) (b : Y)
: apD (Smash_ind Psm Pl Pr Pgl Pgr) (gluer b) = Pgr b
:= Pushout_ind_beta_pglue P _ _ _ (inr b).
Definition Smash_ind_beta_gluel' {P : Smash X Y → Type}
{Psm : ∀ a b, P (sm a b)} {Pl : P auxl} {Pr : P auxr}
(Pgl : ∀ a, DPath P (gluel a) (Psm a pt) Pl)
(Pgr : ∀ b, DPath P (gluer b) (Psm pt b) Pr) (a b : X)
: apD (Smash_ind Psm Pl Pr Pgl Pgr) (gluel' a b)
= (Pgl a) @Dp ((Pgl b)^D).
Proof.
lhs napply dp_apD_pp.
apply ap011.
1: apply Smash_ind_beta_gluel.
lhs napply dp_apD_V.
apply ap.
apply Smash_ind_beta_gluel.
Defined.
Definition Smash_ind_beta_gluer' {P : Smash X Y → Type}
{Psm : ∀ a b, P (sm a b)} {Pl : P auxl} {Pr : P auxr}
(Pgl : ∀ a, DPath P (gluel a) (Psm a pt) Pl)
(Pgr : ∀ b, DPath P (gluer b) (Psm pt b) Pr) (a b : Y)
: apD (Smash_ind Psm Pl Pr Pgl Pgr) (gluer' a b)
= (Pgr a) @Dp ((Pgr b)^D).
Proof.
lhs napply dp_apD_pp.
apply ap011.
1: apply Smash_ind_beta_gluer.
lhs napply dp_apD_V.
apply ap.
apply Smash_ind_beta_gluer.
Defined.
Definition Smash_ind_beta_glue {P : Smash X Y → Type}
{Psm : ∀ a b, P (sm a b)} {Pl : P auxl} {Pr : P auxr}
(Pgl : ∀ a, DPath P (gluel a) (Psm a pt) Pl)
(Pgr : ∀ b, DPath P (gluer b) (Psm pt b) Pr) (a : X) (b : Y)
: apD (Smash_ind Psm Pl Pr Pgl Pgr) (glue a b)
= ((Pgl a) @Dp ((Pgl pt)^D)) @Dp ((Pgr pt) @Dp ((Pgr b)^D)).
Proof.
lhs napply dp_apD_pp.
apply ap011.
- apply Smash_ind_beta_gluel'.
- apply Smash_ind_beta_gluer'.
Defined.
Definition Smash_rec {P : Type} (Psm : X → Y → P) (Pl Pr : P)
(Pgl : ∀ a, Psm a pt = Pl) (Pgr : ∀ b, Psm pt b = Pr)
: Smash X Y → P
:= Smash_ind Psm Pl Pr (fun x ⇒ dp_const (Pgl x)) (fun x ⇒ dp_const (Pgr x)).
(* Version of smash_rec that forces (Pgl pt) and (Pgr pt) to be idpath *)
Definition Smash_rec' {P : Type} {Psm : X → Y → P}
(Pgl : ∀ a, Psm a pt = Psm pt pt) (Pgr : ∀ b, Psm pt b = Psm pt pt)
(ql : Pgl pt = 1) (qr : Pgr pt = 1)
: Smash X Y → P
:= Smash_rec Psm (Psm pt pt) (Psm pt pt) Pgl Pgr.
Definition Smash_rec_beta_gluel {P : Type} {Psm : X → Y → P} {Pl Pr : P}
(Pgl : ∀ a, Psm a pt = Pl) (Pgr : ∀ b, Psm pt b = Pr) (a : X)
: ap (Smash_rec Psm Pl Pr Pgl Pgr) (gluel a) = Pgl a.
Proof.
rhs_V napply (eissect dp_const).
apply moveL_equiv_V.
lhs_V napply dp_apD_const.
napply Smash_ind_beta_gluel.
Defined.
Definition Smash_rec_beta_gluer {P : Type} {Psm : X → Y → P} {Pl Pr : P}
(Pgl : ∀ a, Psm a pt = Pl) (Pgr : ∀ b, Psm pt b = Pr) (b : Y)
: ap (Smash_rec Psm Pl Pr Pgl Pgr) (gluer b) = Pgr b.
Proof.
rhs_V napply (eissect dp_const).
apply moveL_equiv_V.
lhs_V napply dp_apD_const.
napply Smash_ind_beta_gluer.
Defined.
Definition Smash_rec_beta_gluel' {P : Type} {Psm : X → Y → P} {Pl Pr : P}
(Pgl : ∀ a, Psm a pt = Pl) (Pgr : ∀ b, Psm pt b = Pr) (a b : X)
: ap (Smash_rec Psm Pl Pr Pgl Pgr) (gluel' a b) = Pgl a @ (Pgl b)^.
Proof.
lhs napply ap_pV.
f_ap.
1: apply Smash_rec_beta_gluel.
apply inverse2.
apply Smash_rec_beta_gluel.
Defined.
Definition Smash_rec_beta_gluer' {P : Type} {Psm : X → Y → P} {Pl Pr : P}
(Pgl : ∀ a, Psm a pt = Pl) (Pgr : ∀ b, Psm pt b = Pr) (a b : Y)
: ap (Smash_rec Psm Pl Pr Pgl Pgr) (gluer' a b) = Pgr a @ (Pgr b)^.
Proof.
lhs napply ap_pV.
f_ap.
1: apply Smash_rec_beta_gluer.
apply inverse2.
apply Smash_rec_beta_gluer.
Defined.
Definition Smash_rec_beta_glue {P : Type} {Psm : X → Y → P} {Pl Pr : P}
(Pgl : ∀ a, Psm a pt = Pl) (Pgr : ∀ b, Psm pt b = Pr) (a : X)
(b : Y) : ap (Smash_rec Psm Pl Pr Pgl Pgr) (glue a b)
= ((Pgl a) @ (Pgl pt)^) @ (Pgr pt @ (Pgr b)^).
Proof.
lhs napply ap_pp.
f_ap.
- apply Smash_rec_beta_gluel'.
- apply Smash_rec_beta_gluer'.
Defined.
End Smash.
Arguments sm : simpl never.
Arguments auxl : simpl never.
Arguments gluel : simpl never.
Arguments gluer : simpl never.
Require Import Types.Sum Types.Bool Types.Paths Types.Forall.
Require Import WildCat.Core WildCat.Bifunctor WildCat.Equiv.
Require Import Colimits.Pushout.
Require Import Cubical.DPath.
Require Import Pointed.Core.
Local Open Scope pointed_scope.
Local Open Scope dpath_scope.
Local Open Scope path_scope.
(* Definition of smash product *)
Definition sum_to_prod (X Y : pType) : X + Y → X × Y
:= sum_ind _ (fun x ⇒ (x, point Y)) (fun y ⇒ (point X, y)).
Definition sum_to_bool X Y : X + Y → Bool
:= sum_ind _ (fun _ ⇒ false) (fun _ ⇒ true).
Definition Smash@{u v w | u ≤ w, v ≤ w} (X : pType@{u}) (Y : pType@{v}) : pType@{w}
:= [Pushout@{w w w w} (sum_to_prod@{w w w} X Y) (sum_to_bool@{u v w} X Y), pushl (point X, point Y)].
Section Smash.
Context {X Y : pType}.
Definition sm (x : X) (y : Y) : Smash X Y := pushl (x, y).
Definition auxl : Smash X Y := pushr false.
Definition auxr : Smash X Y := pushr true.
Definition gluel (x : X) : sm x pt = auxl
:= pglue (f:=sum_to_prod X Y) (g:=sum_to_bool X Y) (inl x).
Definition gluer (y : Y) : sm pt y = auxr
:= pglue (f:=sum_to_prod X Y) (g:=sum_to_bool X Y) (inr y).
Definition gluel' (x x' : X) : sm x pt = sm x' pt
:= gluel x @ (gluel x')^.
Definition gluer' (y y' : Y) : sm pt y = sm pt y'
:= gluer y @ (gluer y')^.
Definition glue (x : X) (y : Y) : sm x pt = sm pt y
:= gluel' x pt @ gluer' pt y.
Definition glue_pt_left (y : Y) : glue pt y = gluer' pt y.
Proof.
refine (_ @ concat_1p _).
apply whiskerR, concat_pV.
Defined.
Definition glue_pt_right (x : X) : glue x pt = gluel' x pt.
Proof.
refine (_ @ concat_p1 _).
apply whiskerL, concat_pV.
Defined.
Definition ap_sm_left {x x' : X} (p : x = x')
: ap (fun t ⇒ sm t pt) p = gluel' x x'.
Proof.
destruct p.
symmetry.
apply concat_pV.
Defined.
Definition ap_sm_right {y y' : Y} (p : y = y')
: ap (sm pt) p = gluer' y y'.
Proof.
destruct p.
symmetry.
apply concat_pV.
Defined.
Definition Smash_ind {P : Smash X Y → Type}
(Psm : ∀ a b, P (sm a b)) (Pl : P auxl) (Pr : P auxr)
(Pgl : ∀ a, DPath P (gluel a) (Psm a pt) Pl)
(Pgr : ∀ b, DPath P (gluer b) (Psm pt b) Pr)
: ∀ x : Smash X Y, P x.
Proof.
srapply Pushout_ind.
+ intros [a b].
apply Psm.
+ exact (Bool_ind _ Pr Pl).
+ srapply sum_ind.
- exact Pgl.
- exact Pgr.
Defined.
Definition Smash_ind_beta_gluel {P : Smash X Y → Type}
{Psm : ∀ a b, P (sm a b)} {Pl : P auxl} {Pr : P auxr}
(Pgl : ∀ a, DPath P (gluel a) (Psm a pt) Pl)
(Pgr : ∀ b, DPath P (gluer b) (Psm pt b) Pr) (a : X)
: apD (Smash_ind Psm Pl Pr Pgl Pgr) (gluel a) = Pgl a
:= Pushout_ind_beta_pglue P _ _ _ (inl a).
Definition Smash_ind_beta_gluer {P : Smash X Y → Type}
{Psm : ∀ a b, P (sm a b)} {Pl : P auxl} {Pr : P auxr}
(Pgl : ∀ a, DPath P (gluel a) (Psm a pt) Pl)
(Pgr : ∀ b, DPath P (gluer b) (Psm pt b) Pr) (b : Y)
: apD (Smash_ind Psm Pl Pr Pgl Pgr) (gluer b) = Pgr b
:= Pushout_ind_beta_pglue P _ _ _ (inr b).
Definition Smash_ind_beta_gluel' {P : Smash X Y → Type}
{Psm : ∀ a b, P (sm a b)} {Pl : P auxl} {Pr : P auxr}
(Pgl : ∀ a, DPath P (gluel a) (Psm a pt) Pl)
(Pgr : ∀ b, DPath P (gluer b) (Psm pt b) Pr) (a b : X)
: apD (Smash_ind Psm Pl Pr Pgl Pgr) (gluel' a b)
= (Pgl a) @Dp ((Pgl b)^D).
Proof.
lhs napply dp_apD_pp.
apply ap011.
1: apply Smash_ind_beta_gluel.
lhs napply dp_apD_V.
apply ap.
apply Smash_ind_beta_gluel.
Defined.
Definition Smash_ind_beta_gluer' {P : Smash X Y → Type}
{Psm : ∀ a b, P (sm a b)} {Pl : P auxl} {Pr : P auxr}
(Pgl : ∀ a, DPath P (gluel a) (Psm a pt) Pl)
(Pgr : ∀ b, DPath P (gluer b) (Psm pt b) Pr) (a b : Y)
: apD (Smash_ind Psm Pl Pr Pgl Pgr) (gluer' a b)
= (Pgr a) @Dp ((Pgr b)^D).
Proof.
lhs napply dp_apD_pp.
apply ap011.
1: apply Smash_ind_beta_gluer.
lhs napply dp_apD_V.
apply ap.
apply Smash_ind_beta_gluer.
Defined.
Definition Smash_ind_beta_glue {P : Smash X Y → Type}
{Psm : ∀ a b, P (sm a b)} {Pl : P auxl} {Pr : P auxr}
(Pgl : ∀ a, DPath P (gluel a) (Psm a pt) Pl)
(Pgr : ∀ b, DPath P (gluer b) (Psm pt b) Pr) (a : X) (b : Y)
: apD (Smash_ind Psm Pl Pr Pgl Pgr) (glue a b)
= ((Pgl a) @Dp ((Pgl pt)^D)) @Dp ((Pgr pt) @Dp ((Pgr b)^D)).
Proof.
lhs napply dp_apD_pp.
apply ap011.
- apply Smash_ind_beta_gluel'.
- apply Smash_ind_beta_gluer'.
Defined.
Definition Smash_rec {P : Type} (Psm : X → Y → P) (Pl Pr : P)
(Pgl : ∀ a, Psm a pt = Pl) (Pgr : ∀ b, Psm pt b = Pr)
: Smash X Y → P
:= Smash_ind Psm Pl Pr (fun x ⇒ dp_const (Pgl x)) (fun x ⇒ dp_const (Pgr x)).
(* Version of smash_rec that forces (Pgl pt) and (Pgr pt) to be idpath *)
Definition Smash_rec' {P : Type} {Psm : X → Y → P}
(Pgl : ∀ a, Psm a pt = Psm pt pt) (Pgr : ∀ b, Psm pt b = Psm pt pt)
(ql : Pgl pt = 1) (qr : Pgr pt = 1)
: Smash X Y → P
:= Smash_rec Psm (Psm pt pt) (Psm pt pt) Pgl Pgr.
Definition Smash_rec_beta_gluel {P : Type} {Psm : X → Y → P} {Pl Pr : P}
(Pgl : ∀ a, Psm a pt = Pl) (Pgr : ∀ b, Psm pt b = Pr) (a : X)
: ap (Smash_rec Psm Pl Pr Pgl Pgr) (gluel a) = Pgl a.
Proof.
rhs_V napply (eissect dp_const).
apply moveL_equiv_V.
lhs_V napply dp_apD_const.
napply Smash_ind_beta_gluel.
Defined.
Definition Smash_rec_beta_gluer {P : Type} {Psm : X → Y → P} {Pl Pr : P}
(Pgl : ∀ a, Psm a pt = Pl) (Pgr : ∀ b, Psm pt b = Pr) (b : Y)
: ap (Smash_rec Psm Pl Pr Pgl Pgr) (gluer b) = Pgr b.
Proof.
rhs_V napply (eissect dp_const).
apply moveL_equiv_V.
lhs_V napply dp_apD_const.
napply Smash_ind_beta_gluer.
Defined.
Definition Smash_rec_beta_gluel' {P : Type} {Psm : X → Y → P} {Pl Pr : P}
(Pgl : ∀ a, Psm a pt = Pl) (Pgr : ∀ b, Psm pt b = Pr) (a b : X)
: ap (Smash_rec Psm Pl Pr Pgl Pgr) (gluel' a b) = Pgl a @ (Pgl b)^.
Proof.
lhs napply ap_pV.
f_ap.
1: apply Smash_rec_beta_gluel.
apply inverse2.
apply Smash_rec_beta_gluel.
Defined.
Definition Smash_rec_beta_gluer' {P : Type} {Psm : X → Y → P} {Pl Pr : P}
(Pgl : ∀ a, Psm a pt = Pl) (Pgr : ∀ b, Psm pt b = Pr) (a b : Y)
: ap (Smash_rec Psm Pl Pr Pgl Pgr) (gluer' a b) = Pgr a @ (Pgr b)^.
Proof.
lhs napply ap_pV.
f_ap.
1: apply Smash_rec_beta_gluer.
apply inverse2.
apply Smash_rec_beta_gluer.
Defined.
Definition Smash_rec_beta_glue {P : Type} {Psm : X → Y → P} {Pl Pr : P}
(Pgl : ∀ a, Psm a pt = Pl) (Pgr : ∀ b, Psm pt b = Pr) (a : X)
(b : Y) : ap (Smash_rec Psm Pl Pr Pgl Pgr) (glue a b)
= ((Pgl a) @ (Pgl pt)^) @ (Pgr pt @ (Pgr b)^).
Proof.
lhs napply ap_pp.
f_ap.
- apply Smash_rec_beta_gluel'.
- apply Smash_rec_beta_gluer'.
Defined.
End Smash.
Arguments sm : simpl never.
Arguments auxl : simpl never.
Arguments gluel : simpl never.
Arguments gluer : simpl never.
Miscellaneous lemmas about Smash
Definition Smash_ind_FlFr {A B : pType} {P : Type} (f g : Smash A B → P)
(Hsm : ∀ a b, f (sm a b) = g (sm a b))
(Hl : f auxl = g auxl) (Hr : f auxr = g auxr)
(Hgluel : ∀ a, ap f (gluel a) @ Hl = Hsm a pt @ ap g (gluel a))
(Hgluer : ∀ b, ap f (gluer b) @ Hr = Hsm pt b @ ap g (gluer b))
: f == g.
Proof.
snapply (Smash_ind Hsm Hl Hr).
- intros a.
transport_paths FlFr.
exact (Hgluel a).
- intros b.
transport_paths FlFr.
exact (Hgluer b).
Defined.
(Hsm : ∀ a b, f (sm a b) = g (sm a b))
(Hl : f auxl = g auxl) (Hr : f auxr = g auxr)
(Hgluel : ∀ a, ap f (gluel a) @ Hl = Hsm a pt @ ap g (gluel a))
(Hgluer : ∀ b, ap f (gluer b) @ Hr = Hsm pt b @ ap g (gluer b))
: f == g.
Proof.
snapply (Smash_ind Hsm Hl Hr).
- intros a.
transport_paths FlFr.
exact (Hgluel a).
- intros b.
transport_paths FlFr.
exact (Hgluer b).
Defined.
A version of Smash_indj specifically for proving that the composition of two functions is the identity map.
Definition Smash_ind_FFlr {A B : pType} {P : Type}
(f : Smash A B → P) (g : P → Smash A B)
(Hsm : ∀ a b, g (f (sm a b)) = sm a b)
(Hl : g (f auxl) = auxl) (Hr : g (f auxr) = auxr)
(Hgluel : ∀ a, ap g (ap f (gluel a)) @ Hl = Hsm a pt @ gluel a)
(Hgluer : ∀ b, ap g (ap f (gluer b)) @ Hr = Hsm pt b @ gluer b)
: g o f == idmap.
Proof.
snapply (Smash_ind Hsm Hl Hr).
- intros a; cbn beta.
transport_paths FFlr.
exact (Hgluel a).
- intros b; cbn beta.
transport_paths FFlr.
exact (Hgluer b).
Defined.
(f : Smash A B → P) (g : P → Smash A B)
(Hsm : ∀ a b, g (f (sm a b)) = sm a b)
(Hl : g (f auxl) = auxl) (Hr : g (f auxr) = auxr)
(Hgluel : ∀ a, ap g (ap f (gluel a)) @ Hl = Hsm a pt @ gluel a)
(Hgluer : ∀ b, ap g (ap f (gluer b)) @ Hr = Hsm pt b @ gluer b)
: g o f == idmap.
Proof.
snapply (Smash_ind Hsm Hl Hr).
- intros a; cbn beta.
transport_paths FFlr.
exact (Hgluel a).
- intros b; cbn beta.
transport_paths FFlr.
exact (Hgluer b).
Defined.
Definition functor_smash {A B X Y : pType} (f : A $-> X) (g : B $-> Y)
: Smash A B $-> Smash X Y.
Proof.
srapply Build_pMap.
- snapply (Smash_rec (fun a b ⇒ sm (f a) (g b)) auxl auxr).
+ intro a; cbn beta.
rhs_V napply (gluel (f a)).
exact (ap011 _ 1 (point_eq g)).
+ intro b; cbn beta.
rhs_V napply (gluer (g b)).
exact (ap011 _ (point_eq f) 1).
- exact (ap011 _ (point_eq f) (point_eq g)).
Defined.
Definition functor_smash_idmap (X Y : pType)
: functor_smash (@pmap_idmap X) (@pmap_idmap Y) $== pmap_idmap.
Proof.
snapply Build_pHomotopy.
{ snapply Smash_ind_FlFr.
1-3: reflexivity.
- intros x.
apply equiv_p1_1q.
rhs napply ap_idmap.
lhs napply Smash_rec_beta_gluel.
apply concat_1p.
- intros y.
apply equiv_p1_1q.
rhs napply ap_idmap.
lhs napply Smash_rec_beta_gluer.
apply concat_1p. }
reflexivity.
Defined.
Definition functor_smash_compose {X Y A B C D : pType}
(f : X $-> A) (g : Y $-> B) (h : A $-> C) (k : B $-> D)
: functor_smash (h $o f) (k $o g) $== functor_smash h k $o functor_smash f g.
Proof.
pointed_reduce.
snapply Build_pHomotopy.
{ snapply Smash_ind_FlFr.
1-3: reflexivity.
- intros x.
apply equiv_p1_1q.
lhs napply Smash_rec_beta_gluel.
symmetry.
lhs napply (ap_compose (functor_smash _ _) _ (gluel x)).
lhs napply ap.
2: napply Smash_rec_beta_gluel.
lhs napply Smash_rec_beta_gluel.
apply concat_1p.
- intros y.
apply equiv_p1_1q.
lhs napply Smash_rec_beta_gluer.
symmetry.
lhs napply (ap_compose (functor_smash _ _) _ (gluer y)).
lhs napply ap.
2: napply Smash_rec_beta_gluer.
lhs napply Smash_rec_beta_gluer.
apply concat_1p. }
reflexivity.
Defined.
Definition functor_smash_homotopic {X Y A B : pType}
{f h : X $-> A} {g k : Y $-> B}
(p : f $== h) (q : g $== k)
: functor_smash f g $== functor_smash h k.
Proof.
pointed_reduce.
snapply Build_pHomotopy.
{ snapply Smash_ind_FlFr.
1: exact (fun x y ⇒ ap011 _ (p x) (q y)).
1,2: reflexivity.
- intros x.
lhs napply concat_p1.
lhs napply Smash_rec_beta_gluel.
rhs napply whiskerL.
2: napply Smash_rec_beta_gluel.
simpl; induction (p x); simpl.
rhs_V napply concat_pp_p.
apply whiskerR.
napply ap_pp.
- intros y.
lhs napply concat_p1.
lhs napply Smash_rec_beta_gluer.
rhs napply whiskerL.
2: napply Smash_rec_beta_gluer.
simpl; induction (q y); simpl.
rhs_V napply concat_pp_p.
apply whiskerR.
exact (ap011_pp _ _ _ 1 1). }
exact (ap022 _ (concat_p1 (p pt))^ (concat_p1 (q pt))^ @ (concat_p1 _)^).
Defined.
#[export] Instance is0bifunctor_smash : Is0Bifunctor Smash.
Proof.
snapply Build_Is0Bifunctor'.
1,2: exact _.
napply Build_Is0Functor.
intros [X Y] [A B] [f g].
exact (functor_smash f g).
Defined.
#[export] Instance is1bifunctor_smash : Is1Bifunctor Smash.
Proof.
snapply Build_Is1Bifunctor'.
snapply Build_Is1Functor.
- intros [X Y] [A B] [f g] [h i] [p q].
exact (functor_smash_homotopic p q).
- intros [X Y].
exact (functor_smash_idmap X Y).
- intros [X Y] [A B] [C D] [f g] [h i].
exact (functor_smash_compose f g h i).
Defined.
Definition pswap (X Y : pType) : Smash X Y $-> Smash Y X
:= Build_pMap (Smash_rec (flip sm) auxr auxl gluer gluel) 1.
Definition pswap_pswap {X Y : pType}
: pswap X Y $o pswap Y X $== pmap_idmap.
Proof.
snapply Build_pHomotopy.
- snapply Smash_ind_FFlr.
1-3: reflexivity.
+ intros y.
apply equiv_p1_1q.
lhs napply ap.
1: apply Smash_rec_beta_gluel.
napply Smash_rec_beta_gluer.
+ intros x.
apply equiv_p1_1q.
lhs napply ap.
1: apply Smash_rec_beta_gluer.
napply Smash_rec_beta_gluel.
- reflexivity.
Defined.
Definition pequiv_pswap {X Y : pType} : Smash X Y $<~> Smash Y X.
Proof.
snapply cate_adjointify.
1,2: exact (pswap _ _).
1,2: exact pswap_pswap.
Defined.
Definition pswap_natural {A B X Y : pType} (f : A $-> X) (g : B $-> Y)
: pswap X Y $o functor_smash f g $== functor_smash g f $o pswap A B.
Proof.
pointed_reduce.
snapply Build_pHomotopy.
- snapply Smash_ind_FlFr.
1-3: reflexivity.
+ intros a.
apply equiv_p1_1q.
rhs napply (ap_compose (pswap _ _) _ (gluel a)).
rhs napply ap.
2: apply Smash_rec_beta_gluel.
rhs napply Smash_rec_beta_gluer.
lhs napply (ap_compose (functor_smash _ _) (pswap _ _) (gluel a)).
lhs napply ap.
1: apply Smash_rec_beta_gluel.
simpl.
lhs napply ap.
1: apply concat_1p.
rhs napply concat_1p.
napply Smash_rec_beta_gluel.
+ intros b.
apply equiv_p1_1q.
rhs napply (ap_compose (pswap _ _) (functor_smash _ _) (gluer b)).
rhs napply ap.
2: apply Smash_rec_beta_gluer.
rhs napply Smash_rec_beta_gluel.
lhs napply (ap_compose (functor_smash _ _) (pswap _ _) (gluer b)).
lhs napply ap.
1: apply Smash_rec_beta_gluer.
lhs napply ap.
1: apply concat_1p.
rhs napply concat_1p.
napply Smash_rec_beta_gluer.
- reflexivity.
Defined.