Datasets:

phanerozoic

/

Coq-HoTT

like 2

Splitting f := (split_idem_retract ; split_idem_splits).

Definition

Require Import HoTT.Basics HoTT.Types. Require Import HFiber Constant. Require Import Truncations.Core Modalities.Modality. Require Import Homotopy.IdentitySystems.

Idempotents.v

split_idem_split

(x : X) : ap split_idem_sect (split_idem_issect (split_idem_retr x)) = I x. Proof. unfold split_idem_issect, nudge_eq. repeat (rewrite !ap_pp, ?ap_V, !sect_path_split_idem; simpl). apply moveR_Vp, whiskerR; symmetry; apply J. Qed.

Definition

Require Import HoTT.Basics HoTT.Types. Require Import HFiber Constant. Require Import Truncations.Core Modalities.Modality. Require Import Homotopy.IdentitySystems.

Idempotents.v

split_idem_preidem

split_idem_retract' `{fs : Funext} {X : Type} : QuasiIdempotent X -> RetractOf X := fun f => split_idem_retract f.

Definition

Require Import HoTT.Basics HoTT.Types. Require Import HFiber Constant. Require Import Truncations.Core Modalities.Modality. Require Import Homotopy.IdentitySystems.

Idempotents.v

split_idem_retract'

split_idem_split' `{fs : Funext} {X : Type} (f : QuasiIdempotent X) : Splitting f := split_idem_split f.

Definition

Require Import HoTT.Basics HoTT.Types. Require Import HFiber Constant. Require Import Truncations.Core Modalities.Modality. Require Import Homotopy.IdentitySystems.

Idempotents.v

split_idem_split'

(a : split_idem (s o r)) : H (r (s (r (split_idem_sect (s o r) a)))) @ H (r (split_idem_sect (s o r) a)) = ap (r o split_idem_sect (s o r)) (ap (split_idem_retr (s o r)) (1 @ ap (split_idem_sect (s o r)) (split_idem_issect (s o r) a)) @ split_idem_issect (s o r) a). Proof. rewrite ap_pp. rewrite <- ap_compose; simpl. rewrite concat_1p. rewrite <- (ap_compose (split_idem_sect (s o r)) (r o s o r) (split_idem_issect (s o r) a)). rewrite (ap_compose _ (r o s o r) (split_idem_issect (s o r) a)). rewrite (ap_compose _ r (split_idem_issect (s o r) a)). unfold split_idem_issect, nudge_eq; repeat (rewrite !ap_pp, ?ap_V, !sect_path_split_idem; simpl). unfold isidem; fold r s H. rewrite !concat_pp_p. rewrite <- !ap_compose. rewrite <- (ap_compose (s o r) r). rewrite <- (ap_compose (s o r) (r o s o r)). rewrite (concat_p_Vp (ap (r o s o r) (a.2 0))). rewrite_moveL_Vp_p. rewrite (ap_compose (r o s o r) (r o s) (a.2 0)). rewrite (concat_A1p H (ap (r o s o r) (a.2 0))). rewrite (ap_compose r (r o s) (a.2 0)). rewrite (concat_pA1_p H (ap r (a.2 0))). apply whiskerR. refine (cancelR _ _ (H (r (a.1 1%nat))) _). rewrite (concat_pA1_p H (H (r (a 1%nat)))). rewrite !concat_pp_p; symmetry; refine (_ @ concat_pp_p _ _ _). exact (concat_A1p (fun x => H (r (s x)) @ H x) (H (r (a 1%nat)))). Qed.

Lemma

Require Import HoTT.Basics HoTT.Types. Require Import HFiber Constant. Require Import Truncations.Core Modalities.Modality. Require Import Homotopy.IdentitySystems.

Idempotents.v

equiv_split_idem_retract_isadj

split_idem (s o r) <~> A. Proof. simple refine (Build_Equiv _ _ (r o split_idem_sect (s o r)) (Build_IsEquiv _ _ _ (split_idem_retr (s o r) o s) _ _ _)). - intros a; simpl. refine (H _ @ H _). - intros a; simpl. refine (_ @ split_idem_issect (s o r) a). apply ap. refine ((split_idem_splits (s o r) _)^ @ _). apply ap, split_idem_issect; exact _. - intros a; simpl; apply equiv_split_idem_retract_isadj. Defined.

Definition

Require Import HoTT.Basics HoTT.Types. Require Import HFiber Constant. Require Import Truncations.Core Modalities.Modality. Require Import Homotopy.IdentitySystems.

Idempotents.v

equiv_split_idem_retract

(x : X) : equiv_split_idem_retract (split_idem_retr (s o r) x) = r x := H (r x).

Definition

Require Import HoTT.Basics HoTT.Types. Require Import HFiber Constant. Require Import Truncations.Core Modalities.Modality. Require Import Homotopy.IdentitySystems.

Idempotents.v

equiv_split_idem_retract_retr

(a : A) : split_idem_sect (s o r) (equiv_split_idem_retract^-1 a) = s a := ap s (H a).

Definition

Require Import HoTT.Basics HoTT.Types. Require Import HFiber Constant. Require Import Truncations.Core Modalities.Modality. Require Import Homotopy.IdentitySystems.

Idempotents.v

equiv_split_idem_retract_sect

(a : A) : ap equiv_split_idem_retract (split_idem_issect (s o r) (equiv_split_idem_retract^-1 a)) @ eisretr equiv_split_idem_retract a = equiv_split_idem_retract_retr (split_idem_sect (s o r) (equiv_split_idem_retract^-1 a)) @ ap r (equiv_split_idem_retract_sect a) @ H a. Proof. simpl. unfold equiv_split_idem_retract_retr, equiv_split_idem_retract_sect. rewrite ap_compose. unfold split_idem_issect, nudge_eq. repeat (rewrite !ap_pp, ?ap_V, !sect_path_split_idem; simpl). unfold isidem; fold A r s H. Open Scope long_path_scope. rewrite !concat_pp_p; apply moveR_Vp; rewrite !concat_p_pp. do 4 rewrite <- ap_compose. rewrite <- (ap_compose (s o r o s) r (H (r (s a)))). rewrite <- (ap_pp (r o s) _ _). rewrite <- (concat_A1p H (H (r (s a)))). rewrite ap_pp. rewrite <- (ap_compose (r o s) (r o s) _). rewrite !concat_pp_p; apply whiskerL; rewrite !concat_p_pp. rewrite (concat_A1p H (H (r (s a)))). rewrite !concat_pp_p; apply whiskerL. symmetry; refine (concat_A1p H (H a)). Close Scope long_path_scope. Qed.

Definition

Require Import HoTT.Basics HoTT.Types. Require Import HFiber Constant. Require Import Truncations.Core Modalities.Modality. Require Import Homotopy.IdentitySystems.

Idempotents.v

equiv_split_idem_retract_issect

(x : X) : split_idem_splits (s o r) x = ap (split_idem_sect (s o r)) (eissect equiv_split_idem_retract (split_idem_retr (s o r) x))^ @ equiv_split_idem_retract_sect (equiv_split_idem_retract (split_idem_retr (s o r) x)) @ ap s (equiv_split_idem_retract_retr x). Proof. simpl. unfold equiv_split_idem_retract_retr, equiv_split_idem_retract_sect, split_idem_splits. rewrite concat_1p, concat_pp_p, ap_V; apply moveL_Vp; rewrite concat_p1. unfold split_idem_issect, nudge_eq. repeat (rewrite !ap_pp, ?ap_V, !sect_path_split_idem; simpl). unfold isidem; fold A r s H. Open Scope long_path_scope. rewrite !concat_p_pp. rewrite <- !ap_compose; simpl. apply whiskerR. refine (_ @ (concat_1p _)); apply whiskerR. apply moveR_pV; rewrite concat_1p, concat_pp_p; apply moveR_Vp. rewrite <- (ap_compose (s o r o s) (s o r)). rewrite (ap_compose (r o s) s _). rewrite (ap_compose (r o s) s _). rewrite (ap_compose (r o s o r o s) s _). rewrite <- !ap_pp; apply ap. refine (cancelR _ _ (H (r x)) _). rewrite (concat_pA1_p H (H (r x)) _). rewrite (concat_pA1_p H (H (r x)) _). refine ((concat_A1p H (H (r (s (r x)))) @@ 1) @ _). rewrite (ap_compose (r o s) (r o s) _). rewrite (concat_A1p H (ap (r o s) (H (r x)))). rewrite !concat_pp_p; apply whiskerL. symmetry; refine (concat_A1p H (H (r x))). Close Scope long_path_scope. Qed.

Definition

Require Import HoTT.Basics HoTT.Types. Require Import HFiber Constant. Require Import Truncations.Core Modalities.Modality. Require Import Homotopy.IdentitySystems.

Idempotents.v

equiv_split_idem_retract_splits

RetractOf (QuasiIdempotent X). Proof. refine (Build_RetractOf (QuasiIdempotent X) (RetractOf X) split_idem_retract' qidem_retract _). intros R. exact (@path_retractof _ _ (split_idem_retract' (qidem_retract R)) R (equiv_split_idem_retract R) (equiv_split_idem_retract_retr R) (equiv_split_idem_retract_sect R) (equiv_split_idem_retract_issect R)). Defined.

Definition

Require Import HoTT.Basics HoTT.Types. Require Import HFiber Constant. Require Import Truncations.Core Modalities.Modality. Require Import Homotopy.IdentitySystems.

Idempotents.v

retract_retractof_qidem

(f : X -> X) : RetractOf { I : IsPreIdempotent f & IsQuasiIdempotent f }. Proof. simple refine (@equiv_retractof' _ (@retractof_equiv' (hfiber quasiidempotent_pr1 f) _ _ (retractof_hfiber retract_retractof_qidem quasiidempotent_pr1 retract_idem (fun _ => 1) f)) (Splitting f) _). - refine ((hfiber_fibration f (fun g => { I : IsPreIdempotent g & @IsQuasiIdempotent _ g I }))^-1 oE _). unfold hfiber. refine (equiv_functor_sigma' (equiv_sigma_assoc _ _)^-1 (fun a => _)); simpl. destruct a as [[g I] J]; unfold quasiidempotent_pr1; simpl. apply equiv_idmap. - simpl. unfold hfiber, Splitting. refine (equiv_functor_sigma_id _); intros R; simpl. apply equiv_ap10. Defined.

Definition

Require Import HoTT.Basics HoTT.Types. Require Import HFiber Constant. Require Import Truncations.Core Modalities.Modality. Require Import Homotopy.IdentitySystems.

Idempotents.v

splitting_retractof_isqidem

(f : PreIdempotent X) := { S : Splitting f & forall x, ap f (S.

Definition

Require Import HoTT.Basics HoTT.Types. Require Import HFiber Constant. Require Import Truncations.Core Modalities.Modality. Require Import Homotopy.IdentitySystems.

Idempotents.v

Splitting_PreIdempotent

(f : PreIdempotent X) : RetractOf (IsQuasiIdempotent f). Proof. simple refine (@equiv_retractof' _ (@retractof_equiv' (hfiber (@pr1 _ (fun fi => @IsQuasiIdempotent _ fi.1 fi.2)) f) _ _ (retractof_hfiber retract_retractof_qidem pr1 preidem_retract _ f)) (Splitting_PreIdempotent f) _). - symmetry; refine (hfiber_fibration f _). - intros [[g I] J]; simpl. refine (path_sigma' _ 1 _); simpl. apply path_forall; intros x; apply split_idem_preidem. - simpl; unfold hfiber, Splitting. refine (equiv_sigma_assoc _ _ oE _). apply equiv_functor_sigma_id; intros R; simpl. refine (_ oE (equiv_path_sigma _ _ _)^-1); simpl. refine (equiv_functor_sigma' (equiv_ap10 _ _) _); intros H; simpl. destruct f as [f I]; simpl in *. destruct H; simpl. refine (_ oE (equiv_path_forall _ _)^-1); unfold pointwise_paths. apply equiv_functor_forall_id; intros x; simpl. unfold isidem. apply equiv_concat_l. refine (concat_p1 _ @ concat_1p _). Defined.

Definition

Require Import HoTT.Basics HoTT.Types. Require Import HFiber Constant. Require Import Truncations.Core Modalities.Modality. Require Import Homotopy.IdentitySystems.

Idempotents.v

splitting_preidem_retractof_qidem

{X : Type} (f : X -> X) : Splitting f -> IsIdempotent f := (equiv_split_idem_retract (splitting_retractof_isqidem f))^-1.

Definition

Require Import HoTT.Basics HoTT.Types. Require Import HFiber Constant. Require Import Truncations.Core Modalities.Modality. Require Import Homotopy.IdentitySystems.

Idempotents.v

Build_IsIdempotent

{X : Type} (f : X -> X) `{IsQuasiIdempotent _ f} : IsIdempotent f := Build_IsIdempotent f (split_idem_split f). Global Instance ispreidem_isidem {X : Type} (f : X -> X) `{IsIdempotent _ f} : IsPreIdempotent f. Proof. refine (split_idem_sect (retract_idem (splitting_retractof_isqidem f)) _).1. assumption. Defined.

Definition

Require Import HoTT.Basics HoTT.Types. Require Import HFiber Constant. Require Import Truncations.Core Modalities.Modality. Require Import Homotopy.IdentitySystems.

Idempotents.v

isidem_isqidem

(X : Type) := { f : X -> X & IsIdempotent f }.

Definition

Require Import HoTT.Basics HoTT.Types. Require Import HFiber Constant. Require Import Truncations.Core Modalities.Modality. Require Import Homotopy.IdentitySystems.

Idempotents.v

Idempotent

{X : Type} : Idempotent X -> (X -> X) := pr1.

Definition

Require Import HoTT.Basics HoTT.Types. Require Import HFiber Constant. Require Import Truncations.Core Modalities.Modality. Require Import Homotopy.IdentitySystems.

Idempotents.v

idempotent_pr1

(X : Type) : Idempotent X <~> RetractOf X. Proof. transitivity ({ f : X -> X & Splitting f }). - unfold Idempotent. refine (equiv_functor_sigma' (equiv_idmap _) _); intros f; simpl. refine (equiv_split_idem_retract (splitting_retractof_isqidem f)). - unfold Splitting. refine (_ oE equiv_sigma_symm _). apply equiv_sigma_contr; intros R. apply contr_basedhomotopy. Defined.

Definition

Require Import HoTT.Basics HoTT.Types. Require Import HFiber Constant. Require Import Truncations.Core Modalities.Modality. Require Import Homotopy.IdentitySystems.

Idempotents.v

equiv_idempotent_retractof

(X : Type@{i}) : Idempotent@{i i j} X := (idmap ; isidem_idmap X).

Definition

Require Import HoTT.Basics HoTT.Types. Require Import HFiber Constant. Require Import Truncations.Core Modalities.Modality. Require Import Homotopy.IdentitySystems.

Idempotents.v

idem_idmap

{ua : Univalence} (X : Type) : Contr (Splitting_PreIdempotent (preidem_idmap X)). Proof. refine (contr_equiv' {Y : Type & X <~> Y} _). transitivity { S : Splitting (preidem_idmap X) & forall x : X, (retract_issect S.1) (retract_retr S.1 x) = ap (retract_retr S.1) (S.2 x) }. 1:make_equiv. apply equiv_functor_sigma_id; intros [[Y r s eta] ep]; cbn in *. apply equiv_functor_forall_id; intros x. unfold ispreidem_idmap; simpl. rewrite ap_idmap, !concat_pp_p. refine (equiv_moveR_Vp _ _ _ oE _). rewrite concat_p1, concat_p_pp. refine (equiv_concat_r (concat_1p _) _ oE _). refine (equiv_whiskerR _ _ _ oE _). refine (equiv_moveR_Vp _ _ _ oE _). rewrite concat_p1. pose (isequiv_adjointify s r ep eta). refine (_ oE equiv_ap (ap s) _ _). apply equiv_concat_r. refine (cancelR _ _ (ep x) _). rewrite <- ap_compose. refine (concat_A1p ep (ep x)). Qed.

Definition

Require Import HoTT.Basics HoTT.Types. Require Import HFiber Constant. Require Import Truncations.Core Modalities.Modality. Require Import Homotopy.IdentitySystems.

Idempotents.v

contr_splitting_preidem_idmap

{X Y : Type} (f : X -> Y) := {y : Y & forall x:X, f x = y}.

Definition

Require Import HoTT.Basics. Require Import Types.Sigma.

NullHomotopy.v

NullHomotopy

{n : trunc_index} {X Y : Type} (f : X -> Y) `{IsTrunc n Y} : IsTrunc n (NullHomotopy f). Proof. apply @istrunc_sigma; auto. intros y. apply (@istrunc_forall _). intros x. apply istrunc_paths'. Defined.

Lemma

Require Import HoTT.Basics. Require Import Types.Sigma.

NullHomotopy.v

istrunc_nullhomotopy

{X Y : Type} {f g : X -> Y} (p : f == g) : NullHomotopy f -> NullHomotopy g. Proof. intros [y e]. exists y. intros x; exact ((p x)^ @ e x). Defined.

Definition

Require Import HoTT.Basics. Require Import Types.Sigma.

NullHomotopy.v

nullhomotopy_homotopic

{X Y Z : Type} (f : X -> Y) (g : Y -> Z) : NullHomotopy g -> NullHomotopy (g o f). Proof. intros [z e]. exists z. intros x; exact (e (f x)). Defined.

Definition

Require Import HoTT.Basics. Require Import Types.Sigma.

NullHomotopy.v

nullhomotopy_composeR

{X Y Z : Type} (f : X -> Y) (g : Y -> Z) : NullHomotopy f -> NullHomotopy (g o f). Proof. intros [y e]. exists (g y). intros x; exact (ap g (e x)). Defined.

Definition

Require Import HoTT.Basics. Require Import Types.Sigma.

NullHomotopy.v

nullhomotopy_composeL

{X Y Z : Type} (f : X -> Y) (g : Y -> Z) `{IsEquiv _ _ g} : NullHomotopy (g o f) -> NullHomotopy f. Proof. intros [z e]. exists (g^-1 z). intros x; apply moveL_equiv_V, e. Defined.

Definition

Require Import HoTT.Basics. Require Import Types.Sigma.

NullHomotopy.v

cancelL_nullhomotopy_equiv

{X Y Z : Type} (f : X -> Y) (g : Y -> Z) `{IsEquiv _ _ f} : NullHomotopy (g o f) -> NullHomotopy g. Proof. intros [z e]. exists z. intros y; transitivity (g (f (f^-1 y))). - symmetry; apply ap, eisretr. - apply e. Defined.

Definition

Require Import HoTT.Basics. Require Import Types.Sigma.

NullHomotopy.v

cancelR_nullhomotopy_equiv

{X Y : Type} (f : X -> Y) (x1 x2 : X) : NullHomotopy f -> NullHomotopy (@ap _ _ f x1 x2). Proof. intros [y e]. unshelve eexists. - exact (e x1 @ (e x2)^). - intros p. apply moveL_pV. refine (concat_Ap e p @ _). refine (_ @ concat_p1 _); apply ap. apply ap_const. Defined.

Definition

Require Import HoTT.Basics. Require Import Types.Sigma.

NullHomotopy.v

nullhomotopy_ap

(O : Modality) (X : Type) : Type := forall A, In O A -> forall B, In O B -> forall f : X -> B, forall p : A -> B, IsSurjection p -> merely (exists s : X -> A, p o s == f).

Definition

Require Import Basics Types. Require Import Truncations.Core Truncations.SeparatedTrunc. Require Import Modalities.Modality Modalities.Identity. Require Import Limits.Pullback.

Projective.v

IsOProjective

(O : Modality) (X : Type) `{In O X} : IsOProjective O X <-> (forall (Y : Type), In O Y -> forall (p : Y -> X), IsSurjection p -> merely (exists s : X -> Y, p o s == idmap)). Proof. split. - intros isprojX Y oY p S; unfold IsOProjective in isprojX. exact (isprojX Y _ X _ idmap p S). - intro splits. unfold IsOProjective. intros A oA B oB f p S. pose proof (splits (Pullback p f) _ pullback_pr2 _) as s'. strip_truncations. destruct s' as [s E]. refine (tr (pullback_pr1 o s; _)). intro x. refine (pullback_commsq p f (s x) @ _). apply (ap f). apply E. Defined.

Proposition

Require Import Basics Types. Require Import Truncations.Core Truncations.SeparatedTrunc. Require Import Modalities.Modality Modalities.Identity. Require Import Limits.Pullback.

Projective.v

iff_isoprojective_surjections_split

`{Funext} (O : Modality) (X : Type) `{In O X} : IsOProjective O X <~> (forall (Y : Type), In O Y -> forall (p : Y -> X), IsSurjection p -> merely (exists s : X -> Y, p o s == idmap)). Proof. exact (equiv_iff_hprop_uncurried (iff_isoprojective_surjections_split O X)). Defined.

Corollary

Require Import Basics Types. Require Import Truncations.Core Truncations.SeparatedTrunc. Require Import Modalities.Modality Modalities.Identity. Require Import Limits.Pullback.

Projective.v

equiv_isoprojective_surjections_split

(O : Modality) (X : Type) : HasOChoice O X -> IsOProjective O X. Proof. intros chX A ? B ? f p S. assert (g : merely (forall x:X, hfiber p (f x))). - rapply chX. intro x. exact (center _). - strip_truncations; apply tr. exists (fun x:X => pr1 (g x)). intro x. exact (g x).2. Defined.

Proposition

Require Import Basics Types. Require Import Truncations.Core Truncations.SeparatedTrunc. Require Import Modalities.Modality Modalities.Identity. Require Import Limits.Pullback.

Projective.v

isoprojective_ochoice

(O : Modality) (X : Type) `{In O X} : IsOProjective O X -> HasOChoice O X. Proof. refine (_ o fst (iff_isoprojective_surjections_split O X)). intros splits P oP S. specialize splits with {x : X & P x} pr1. pose proof (splits _ (fst (iff_merely_issurjection P) S)) as M. clear S splits. strip_truncations; apply tr. destruct M as [s p]. intro x. exact (transport _ (p x) (s x).2). Defined.

Proposition

Require Import Basics Types. Require Import Truncations.Core Truncations.SeparatedTrunc. Require Import Modalities.Modality Modalities.Identity. Require Import Limits.Pullback.

Projective.v

hasochoice_oprojective

(O : Modality) (X : Type) `{In O X} : IsOProjective O X <-> HasOChoice O X. Proof. split. - apply hasochoice_oprojective. exact _. - apply isoprojective_ochoice. Defined.

Proposition

Require Import Basics Types. Require Import Truncations.Core Truncations.SeparatedTrunc. Require Import Modalities.Modality Modalities.Identity. Require Import Limits.Pullback.

Projective.v

iff_isoprojective_hasochoice

`{Funext} (O : Modality) (X : Type) `{In O X} : IsOProjective O X <~> HasOChoice O X. Proof. refine (equiv_iff_hprop_uncurried (iff_isoprojective_hasochoice O X)). apply istrunc_forall. Defined.

Proposition

Require Import Basics Types. Require Import Truncations.Core Truncations.SeparatedTrunc. Require Import Modalities.Modality Modalities.Identity. Require Import Limits.Pullback.

Projective.v

equiv_isoprojective_hasochoice

IsProjective Unit. Proof. apply (isoprojective_ochoice purely Unit). intros P trP S. specialize S with tt. strip_truncations; apply tr. apply Unit_ind. exact S. Defined.

Proposition

Require Import Basics Types. Require Import Truncations.Core Truncations.SeparatedTrunc. Require Import Modalities.Modality Modalities.Identity. Require Import Limits.Pullback.

Projective.v

isprojective_unit

`{Univalence} (A : Type) : merely (exists X:HSet, exists p : X -> A, IsSurjection p). Proof. pose (X := Build_HSet (Tr 0 A)). pose proof ((equiv_isoprojective_hasochoice _ X)^-1 (AC X)) as P. pose proof (P A _ X _ idmap tr _) as F; clear P. strip_truncations. destruct F as [f p]. refine (tr (X; (f; BuildIsSurjection f _))). intro a; unfold hfiber. apply equiv_O_sigma_O. refine (tr (tr a; _)). rapply (equiv_path_Tr _ _)^-1%equiv. apply p. Defined.

Proposition

Require Import Basics Types. Require Import Truncations.Core Truncations.SeparatedTrunc. Require Import Modalities.Modality Modalities.Identity. Require Import Limits.Pullback.

Projective.v

projective_cover_AC

`{Univalence} (X : Type) : IsProjective X <~> IsHSet X. Proof. apply equiv_iff_hprop. - intro isprojX. unfold IsOProjective in isprojX. pose proof (projective_cover_AC X) as P; strip_truncations. destruct P as [P [p issurj_p]]. pose proof (isprojX P _ X _ idmap p issurj_p) as S; strip_truncations. exact (inO_retract_inO (Tr 0) X P S.1 p S.2). - intro ishsetX. apply (equiv_isoprojective_hasochoice purely X)^-1. rapply AC. Defined.

Theorem

Require Import Basics Types. Require Import Truncations.Core Truncations.SeparatedTrunc. Require Import Modalities.Modality Modalities.Identity. Require Import Limits.Pullback.

Projective.v

equiv_isprojective_ishset_AC

`{Funext} A B x P f g e Px : @transport (forall a : A, B a) (fun f => P (f x)) f g (@path_forall _ _ _ _ _ e) Px = @transport (B x) P (f x) (g x) (e x) Px. Proof. path_forall_beta_t. Defined.

Lemma

Require Import Basics.Overture Basics.Tactics Basics.PathGroupoids Basics.Contractible Basics.Equivalences. Require Import Types.Prod Types.Forall.

Tactics.v

path_forall_1_beta

path_forall_recr_beta' `{Funext} A B x0 P f g e Px : @transport (forall a : A, B a) (fun f => P f (f x0)) f g (@path_forall _ _ _ _ _ e) Px = @transport ((forall a, B a) * B x0) (fun x => P (fst x) (snd x)) (f, f x0) (g, g x0) (path_prod' (@path_forall _ _ _ _ _ e) (e x0)) Px. Proof. path_forall_beta_t. Defined.

Lemma

Require Import Basics.Overture Basics.Tactics Basics.PathGroupoids Basics.Contractible Basics.Equivalences. Require Import Types.Prod Types.Forall.

Tactics.v

path_forall_recr_beta'

`{Funext} A B x0 P f g e Px : @transport (forall a : A, B a) (fun f => P f (f x0)) f g (@path_forall _ _ _ _ _ e) Px = @transport (forall x : A, B x) (fun x => P x (g x0)) f g (@path_forall H A B f g e) (@transport (B x0) (fun y => P f y) (f x0) (g x0) (e x0) Px). Proof. etransitivity. - apply '. - refine (transport_path_prod' _ _ _ _). Defined.

Lemma

Require Import Basics.Overture Basics.Tactics Basics.PathGroupoids Basics.Contractible Basics.Equivalences. Require Import Types.Prod Types.Forall.

Tactics.v

path_forall_recr_beta

path_forall_2_beta' `{Funext} A B x0 x1 P f g e Px : @transport (forall a : A, B a) (fun f => P (f x0) (f x1)) f g (@path_forall _ _ _ _ _ e) Px = @transport (B x0 * B x1) (fun x => P (fst x) (snd x)) (f x0, f x1) (g x0, g x1) (path_prod' (e x0) (e x1)) Px. Proof. transport_path_forall_hammer. repeat match goal with | [ |- context[e ?x] ] => induction (e x) end; cbn. reflexivity. Qed.

Lemma

Require Import Basics.Overture Basics.Tactics Basics.PathGroupoids Basics.Contractible Basics.Equivalences. Require Import Types.Prod Types.Forall.

Tactics.v

path_forall_2_beta'

`{Funext} A B x0 x1 P f g e Px : @transport (forall a : A, B a) (fun f => P (f x0) (f x1)) f g (@path_forall _ _ _ _ _ e) Px = transport (fun y : B x1 => P (g x0) y) (e x1) (transport (fun y : B x0 => P y (f x1)) (e x0) Px). Proof. transport_path_forall_hammer. reflexivity. Qed.

Lemma

Require Import Basics.Overture Basics.Tactics Basics.PathGroupoids Basics.Contractible Basics.Equivalences. Require Import Types.Prod Types.Forall.

Tactics.v

path_forall_2_beta

{T} {x y : T} (H0 : x = y) : (H0 = match H0 in (_ = y) return (x = y) with | idpath => idpath end) := match H0 with idpath => idpath end.

Definition

Require Import Basics.Overture Basics.Tactics Basics.PathGroupoids Basics.Contractible Basics.Equivalences. Require Import Types.Prod Types.Forall.

Tactics.v

match_eta

{T} {x : T} (E : x = x) : (match E in (_ = y) return (x = y) with | idpath => idpath end = idpath) -> idpath = E := fun H => ((H # match_eta E) ^)%path.

Definition

Require Import Basics.Overture Basics.Tactics Basics.PathGroupoids Basics.Contractible Basics.Equivalences. Require Import Types.Prod Types.Forall.

Tactics.v

match_eta1

{T} {x : T} (E : x = x) : (idpath = match E in (_ = y) return (x = y) with | idpath => idpath end) -> idpath = E := fun H => match_eta1 E (H ^)%path.

Definition

Require Import Basics.Overture Basics.Tactics Basics.PathGroupoids Basics.Contractible Basics.Equivalences. Require Import Types.Prod Types.Forall.

Tactics.v

match_eta2

{A : Type} {x y : A} (P : A -> Type) (u : P y) (p : x = y) : internal_paths_rew_r P u p = transport P p^ u. Proof. destruct p; reflexivity. Defined.

Definition

Require Import Basics.Overture Basics.Tactics Basics.PathGroupoids Basics.Contractible Basics.Equivalences. Require Import Types.Prod Types.Forall.

Tactics.v

internal_paths_rew_r_to_transport

(X : Type) : ooGroup := Build_ooGroup [BAut X, _] _.

Definition

Require Import Basics. Require Import Truncations. Require Import Algebra.ooGroup. Require Import Universes.BAut. Require Import Pointed.Core.

Algebra\Aut.v

Aut

(G : ooGroup) := classifying_space G -> Type.

Definition

Require Import Basics. Require Import Algebra.ooGroup.

Algebra\ooAction.v

ooAction

{G} : ooAction G -> Type := fun X => X (point _).

Definition

Require Import Basics. Require Import Algebra.ooGroup.

Algebra\ooAction.v

action_space

Record

Require Import Basics Types. Require Import Pointed. Require Import Truncations.Core Truncations.Connectedness. Require Import Homotopy.ClassifyingSpace. Require Import Algebra.Groups. Require Import WildCat.

Algebra\ooGroup.v

ooGroup

(G : ooGroup) : Type := point (B G) = point (B G).

Definition

Require Import Basics Types. Require Import Pointed. Require Import Truncations.Core Truncations.Connectedness. Require Import Homotopy.ClassifyingSpace. Require Import Algebra.Groups. Require Import WildCat.

Algebra\ooGroup.v

group_type

(X : pType) : ooGroup. Proof. pose (BG := [{ x:X & merely (x = pt) }, exist (fun x:X => merely (x = pt)) pt (tr 1)]). cut (IsConnected 0 BG). { exact (Build_ooGroup BG). } cut (IsSurjection (unit_name (point BG))). { intros; refine (conn_pointed_type pt). } apply BuildIsSurjection; simpl; intros [x p]. strip_truncations; apply tr; exists tt. apply path_sigma_hprop; simpl. exact (p^). Defined.

Definition

Require Import Basics Types. Require Import Pointed. Require Import Truncations.Core Truncations.Connectedness. Require Import Homotopy.ClassifyingSpace. Require Import Algebra.Groups. Require Import WildCat.

Algebra\ooGroup.v

group_loops

(X : pType) : loops X <~> group_loops X. Proof. unfold loops, group_type. simpl. exact (equiv_path_sigma_hprop (point X ; tr 1) (point X ; tr 1)). Defined.

Definition

Require Import Basics Types. Require Import Pointed. Require Import Truncations.Core Truncations.Connectedness. Require Import Homotopy.ClassifyingSpace. Require Import Algebra.Groups. Require Import WildCat.

Algebra\ooGroup.v

loops_group

(G H : ooGroup) := B G ->* B H.

Definition

Require Import Basics Types. Require Import Pointed. Require Import Truncations.Core Truncations.Connectedness. Require Import Homotopy.ClassifyingSpace. Require Import Algebra.Groups. Require Import WildCat.

Algebra\ooGroup.v

ooGroupHom

{G H} (phi : ooGroupHom G H) : G -> H := fmap loops phi.

Definition

Require Import Basics Types. Require Import Pointed. Require Import Truncations.Core Truncations.Connectedness. Require Import Homotopy.ClassifyingSpace. Require Import Algebra.Groups. Require Import WildCat.

Algebra\ooGroup.v

grouphom_fun

{X Y : pType} (f : X ->* Y) : ooGroupHom (group_loops X) (group_loops Y). Proof. simple refine (Build_pMap _ _ _ _); simpl. - intros [x p]. exists (f x). strip_truncations; apply tr. exact (ap f p @ point_eq f). - apply path_sigma_hprop; simpl. apply point_eq. Defined.

Definition

Require Import Basics Types. Require Import Pointed. Require Import Truncations.Core Truncations.Connectedness. Require Import Homotopy.ClassifyingSpace. Require Import Algebra.Groups. Require Import WildCat.

Algebra\ooGroup.v

group_loops_functor

{X Y : pType} (f : X ->* Y) : fmap loops (group_loops_functor f) o loops_group X == loops_group Y o fmap loops f. Proof. intros x. apply (equiv_inj (equiv_path_sigma_hprop _ _)^-1). simpl. unfold pr1_path; rewrite !ap_pp. rewrite ap_V, !ap_pr1_path_sigma_hprop. apply whiskerL, whiskerR. transitivity (ap (fun X0 : {x0 : X & merely (x0 = point X)} => f X0.1) (path_sigma_hprop (point X; tr 1) (point X; tr 1) x)). - match goal with |- ap ?f (ap ?g ?p) = ?z => symmetry; refine (ap_compose g f p) end. - rewrite ap_compose; apply ap. apply ap_pr1_path_sigma_hprop. Qed.

Definition

Require Import Basics Types. Require Import Pointed. Require Import Truncations.Core Truncations.Connectedness. Require Import Homotopy.ClassifyingSpace. Require Import Algebra.Groups. Require Import WildCat.

Algebra\ooGroup.v

loops_functor_group

{G H K : ooGroup} (psi : ooGroupHom H K) (phi : ooGroupHom G H) : ooGroupHom G K := pmap_compose psi phi.

Definition

Require Import Basics Types. Require Import Pointed. Require Import Truncations.Core Truncations.Connectedness. Require Import Homotopy.ClassifyingSpace. Require Import Algebra.Groups. Require Import WildCat.

Algebra\ooGroup.v

grouphom_compose

{X Y Z : pType} (psi : Y ->* Z) (phi : X ->* Y) : grouphom_compose (group_loops_functor psi) (group_loops_functor phi) == group_loops_functor (pmap_compose psi phi). Proof. intros g. unfold grouphom_fun, grouphom_compose. refine (pointed_htpy (fmap_comp loops _ _) g @ _). pose (p := eisretr (loops_group X) g). change (fmap loops (group_loops_functor psi) (fmap loops (group_loops_functor phi) g) = fmap loops (group_loops_functor (pmap_compose psi phi)) g). rewrite <- p. rewrite !loops_functor_group. apply ap. symmetry; rapply (fmap_comp loops). Qed.

Definition

Require Import Basics Types. Require Import Pointed. Require Import Truncations.Core Truncations.Connectedness. Require Import Homotopy.ClassifyingSpace. Require Import Algebra.Groups. Require Import WildCat.

Algebra\ooGroup.v

group_loops_functor_compose

(G : ooGroup) : ooGroupHom G G := pmap_idmap.

Definition

Require Import Basics Types. Require Import Pointed. Require Import Truncations.Core Truncations.Connectedness. Require Import Homotopy.ClassifyingSpace. Require Import Algebra.Groups. Require Import WildCat.

Algebra\ooGroup.v

grouphom_idmap

{X : pType} : grouphom_idmap (group_loops X) == group_loops_functor (Id (A:=pType) _). Proof. intros g. refine (fmap_id loops _ g @ _). rewrite <- (eisretr (loops_group X) g). unfold grouphom_fun, grouphom_idmap. rewrite !loops_functor_group. exact (ap (loops_group X) (fmap_id loops _ _)^). Qed.

Definition

Require Import Basics Types. Require Import Pointed. Require Import Truncations.Core Truncations.Connectedness. Require Import Homotopy.ClassifyingSpace. Require Import Algebra.Groups. Require Import WildCat.

Algebra\ooGroup.v

group_loops_functor_idmap

{G H K : ooGroup} (psi : ooGroupHom H K) (phi : ooGroupHom G H) : grouphom_compose psi phi == psi o phi. Proof. intros g; grouphom_reduce. exact (ap_compose phi psi g). Qed.

Definition

Require Import Basics Types. Require Import Pointed. Require Import Truncations.Core Truncations.Connectedness. Require Import Homotopy.ClassifyingSpace. Require Import Algebra.Groups. Require Import WildCat.

Algebra\ooGroup.v

compose_grouphom

(G : ooGroup) : grouphom_idmap G == idmap. Proof. intros g; grouphom_reduce. exact (ap_idmap g). Qed.

Definition

Require Import Basics Types. Require Import Pointed. Require Import Truncations.Core Truncations.Connectedness. Require Import Homotopy.ClassifyingSpace. Require Import Algebra.Groups. Require Import WildCat.

Algebra\ooGroup.v

idmap_grouphom

{G H} (phi : ooGroupHom G H) (g1 g2 : G) : phi (g1 @ g2) = phi g1 @ phi g2. Proof. grouphom_reduce. exact (ap_pp phi g1 g2). Qed.

Definition

Require Import Basics Types. Require Import Pointed. Require Import Truncations.Core Truncations.Connectedness. Require Import Homotopy.ClassifyingSpace. Require Import Algebra.Groups. Require Import WildCat.

Algebra\ooGroup.v

grouphom_pp

{G H} (phi : ooGroupHom G H) (g : G) : phi g^ = (phi g)^. Proof. grouphom_reduce. exact (ap_V phi g). Qed.

Definition

Require Import Basics Types. Require Import Pointed. Require Import Truncations.Core Truncations.Connectedness. Require Import Homotopy.ClassifyingSpace. Require Import Algebra.Groups. Require Import WildCat.

Algebra\ooGroup.v

grouphom_V

{G H} (phi : ooGroupHom G H) : phi 1 = 1. Proof. grouphom_reduce. reflexivity. Qed.

Definition

Require Import Basics Types. Require Import Pointed. Require Import Truncations.Core Truncations.Connectedness. Require Import Homotopy.ClassifyingSpace. Require Import Algebra.Groups. Require Import WildCat.

Algebra\ooGroup.v

grouphom_1

{G H} (phi : ooGroupHom G H) (g1 g2 g3 : G) : grouphom_pp phi (g1 @ g2) g3 @ whiskerR (grouphom_pp phi g1 g2) (phi g3) @ concat_pp_p (phi g1) (phi g2) (phi g3) = ap phi (concat_pp_p g1 g2 g3) @ grouphom_pp phi g1 (g2 @ g3) @ whiskerL (phi g1) (grouphom_pp phi g2 g3). Proof. grouphom_reduce. Abort. Section Subgroups. Context {G H : ooGroup} (incl : ooGroupHom H G) `{IsEmbedding incl}.

Definition

Require Import Basics Types. Require Import Pointed. Require Import Truncations.Core Truncations.Connectedness. Require Import Homotopy.ClassifyingSpace. Require Import Algebra.Groups. Require Import WildCat.

Algebra\ooGroup.v

grouphom_pp_p

G -> G -> Type := fun g1 g2 => hfiber incl (g1 @ g2^). Global Instance ishprop_in_coset : is_mere_relation G . Proof. exact _. Defined.

Definition

Require Import Basics Types. Require Import Pointed. Require Import Truncations.Core Truncations.Connectedness. Require Import Homotopy.ClassifyingSpace. Require Import Algebra.Groups. Require Import WildCat.

Algebra\ooGroup.v

in_coset

(g : G) : { g' : G & in_coset g g'} <~> H. Proof. simple refine (equiv_adjointify _ _ _ _). - intros [? [h ?]]; exact h. - intros h; exists (incl h^ @ g); exists h; simpl. abstract (rewrite inv_pp, grouphom_V, inv_V, concat_p_Vp; reflexivity). - intros h; reflexivity. - intros [g' [h p]]. apply path_sigma_hprop; simpl. refine ((grouphom_V incl h @@ 1) @ _). apply moveR_Vp, moveL_pM. exact (p^). Defined.

Definition

Require Import Basics Types. Require Import Pointed. Require Import Truncations.Core Truncations.Connectedness. Require Import Homotopy.ClassifyingSpace. Require Import Algebra.Groups. Require Import WildCat.

Algebra\ooGroup.v

equiv_coset_subgroup

Quotient in_coset.

Definition

Require Import Basics Types. Require Import Pointed. Require Import Truncations.Core Truncations.Connectedness. Require Import Homotopy.ClassifyingSpace. Require Import Algebra.Groups. Require Import WildCat.

Algebra\ooGroup.v

cosets

Group -> ooGroup := fun G => Build_ooGroup (pClassifyingSpace G) _. Global Instance is0functor_group_to_oogroup : Is0Functor . Proof. snrapply Build_Is0Functor. intros G H f. by rapply (fmap pClassifyingSpace). Defined.

Definition

Require Import Basics Types. Require Import Pointed. Require Import Truncations.Core Truncations.Connectedness. Require Import Homotopy.ClassifyingSpace. Require Import Algebra.Groups. Require Import WildCat.

Algebra\ooGroup.v

group_to_oogroup

{ abgroup_group : Group; abgroup_commutative : Commutative (@group_sgop abgroup_group); }.

Record

Require Import Basics Types. Require Import Spaces.Nat.Core Spaces.Int. Require Import Algebra.Groups.QuotientGroup. Require Import WildCat.

Algebra\AbGroups\AbelianGroup.v

AbGroup

Build_AbGroup' (G : Type) `{Zero G, Negate G, Plus G, IsHSet G} (comm : Commutative (A:=G) (+)) (assoc : Associative (A:=G) (+)) (unit_l : LeftIdentity (A:=G) (+) 0) (inv_l : LeftInverse (A:=G) (+) (-) 0) : AbGroup. Proof. snrapply Build_AbGroup. - rapply (Build_Group G). repeat split; only 1-3, 5: exact _. + intros x. lhs nrapply comm. exact (unit_l x). + intros x. lhs nrapply comm. exact (inv_l x). - exact comm. Defined.

Definition

Require Import Basics Types. Require Import Spaces.Nat.Core Spaces.Int. Require Import Algebra.Groups.QuotientGroup. Require Import WildCat.

Algebra\AbGroups\AbelianGroup.v

Build_AbGroup'

_ <~> AbGroup := ltac:(issig).

Definition

Require Import Basics Types. Require Import Spaces.Nat.Core Spaces.Int. Require Import Algebra.Groups.QuotientGroup. Require Import WildCat.

Algebra\AbGroups\AbelianGroup.v

issig_abgroup

{A : AbGroup} (x y : A) : x + y = y + x := commutativity x y.

Definition

Require Import Basics Types. Require Import Spaces.Nat.Core Spaces.Int. Require Import Algebra.Groups.QuotientGroup. Require Import WildCat.

Algebra\AbGroups\AbelianGroup.v

ab_comm

{A : AbGroup} (x y : A) : - (x + y) = -x - y. Proof. lhs nrapply grp_inv_op. apply ab_comm. Defined.

Definition

Require Import Basics Types. Require Import Spaces.Nat.Core Spaces.Int. Require Import Algebra.Groups.QuotientGroup. Require Import WildCat.

Algebra\AbGroups\AbelianGroup.v

ab_neg_op

`{Univalence} {A B : AbGroup@{u}} : GroupIsomorphism A B <~> (A = B). Proof. refine (equiv_ap_inv issig_abgroup _ _ oE _). refine (equiv_path_sigma_hprop _ _ oE _). exact equiv_path_group. Defined.

Definition

Require Import Basics Types. Require Import Spaces.Nat.Core Spaces.Int. Require Import Algebra.Groups.QuotientGroup. Require Import WildCat.

Algebra\AbGroups\AbelianGroup.v

equiv_path_abgroup

`{Univalence} {A B : AbGroup} : (A = B :> AbGroup) <~> (A = B :> Group) := equiv_path_group oE equiv_path_abgroup^-1. Global Instance isabgroup_subgroup (G : AbGroup) (H : Subgroup G) : IsAbGroup H. Proof. nrapply Build_IsAbGroup. 1: exact _. intros x y. apply path_sigma_hprop. cbn. apply commutativity. Defined.

Definition

Require Import Basics Types. Require Import Spaces.Nat.Core Spaces.Int. Require Import Algebra.Groups.QuotientGroup. Require Import WildCat.

Algebra\AbGroups\AbelianGroup.v

equiv_path_abgroup_group

(G : AbGroup) : Subgroup G -> AbGroup := fun H => Build_AbGroup H _. #[warnings="-uniform-inheritance"] Coercion : Subgroup >-> AbGroup. Global Instance isnormal_ab_subgroup (G : AbGroup) (H : Subgroup G) : IsNormalSubgroup H. Proof. intros x y h. by rewrite ab_comm. Defined.

Definition

Require Import Basics Types. Require Import Spaces.Nat.Core Spaces.Int. Require Import Algebra.Groups.QuotientGroup. Require Import WildCat.

Algebra\AbGroups\AbelianGroup.v

abgroup_subgroup

(G : AbGroup) (H : Subgroup G) : AbGroup := (Build_AbGroup (QuotientGroup' G H (isnormal_ab_subgroup G H)) _).

Definition

Require Import Basics Types. Require Import Spaces.Nat.Core Spaces.Int. Require Import Algebra.Groups.QuotientGroup. Require Import WildCat.

Algebra\AbGroups\AbelianGroup.v

QuotientAbGroup

{G : AbGroup} (N : Subgroup G) (H : AbGroup) (f : GroupHomomorphism G H) (h : forall n : G, N n -> f n = mon_unit) : GroupHomomorphism (QuotientAbGroup G N) H := grp_quotient_rec G (Build_NormalSubgroup G N _) f h.

Definition

Require Import Basics Types. Require Import Spaces.Nat.Core Spaces.Int. Require Import Algebra.Groups.QuotientGroup. Require Import WildCat.

Algebra\AbGroups\AbelianGroup.v

quotient_abgroup_rec

{F : Funext} {G : AbGroup} (N : Subgroup G) (H : Group) : {f : GroupHomomorphism G H & forall (n : G), N n -> f n = mon_unit} <~> (GroupHomomorphism (QuotientAbGroup G N) H). Proof. exact (equiv_grp_quotient_ump (Build_NormalSubgroup G N _) _). Defined.

Theorem

Require Import Basics Types. Require Import Spaces.Nat.Core Spaces.Int. Require Import Algebra.Groups.QuotientGroup. Require Import WildCat.

Algebra\AbGroups\AbelianGroup.v

equiv_quotient_abgroup_ump

AbGroup. Proof. rapply (Build_AbGroup grp_trivial). by intros []. Defined.

Definition

Require Import Basics Types. Require Import Spaces.Nat.Core Spaces.Int. Require Import Algebra.Groups.QuotientGroup. Require Import WildCat.

Algebra\AbGroups\AbelianGroup.v

abgroup_trivial

{A B : AbGroup} (f : A $-> B) : AbGroup := Build_AbGroup (grp_image f) _.

Definition

Require Import Basics Types. Require Import Spaces.Nat.Core Spaces.Int. Require Import Algebra.Groups.QuotientGroup. Require Import WildCat.

Algebra\AbGroups\AbelianGroup.v

abgroup_image

`{Funext} {A B : AbGroup} (f : A $-> B) : GroupIsomorphism (QuotientAbGroup A (grp_kernel f)) (abgroup_image f). Proof. etransitivity. 2: rapply grp_first_iso. apply grp_iso_quotient_normal. Defined.

Definition

Require Import Basics Types. Require Import Spaces.Nat.Core Spaces.Int. Require Import Algebra.Groups.QuotientGroup. Require Import WildCat.

Algebra\AbGroups\AbelianGroup.v

abgroup_first_iso

{A B : AbGroup} (f : A $-> B) : AbGroup := Build_AbGroup (grp_kernel f) _.

Definition

Require Import Basics Types. Require Import Spaces.Nat.Core Spaces.Int. Require Import Algebra.Groups.QuotientGroup. Require Import WildCat.

Algebra\AbGroups\AbelianGroup.v

ab_kernel

`{Univalence} {A B B' : AbGroup} (p : B = B') (f : GroupHomomorphism A B) : transport (Hom A) p f = grp_homo_compose (equiv_path_abgroup^-1 p) f. Proof. induction p. by apply equiv_path_grouphomomorphism. Defined.

Lemma

Require Import Basics Types. Require Import Spaces.Nat.Core Spaces.Int. Require Import Algebra.Groups.QuotientGroup. Require Import WildCat.

Algebra\AbGroups\AbelianGroup.v

transport_abgrouphomomorphism_from_const

`{Univalence} {A B B' : AbGroup} (phi : GroupIsomorphism B B') (f : GroupHomomorphism A B) : transport (Hom A) (equiv_path_abgroup phi) f = grp_homo_compose phi f. Proof. refine (transport_abgrouphomomorphism_from_const _ _ @ _). by rewrite eissect. Defined.

Lemma

Require Import Basics Types. Require Import Spaces.Nat.Core Spaces.Int. Require Import Algebra.Groups.QuotientGroup. Require Import WildCat.

Algebra\AbGroups\AbelianGroup.v

transport_iso_abgrouphomomorphism_from_const

`{Univalence} {A A' B : AbGroup} (p : A = A') (f : GroupHomomorphism A B) : transport (fun G => Hom G B) p f = grp_homo_compose f (grp_iso_inverse (equiv_path_abgroup^-1 p)). Proof. induction p; cbn. by apply equiv_path_grouphomomorphism. Defined.

Lemma

Require Import Basics Types. Require Import Spaces.Nat.Core Spaces.Int. Require Import Algebra.Groups.QuotientGroup. Require Import WildCat.

Algebra\AbGroups\AbelianGroup.v

transport_abgrouphomomorphism_to_const

`{Univalence} {A A' B : AbGroup} (phi : GroupIsomorphism A A') (f : GroupHomomorphism A B) : transport (fun G => Hom G B) (equiv_path_abgroup phi) f = grp_homo_compose f (grp_iso_inverse phi). Proof. refine (transport_abgrouphomomorphism_to_const _ _ @ _). by rewrite eissect. Defined.

Lemma

Require Import Basics Types. Require Import Spaces.Nat.Core Spaces.Int. Require Import Algebra.Groups.QuotientGroup. Require Import WildCat.

Algebra\AbGroups\AbelianGroup.v

transport_iso_abgrouphomomorphism_to_const

{A : AbGroup} : GroupIsomorphism A A. Proof. snrapply Build_GroupIsomorphism. - snrapply Build_GroupHomomorphism. + exact (fun a => -a). + intros x y. refine (grp_inv_op x y @ _). apply commutativity. - srapply isequiv_adjointify. 1: exact (fun a => -a). 1-2: exact negate_involutive. Defined.

Definition

Require Import Basics Types. Require Import Spaces.Nat.Core Spaces.Int. Require Import Algebra.Groups.QuotientGroup. Require Import WildCat.

Algebra\AbGroups\AbelianGroup.v

ab_homo_negation

{A : AbGroup} (n : Int) : GroupHomomorphism A A. Proof. snrapply Build_GroupHomomorphism. 1: exact (fun a => grp_pow a n). intros a b. apply grp_pow_mul, ab_comm. Defined.

Definition

Require Import Basics Types. Require Import Spaces.Nat.Core Spaces.Int. Require Import Algebra.Groups.QuotientGroup. Require Import WildCat.

Algebra\AbGroups\AbelianGroup.v

ab_mul

{A B : AbGroup} (f : GroupHomomorphism A B) (n : Int) : f o ab_mul n == ab_mul n o f := grp_pow_natural f n.

Definition

Require Import Basics Types. Require Import Spaces.Nat.Core Spaces.Int. Require Import Algebra.Groups.QuotientGroup. Require Import WildCat.

Algebra\AbGroups\AbelianGroup.v

ab_mul_natural

{A B : AbGroup} (f : A $-> B) `{IsEmbedding f} : NormalSubgroup B := {| normalsubgroup_subgroup := grp_image_embedding f; normalsubgroup_isnormal := _ |}.

Definition

Require Import Basics Types. Require Import Spaces.Nat.Core Spaces.Int. Require Import Algebra.Groups.QuotientGroup. Require Import WildCat.

Algebra\AbGroups\AbelianGroup.v

ab_image_embedding

{A B : AbGroup} (f : A $-> B) `{IsEmbedding f} : GroupIsomorphism A (ab_image_embedding f) := grp_image_in_embedding f.

Definition

Require Import Basics Types. Require Import Spaces.Nat.Core Spaces.Int. Require Import Algebra.Groups.QuotientGroup. Require Import WildCat.

Algebra\AbGroups\AbelianGroup.v

ab_image_in_embedding

{G : Group@{u}} {A : AbGroup@{u}} (f : GroupHomomorphism G A) : AbGroup := QuotientAbGroup _ (grp_image f).

Definition

Require Import Basics Types. Require Import Spaces.Nat.Core Spaces.Int. Require Import Algebra.Groups.QuotientGroup. Require Import WildCat.

Algebra\AbGroups\AbelianGroup.v

ab_cokernel

{G : Group} {A : AbGroup} (f : G $-> A) `{IsEmbedding f} : AbGroup := QuotientAbGroup _ (grp_image_embedding f).

Definition

Require Import Basics Types. Require Import Spaces.Nat.Core Spaces.Int. Require Import Algebra.Groups.QuotientGroup. Require Import WildCat.

Algebra\AbGroups\AbelianGroup.v

ab_cokernel_embedding

{G: Group} {A B : AbGroup} (f : G $-> A) `{IsEmbedding f} (h : A $-> B) (p : grp_homo_compose h f $== grp_homo_const) : ab_cokernel_embedding f $-> B. Proof. snrapply (grp_quotient_rec _ _ h). intros a [g q]. induction q. exact (p g). Defined.

Definition

Require Import Basics Types. Require Import Spaces.Nat.Core Spaces.Int. Require Import Algebra.Groups.QuotientGroup. Require Import WildCat.

Algebra\AbGroups\AbelianGroup.v

ab_cokernel_embedding_rec