Datasets:

phanerozoic

/

Coq-HoTT

like 2

{A : AbGroup} (m n : nat) (f : forall i j, (i < m)%nat -> (j < n)%nat -> A) : ab_sum m (fun i Hi => ab_sum n (fun j Hj => f i j Hi Hj)) = ab_sum n (fun j Hj => ab_sum m (fun i Hi => f i j Hi Hj)). Proof. induction n as [|n IHn] in m, f |- *. 1: by nrapply ab_sum_zero. lhs nrapply ab_sum_plus; cbn; f_ap. Defined.

Definition

Require Import Basics.Overture Basics.Tactics. Require Import Spaces.Nat.Core Spaces.Int. Require Import AbelianGroup.

Algebra\AbGroups\FiniteSum.v

ab_sum_sum

{A : AbGroup} {n : nat} {f g : forall k, (k < n)%nat -> A} (p : forall k Hk, f k Hk = g k Hk) : ab_sum n f = ab_sum n g. Proof. induction n as [|n IHn]. 1: reflexivity. cbn; f_ap. by apply IHn. Defined.

Definition

Require Import Basics.Overture Basics.Tactics. Require Import Spaces.Nat.Core Spaces.Int. Require Import AbelianGroup.

Algebra\AbGroups\FiniteSum.v

path_ab_sum

(S : Type) (F_S : AbGroup) (i : S -> F_S) (A : AbGroup) (g : S -> A) : Type := {f : F_S $-> A & f o i == g}.

Definition

Require Import Basics.Overture Basics.Tactics Basics.Equivalences. Require Import Types.Sigma Types.Forall Types.Paths. Require Import WildCat.Core WildCat.EquivGpd WildCat.Universe. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Abelianization. Require Import Algebra.Groups.FreeGroup. Require Import Spaces.List.Core.

Algebra\AbGroups\FreeAbelianGroup.v

FactorsThroughFreeAbGroup

(S : Type) : AbGroup := abel (FreeGroup S).

Definition

Require Import Basics.Overture Basics.Tactics Basics.Equivalences. Require Import Types.Sigma Types.Forall Types.Paths. Require Import WildCat.Core WildCat.EquivGpd WildCat.Universe. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Abelianization. Require Import Algebra.Groups.FreeGroup. Require Import Spaces.List.Core.

Algebra\AbGroups\FreeAbelianGroup.v

FreeAbGroup

{S : Type} : S -> FreeAbGroup S := abel_unit o freegroup_in.

Definition

Require Import Basics.Overture Basics.Tactics Basics.Equivalences. Require Import Types.Sigma Types.Forall Types.Paths. Require Import WildCat.Core WildCat.EquivGpd WildCat.Universe. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Abelianization. Require Import Algebra.Groups.FreeGroup. Require Import Spaces.List.Core.

Algebra\AbGroups\FreeAbelianGroup.v

freeabgroup_in

{S : Type} {A : AbGroup} (f : S -> A) : FreeAbGroup S $-> A := grp_homo_abel_rec (FreeGroup_rec _ _ f).

Definition

Require Import Basics.Overture Basics.Tactics Basics.Equivalences. Require Import Types.Sigma Types.Forall Types.Paths. Require Import WildCat.Core WildCat.EquivGpd WildCat.Universe. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Abelianization. Require Import Algebra.Groups.FreeGroup. Require Import Spaces.List.Core.

Algebra\AbGroups\FreeAbelianGroup.v

FreeAbGroup_rec

{S : Type} {A : AbGroup} (f : S -> A) : FreeAbGroup_rec f o freeabgroup_in == f := fun _ => idpath.

Definition

Require Import Basics.Overture Basics.Tactics Basics.Equivalences. Require Import Types.Sigma Types.Forall Types.Paths. Require Import WildCat.Core WildCat.EquivGpd WildCat.Universe. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Abelianization. Require Import Algebra.Groups.FreeGroup. Require Import Spaces.List.Core.

Algebra\AbGroups\FreeAbelianGroup.v

FreeAbGroup_rec_beta_in

{X : Type} {A : AbGroup} {f f' : FreeAbGroup X $-> A} (p : forall x, f (freeabgroup_in x) = f' (freeabgroup_in x)) : f $== f'. Proof. snrapply abel_ind_homotopy. snrapply FreeGroup_ind_homotopy. snrapply p. Defined.

Definition

Require Import Basics.Overture Basics.Tactics Basics.Equivalences. Require Import Types.Sigma Types.Forall Types.Paths. Require Import WildCat.Core WildCat.EquivGpd WildCat.Universe. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Abelianization. Require Import Algebra.Groups.FreeGroup. Require Import Spaces.List.Core.

Algebra\AbGroups\FreeAbelianGroup.v

FreeAbGroup_ind_homotopy

{A B : AbGroup} : FreeAbGroup (A * B) -> Type. Proof. intros x. refine ((exists (a1 a2 : A) (b : B), _) + exists (a : A) (b1 b2 : B), _)%type. - refine (- _ + (_ + _) = x). 1-3: apply freeabgroup_in. + exact (a1 + a2, b). + exact (a1, b). + exact (a2, b). - refine (- _ + (_ + _) = x). 1-3: apply freeabgroup_in. + exact (a, b1 + b2). + exact (a, b1). + exact (a, b2). Defined.

Definition

Require Import Basics.Overture Basics.Tactics. Require Import Types.Forall Types.Sigma Types.Prod. Require Import WildCat.Core WildCat.Equiv WildCat.Monoidal WildCat.Bifunctor. Require Import WildCat.NatTrans WildCat.MonoidalTwistConstruction. Require Import Algebra.Groups.Group Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Algebra.AbGroups.AbHom Algebra.AbGroups.FreeAbelianGroup. Require Import Algebra.AbGroups.Abelianization Algebra Algebra.Groups.FreeGroup. Require Import Colimits.Quotient. Require Import Spaces.List.Core Spaces.Int. Require Import AbGroups.Z. Require Import Truncations.

Algebra\AbGroups\TensorProduct.v

family_biadditive_pairs

{A B : AbGroup} : Subgroup (FreeAbGroup (A * B)) := subgroup_generated family_biadditive_pairs.

Definition

Require Import Basics.Overture Basics.Tactics. Require Import Types.Forall Types.Sigma Types.Prod. Require Import WildCat.Core WildCat.Equiv WildCat.Monoidal WildCat.Bifunctor. Require Import WildCat.NatTrans WildCat.MonoidalTwistConstruction. Require Import Algebra.Groups.Group Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Algebra.AbGroups.AbHom Algebra.AbGroups.FreeAbelianGroup. Require Import Algebra.AbGroups.Abelianization Algebra Algebra.Groups.FreeGroup. Require Import Colimits.Quotient. Require Import Spaces.List.Core Spaces.Int. Require Import AbGroups.Z. Require Import Truncations.

Algebra\AbGroups\TensorProduct.v

subgroup_biadditive_pairs

(A B : AbGroup) : AbGroup := QuotientAbGroup (FreeAbGroup (A * B)) subgroup_biadditive_pairs.

Definition

Require Import Basics.Overture Basics.Tactics. Require Import Types.Forall Types.Sigma Types.Prod. Require Import WildCat.Core WildCat.Equiv WildCat.Monoidal WildCat.Bifunctor. Require Import WildCat.NatTrans WildCat.MonoidalTwistConstruction. Require Import Algebra.Groups.Group Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Algebra.AbGroups.AbHom Algebra.AbGroups.FreeAbelianGroup. Require Import Algebra.AbGroups.Abelianization Algebra Algebra.Groups.FreeGroup. Require Import Colimits.Quotient. Require Import Spaces.List.Core Spaces.Int. Require Import AbGroups.Z. Require Import Truncations.

Algebra\AbGroups\TensorProduct.v

ab_tensor_prod

{A B : AbGroup} : A -> B -> ab_tensor_prod A B := fun a b => grp_quotient_map (freeabgroup_in (a, b)).

Definition

Require Import Basics.Overture Basics.Tactics. Require Import Types.Forall Types.Sigma Types.Prod. Require Import WildCat.Core WildCat.Equiv WildCat.Monoidal WildCat.Bifunctor. Require Import WildCat.NatTrans WildCat.MonoidalTwistConstruction. Require Import Algebra.Groups.Group Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Algebra.AbGroups.AbHom Algebra.AbGroups.FreeAbelianGroup. Require Import Algebra.AbGroups.Abelianization Algebra Algebra.Groups.FreeGroup. Require Import Colimits.Quotient. Require Import Spaces.List.Core Spaces.Int. Require Import AbGroups.Z. Require Import Truncations.

Algebra\AbGroups\TensorProduct.v

tensor

{A B : AbGroup} (a : A) (b b' : B) : tensor a (b + b') = tensor a b + tensor a b'. Proof. apply qglue, tr. apply sgt_in. right. by exists a, b, b'. Defined.

Definition

Require Import Basics.Overture Basics.Tactics. Require Import Types.Forall Types.Sigma Types.Prod. Require Import WildCat.Core WildCat.Equiv WildCat.Monoidal WildCat.Bifunctor. Require Import WildCat.NatTrans WildCat.MonoidalTwistConstruction. Require Import Algebra.Groups.Group Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Algebra.AbGroups.AbHom Algebra.AbGroups.FreeAbelianGroup. Require Import Algebra.AbGroups.Abelianization Algebra Algebra.Groups.FreeGroup. Require Import Colimits.Quotient. Require Import Spaces.List.Core Spaces.Int. Require Import AbGroups.Z. Require Import Truncations.

Algebra\AbGroups\TensorProduct.v

tensor_dist_l

{A B : AbGroup} (a a' : A) (b : B) : tensor (a + a') b = tensor a b + tensor a' b. Proof. apply qglue, tr. apply sgt_in. left. by exists a, a', b. Defined.

Definition

Require Import Basics.Overture Basics.Tactics. Require Import Types.Forall Types.Sigma Types.Prod. Require Import WildCat.Core WildCat.Equiv WildCat.Monoidal WildCat.Bifunctor. Require Import WildCat.NatTrans WildCat.MonoidalTwistConstruction. Require Import Algebra.Groups.Group Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Algebra.AbGroups.AbHom Algebra.AbGroups.FreeAbelianGroup. Require Import Algebra.AbGroups.Abelianization Algebra Algebra.Groups.FreeGroup. Require Import Colimits.Quotient. Require Import Spaces.List.Core Spaces.Int. Require Import AbGroups.Z. Require Import Truncations.

Algebra\AbGroups\TensorProduct.v

tensor_dist_r

{A B : AbGroup} (a : A) : B $-> ab_tensor_prod A B. Proof. snrapply Build_GroupHomomorphism. - exact (fun b => tensor a b). - intros b b'. nrapply tensor_dist_l. Defined.

Definition

Require Import Basics.Overture Basics.Tactics. Require Import Types.Forall Types.Sigma Types.Prod. Require Import WildCat.Core WildCat.Equiv WildCat.Monoidal WildCat.Bifunctor. Require Import WildCat.NatTrans WildCat.MonoidalTwistConstruction. Require Import Algebra.Groups.Group Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Algebra.AbGroups.AbHom Algebra.AbGroups.FreeAbelianGroup. Require Import Algebra.AbGroups.Abelianization Algebra Algebra.Groups.FreeGroup. Require Import Colimits.Quotient. Require Import Spaces.List.Core Spaces.Int. Require Import AbGroups.Z. Require Import Truncations.

Algebra\AbGroups\TensorProduct.v

grp_homo_tensor_l

{A B : AbGroup} (b : B) : A $-> ab_tensor_prod A B. Proof. snrapply Build_GroupHomomorphism. - exact (fun a => tensor a b). - intros a a'. nrapply tensor_dist_r. Defined.

Definition

Require Import Basics.Overture Basics.Tactics. Require Import Types.Forall Types.Sigma Types.Prod. Require Import WildCat.Core WildCat.Equiv WildCat.Monoidal WildCat.Bifunctor. Require Import WildCat.NatTrans WildCat.MonoidalTwistConstruction. Require Import Algebra.Groups.Group Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Algebra.AbGroups.AbHom Algebra.AbGroups.FreeAbelianGroup. Require Import Algebra.AbGroups.Abelianization Algebra Algebra.Groups.FreeGroup. Require Import Colimits.Quotient. Require Import Spaces.List.Core Spaces.Int. Require Import AbGroups.Z. Require Import Truncations.

Algebra\AbGroups\TensorProduct.v

grp_homo_tensor_r

{A B : AbGroup} (a : A) (b : B) : tensor (-a) b = - tensor a b := grp_homo_inv (grp_homo_tensor_r b) a.

Definition

Require Import Basics.Overture Basics.Tactics. Require Import Types.Forall Types.Sigma Types.Prod. Require Import WildCat.Core WildCat.Equiv WildCat.Monoidal WildCat.Bifunctor. Require Import WildCat.NatTrans WildCat.MonoidalTwistConstruction. Require Import Algebra.Groups.Group Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Algebra.AbGroups.AbHom Algebra.AbGroups.FreeAbelianGroup. Require Import Algebra.AbGroups.Abelianization Algebra Algebra.Groups.FreeGroup. Require Import Colimits.Quotient. Require Import Spaces.List.Core Spaces.Int. Require Import AbGroups.Z. Require Import Truncations.

Algebra\AbGroups\TensorProduct.v

tensor_neg_l

{A B : AbGroup} (a : A) (b : B) : tensor a (-b) = - tensor a b := grp_homo_inv (grp_homo_tensor_l a) b.

Definition

Require Import Basics.Overture Basics.Tactics. Require Import Types.Forall Types.Sigma Types.Prod. Require Import WildCat.Core WildCat.Equiv WildCat.Monoidal WildCat.Bifunctor. Require Import WildCat.NatTrans WildCat.MonoidalTwistConstruction. Require Import Algebra.Groups.Group Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Algebra.AbGroups.AbHom Algebra.AbGroups.FreeAbelianGroup. Require Import Algebra.AbGroups.Abelianization Algebra Algebra.Groups.FreeGroup. Require Import Colimits.Quotient. Require Import Spaces.List.Core Spaces.Int. Require Import AbGroups.Z. Require Import Truncations.

Algebra\AbGroups\TensorProduct.v

tensor_neg_r

{A B : AbGroup} (b : B) : tensor (A:=A) 0 b = 0 := grp_homo_unit (grp_homo_tensor_r b).

Definition

Require Import Basics.Overture Basics.Tactics. Require Import Types.Forall Types.Sigma Types.Prod. Require Import WildCat.Core WildCat.Equiv WildCat.Monoidal WildCat.Bifunctor. Require Import WildCat.NatTrans WildCat.MonoidalTwistConstruction. Require Import Algebra.Groups.Group Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Algebra.AbGroups.AbHom Algebra.AbGroups.FreeAbelianGroup. Require Import Algebra.AbGroups.Abelianization Algebra Algebra.Groups.FreeGroup. Require Import Colimits.Quotient. Require Import Spaces.List.Core Spaces.Int. Require Import AbGroups.Z. Require Import Truncations.

Algebra\AbGroups\TensorProduct.v

tensor_zero_l

{A B : AbGroup} (a : A) : tensor (B:=B) a 0 = 0 := grp_homo_unit (grp_homo_tensor_l a).

Definition

Require Import Basics.Overture Basics.Tactics. Require Import Types.Forall Types.Sigma Types.Prod. Require Import WildCat.Core WildCat.Equiv WildCat.Monoidal WildCat.Bifunctor. Require Import WildCat.NatTrans WildCat.MonoidalTwistConstruction. Require Import Algebra.Groups.Group Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Algebra.AbGroups.AbHom Algebra.AbGroups.FreeAbelianGroup. Require Import Algebra.AbGroups.Abelianization Algebra Algebra.Groups.FreeGroup. Require Import Colimits.Quotient. Require Import Spaces.List.Core Spaces.Int. Require Import AbGroups.Z. Require Import Truncations.

Algebra\AbGroups\TensorProduct.v

tensor_zero_r

`{Funext} {A B : AbGroup} : A $-> ab_hom B (ab_tensor_prod A B). Proof. snrapply Build_GroupHomomorphism. - intros a. snrapply Build_GroupHomomorphism. + exact (tensor a). + nrapply tensor_dist_l. - intros a a'. apply equiv_path_grouphomomorphism. intros b. nrapply tensor_dist_r. Defined.

Definition

Require Import Basics.Overture Basics.Tactics. Require Import Types.Forall Types.Sigma Types.Prod. Require Import WildCat.Core WildCat.Equiv WildCat.Monoidal WildCat.Bifunctor. Require Import WildCat.NatTrans WildCat.MonoidalTwistConstruction. Require Import Algebra.Groups.Group Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Algebra.AbGroups.AbHom Algebra.AbGroups.FreeAbelianGroup. Require Import Algebra.AbGroups.Abelianization Algebra Algebra.Groups.FreeGroup. Require Import Colimits.Quotient. Require Import Spaces.List.Core Spaces.Int. Require Import AbGroups.Z. Require Import Truncations.

Algebra\AbGroups\TensorProduct.v

grp_homo_tensor

{A B C : AbGroup} (f : A -> B -> C) (l : forall a b b', f a (b + b') = f a b + f a b') (r : forall a a' b, f (a + a') b = f a b + f a' b) (x : FreeAbGroup (A * B)) (insg : subgroup_biadditive_pairs x) : grp_homo_abel_rec (FreeGroup_rec (A * B) C (uncurry f)) x = mon_unit. Proof. set (abel_rec := grp_homo_abel_rec (FreeGroup_rec (A * B) C (uncurry f))). strip_truncations. induction insg as [ x biad | | g h insg_g IHg insg_h IHh ]. - destruct biad as [ [ a [ a' [ b p ] ] ] | [ a [ b [ b' p ] ] ] ]. all: destruct p; simpl. all: apply grp_moveL_M1^-1%equiv; symmetry. 1: apply r. apply l. - nrapply grp_homo_unit. - rewrite grp_homo_op, grp_homo_inv. apply grp_moveL_1M^-1. exact (IHg @ IHh^). Defined.

Definition

Require Import Basics.Overture Basics.Tactics. Require Import Types.Forall Types.Sigma Types.Prod. Require Import WildCat.Core WildCat.Equiv WildCat.Monoidal WildCat.Bifunctor. Require Import WildCat.NatTrans WildCat.MonoidalTwistConstruction. Require Import Algebra.Groups.Group Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Algebra.AbGroups.AbHom Algebra.AbGroups.FreeAbelianGroup. Require Import Algebra.AbGroups.Abelianization Algebra Algebra.Groups.FreeGroup. Require Import Colimits.Quotient. Require Import Spaces.List.Core Spaces.Int. Require Import AbGroups.Z. Require Import Truncations.

Algebra\AbGroups\TensorProduct.v

ab_tensor_prod_rec_helper

{A B C : AbGroup} (f : A -> B -> C) (l : forall a b b', f a (b + b') = f a b + f a b') (r : forall a a' b, f (a + a') b = f a b + f a' b) : ab_tensor_prod A B $-> C. Proof. unfold ab_tensor_prod. snrapply grp_quotient_rec. - snrapply FreeAbGroup_rec. exact (uncurry f). - unfold normalsubgroup_subgroup. apply ab_tensor_prod_rec_helper; assumption. Defined.

Definition

Require Import Basics.Overture Basics.Tactics. Require Import Types.Forall Types.Sigma Types.Prod. Require Import WildCat.Core WildCat.Equiv WildCat.Monoidal WildCat.Bifunctor. Require Import WildCat.NatTrans WildCat.MonoidalTwistConstruction. Require Import Algebra.Groups.Group Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Algebra.AbGroups.AbHom Algebra.AbGroups.FreeAbelianGroup. Require Import Algebra.AbGroups.Abelianization Algebra Algebra.Groups.FreeGroup. Require Import Colimits.Quotient. Require Import Spaces.List.Core Spaces.Int. Require Import AbGroups.Z. Require Import Truncations.

Algebra\AbGroups\TensorProduct.v

ab_tensor_prod_rec

ab_tensor_prod_rec' {A B C : AbGroup} (f : A -> (B $-> C)) (l : forall a a' b, f (a + a') b = f a b + f a' b) : ab_tensor_prod A B $-> C. Proof. refine (ab_tensor_prod_rec f _ l). intro a; apply grp_homo_op. Defined.

Definition

Require Import Basics.Overture Basics.Tactics. Require Import Types.Forall Types.Sigma Types.Prod. Require Import WildCat.Core WildCat.Equiv WildCat.Monoidal WildCat.Bifunctor. Require Import WildCat.NatTrans WildCat.MonoidalTwistConstruction. Require Import Algebra.Groups.Group Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Algebra.AbGroups.AbHom Algebra.AbGroups.FreeAbelianGroup. Require Import Algebra.AbGroups.Abelianization Algebra Algebra.Groups.FreeGroup. Require Import Colimits.Quotient. Require Import Spaces.List.Core Spaces.Int. Require Import AbGroups.Z. Require Import Truncations.

Algebra\AbGroups\TensorProduct.v

ab_tensor_prod_rec'

{A B : AbGroup} (P : ab_tensor_prod A B -> Type) {H : forall x, IsHProp (P x)} (Hin : forall a b, P (tensor a b)) (Hop : forall x y, P x -> P y -> P (x + y)) : forall x, P x. Proof. unfold ab_tensor_prod. srapply grp_quotient_ind_hprop. srapply Abel_ind_hprop; cbn beta. set (tensor_in := grp_quotient_map $o abel_unit : FreeGroup (A * B) $-> ab_tensor_prod A B). change (forall x, P (tensor_in x)). srapply FreeGroup_ind_hprop'; intros w; cbn beta. induction w. - exact (transport P (tensor_zero_l 0) (Hin 0 0)). - change (P (tensor_in (freegroup_eta [a]%list + freegroup_eta w))). rewrite grp_homo_op. destruct a as [[a b]|[a b]]. + change (P (tensor_in (freegroup_in (a, b)) + tensor_in (freegroup_eta w))). apply Hop; trivial. apply Hin. + change (P (tensor_in (- freegroup_in (a, b)) + tensor_in (freegroup_eta w))). rewrite grp_homo_inv. apply Hop; trivial. rewrite <- tensor_neg_l. apply Hin. Defined.

Definition

Require Import Basics.Overture Basics.Tactics. Require Import Types.Forall Types.Sigma Types.Prod. Require Import WildCat.Core WildCat.Equiv WildCat.Monoidal WildCat.Bifunctor. Require Import WildCat.NatTrans WildCat.MonoidalTwistConstruction. Require Import Algebra.Groups.Group Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Algebra.AbGroups.AbHom Algebra.AbGroups.FreeAbelianGroup. Require Import Algebra.AbGroups.Abelianization Algebra Algebra.Groups.FreeGroup. Require Import Colimits.Quotient. Require Import Spaces.List.Core Spaces.Int. Require Import AbGroups.Z. Require Import Truncations.

Algebra\AbGroups\TensorProduct.v

ab_tensor_prod_ind_hprop

{A B G : AbGroup} {f f' : ab_tensor_prod A B $-> G} (H : forall a b, f (tensor a b) = f' (tensor a b)) : f $== f'. Proof. nrapply ab_tensor_prod_ind_hprop. - exact _. - exact H. - intros x y; apply grp_homo_op_agree. Defined.

Definition

Require Import Basics.Overture Basics.Tactics. Require Import Types.Forall Types.Sigma Types.Prod. Require Import WildCat.Core WildCat.Equiv WildCat.Monoidal WildCat.Bifunctor. Require Import WildCat.NatTrans WildCat.MonoidalTwistConstruction. Require Import Algebra.Groups.Group Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Algebra.AbGroups.AbHom Algebra.AbGroups.FreeAbelianGroup. Require Import Algebra.AbGroups.Abelianization Algebra Algebra.Groups.FreeGroup. Require Import Colimits.Quotient. Require Import Spaces.List.Core Spaces.Int. Require Import AbGroups.Z. Require Import Truncations.

Algebra\AbGroups\TensorProduct.v

ab_tensor_prod_ind_homotopy

{A B G : AbGroup} {f f' f'' : ab_tensor_prod A B $-> G} (H : forall a b, f (tensor a b) = f' (tensor a b) + f'' (tensor a b)) : forall x, f x = f' x + f'' x := ab_tensor_prod_ind_homotopy (f':=ab_homo_add f' f'') H.

Definition

Require Import Basics.Overture Basics.Tactics. Require Import Types.Forall Types.Sigma Types.Prod. Require Import WildCat.Core WildCat.Equiv WildCat.Monoidal WildCat.Bifunctor. Require Import WildCat.NatTrans WildCat.MonoidalTwistConstruction. Require Import Algebra.Groups.Group Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Algebra.AbGroups.AbHom Algebra.AbGroups.FreeAbelianGroup. Require Import Algebra.AbGroups.Abelianization Algebra Algebra.Groups.FreeGroup. Require Import Colimits.Quotient. Require Import Spaces.List.Core Spaces.Int. Require Import AbGroups.Z. Require Import Truncations.

Algebra\AbGroups\TensorProduct.v

ab_tensor_prod_ind_homotopy_plus

{A B C : AbGroup} (P : ab_tensor_prod A (ab_tensor_prod B C) -> Type) (H : forall x, IsHProp (P x)) (Hin : forall a b c, P (tensor a (tensor b c))) (Hop : forall x y, P x -> P y -> P (x + y)) : forall x, P x. Proof. rapply (ab_tensor_prod_ind_hprop P). - intros a. rapply (ab_tensor_prod_ind_hprop (fun x => P (tensor _ x))). + nrapply Hin. + intros x y Hx Hy. rewrite tensor_dist_l. by apply Hop. - exact Hop. Defined.

Definition

Require Import Basics.Overture Basics.Tactics. Require Import Types.Forall Types.Sigma Types.Prod. Require Import WildCat.Core WildCat.Equiv WildCat.Monoidal WildCat.Bifunctor. Require Import WildCat.NatTrans WildCat.MonoidalTwistConstruction. Require Import Algebra.Groups.Group Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Algebra.AbGroups.AbHom Algebra.AbGroups.FreeAbelianGroup. Require Import Algebra.AbGroups.Abelianization Algebra Algebra.Groups.FreeGroup. Require Import Colimits.Quotient. Require Import Spaces.List.Core Spaces.Int. Require Import AbGroups.Z. Require Import Truncations.

Algebra\AbGroups\TensorProduct.v

ab_tensor_prod_ind_hprop_triple

{A B C G : AbGroup} {f f' : ab_tensor_prod A (ab_tensor_prod B C) $-> G} (H : forall a b c, f (tensor a (tensor b c)) = f' (tensor a (tensor b c))) : f $== f'. Proof. nrapply ab_tensor_prod_ind_hprop_triple. - exact _. - exact H. - intros x y; apply grp_homo_op_agree. Defined.

Definition

Require Import Basics.Overture Basics.Tactics. Require Import Types.Forall Types.Sigma Types.Prod. Require Import WildCat.Core WildCat.Equiv WildCat.Monoidal WildCat.Bifunctor. Require Import WildCat.NatTrans WildCat.MonoidalTwistConstruction. Require Import Algebra.Groups.Group Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Algebra.AbGroups.AbHom Algebra.AbGroups.FreeAbelianGroup. Require Import Algebra.AbGroups.Abelianization Algebra Algebra.Groups.FreeGroup. Require Import Colimits.Quotient. Require Import Spaces.List.Core Spaces.Int. Require Import AbGroups.Z. Require Import Truncations.

Algebra\AbGroups\TensorProduct.v

ab_tensor_prod_ind_homotopy_triple

{A B C D : AbGroup} (P : ab_tensor_prod A (ab_tensor_prod B (ab_tensor_prod C D)) -> Type) (H : forall x, IsHProp (P x)) (Hin : forall a b c d, P (tensor a (tensor b (tensor c d)))) (Hop : forall x y, P x -> P y -> P (x + y)) : forall x, P x. Proof. rapply (ab_tensor_prod_ind_hprop P). - intros a. nrapply (ab_tensor_prod_ind_hprop_triple (fun x => P (tensor _ x))). + intro x; apply H. + nrapply Hin. + intros x y Hx Hy. rewrite tensor_dist_l. by apply Hop. - exact Hop. Defined.

Definition

Require Import Basics.Overture Basics.Tactics. Require Import Types.Forall Types.Sigma Types.Prod. Require Import WildCat.Core WildCat.Equiv WildCat.Monoidal WildCat.Bifunctor. Require Import WildCat.NatTrans WildCat.MonoidalTwistConstruction. Require Import Algebra.Groups.Group Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Algebra.AbGroups.AbHom Algebra.AbGroups.FreeAbelianGroup. Require Import Algebra.AbGroups.Abelianization Algebra Algebra.Groups.FreeGroup. Require Import Colimits.Quotient. Require Import Spaces.List.Core Spaces.Int. Require Import AbGroups.Z. Require Import Truncations.

Algebra\AbGroups\TensorProduct.v

ab_tensor_prod_ind_hprop_quad

{A B C D G : AbGroup} {f f' : ab_tensor_prod A (ab_tensor_prod B (ab_tensor_prod C D)) $-> G} (H : forall a b c d, f (tensor a (tensor b (tensor c d))) = f' (tensor a (tensor b (tensor c d)))) : f $== f'. Proof. nrapply (ab_tensor_prod_ind_hprop_quad (fun _ => _)). - exact _. - exact H. - intros x y; apply grp_homo_op_agree. Defined.

Definition

Require Import Basics.Overture Basics.Tactics. Require Import Types.Forall Types.Sigma Types.Prod. Require Import WildCat.Core WildCat.Equiv WildCat.Monoidal WildCat.Bifunctor. Require Import WildCat.NatTrans WildCat.MonoidalTwistConstruction. Require Import Algebra.Groups.Group Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Algebra.AbGroups.AbHom Algebra.AbGroups.FreeAbelianGroup. Require Import Algebra.AbGroups.Abelianization Algebra Algebra.Groups.FreeGroup. Require Import Colimits.Quotient. Require Import Spaces.List.Core Spaces.Int. Require Import AbGroups.Z. Require Import Truncations.

Algebra\AbGroups\TensorProduct.v

ab_tensor_prod_ind_homotopy_quad

{A B C : Type} `{SgOp A, SgOp B, SgOp C} (f : A -> B -> C) : _ <~> IsBiadditive f := ltac:(issig). Global Instance istrunc_isbiadditive `{Funext} {A B C : Type} `{SgOp A, SgOp B, SgOp C} (f : A -> B -> C) n `{IsTrunc n.+1 C} : IsTrunc n (IsBiadditive f). Proof. nrapply istrunc_equiv_istrunc. 1: rapply . unfold IsSemiGroupPreserving. exact _. Defined.

Definition

Require Import Basics.Overture Basics.Tactics. Require Import Types.Forall Types.Sigma Types.Prod. Require Import WildCat.Core WildCat.Equiv WildCat.Monoidal WildCat.Bifunctor. Require Import WildCat.NatTrans WildCat.MonoidalTwistConstruction. Require Import Algebra.Groups.Group Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Algebra.AbGroups.AbHom Algebra.AbGroups.FreeAbelianGroup. Require Import Algebra.AbGroups.Abelianization Algebra Algebra.Groups.FreeGroup. Require Import Colimits.Quotient. Require Import Spaces.List.Core Spaces.Int. Require Import AbGroups.Z. Require Import Truncations.

Algebra\AbGroups\TensorProduct.v

issig_IsBiadditive

(A B C : Type) `{SgOp A, SgOp B, SgOp C} := { biadditive_fun :> A -> B -> C; biadditive_isbiadditive :: IsBiadditive biadditive_fun; }.

Record

Require Import Basics.Overture Basics.Tactics. Require Import Types.Forall Types.Sigma Types.Prod. Require Import WildCat.Core WildCat.Equiv WildCat.Monoidal WildCat.Bifunctor. Require Import WildCat.NatTrans WildCat.MonoidalTwistConstruction. Require Import Algebra.Groups.Group Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Algebra.AbGroups.AbHom Algebra.AbGroups.FreeAbelianGroup. Require Import Algebra.AbGroups.Abelianization Algebra Algebra.Groups.FreeGroup. Require Import Colimits.Quotient. Require Import Spaces.List.Core Spaces.Int. Require Import AbGroups.Z. Require Import Truncations.

Algebra\AbGroups\TensorProduct.v

Biadditive

{A B C : Type} `{SgOp A, SgOp B, SgOp C} : _ <~> Biadditive A B C := ltac:(issig).

Definition

Require Import Basics.Overture Basics.Tactics. Require Import Types.Forall Types.Sigma Types.Prod. Require Import WildCat.Core WildCat.Equiv WildCat.Monoidal WildCat.Bifunctor. Require Import WildCat.NatTrans WildCat.MonoidalTwistConstruction. Require Import Algebra.Groups.Group Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Algebra.AbGroups.AbHom Algebra.AbGroups.FreeAbelianGroup. Require Import Algebra.AbGroups.Abelianization Algebra Algebra.Groups.FreeGroup. Require Import Colimits.Quotient. Require Import Spaces.List.Core Spaces.Int. Require Import AbGroups.Z. Require Import Truncations.

Algebra\AbGroups\TensorProduct.v

issig_Biadditive

{A B C : AbGroup} : (ab_tensor_prod A B $-> C) -> Biadditive A B C. Proof. intros f. exists (fun x y => f (tensor x y)). snrapply Build_IsBiadditive. - intros b a a'; simpl. lhs nrapply (ap f). 1: nrapply tensor_dist_r. nrapply grp_homo_op. - intros a a' b; simpl. lhs nrapply (ap f). 1: nrapply tensor_dist_l. nrapply grp_homo_op. Defined.

Definition

Require Import Basics.Overture Basics.Tactics. Require Import Types.Forall Types.Sigma Types.Prod. Require Import WildCat.Core WildCat.Equiv WildCat.Monoidal WildCat.Bifunctor. Require Import WildCat.NatTrans WildCat.MonoidalTwistConstruction. Require Import Algebra.Groups.Group Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Algebra.AbGroups.AbHom Algebra.AbGroups.FreeAbelianGroup. Require Import Algebra.AbGroups.Abelianization Algebra Algebra.Groups.FreeGroup. Require Import Colimits.Quotient. Require Import Spaces.List.Core Spaces.Int. Require Import AbGroups.Z. Require Import Truncations.

Algebra\AbGroups\TensorProduct.v

biadditive_ab_tensor_prod

`{Funext} (A B C : AbGroup) : Biadditive A B C <~> (ab_tensor_prod A B $-> C). Proof. snrapply equiv_adjointify. - intros [f [l r]]. exact (ab_tensor_prod_rec f r (fun a a' b => l b a a')). - snrapply biadditive_ab_tensor_prod. - intros f. snrapply equiv_path_grouphomomorphism. snrapply ab_tensor_prod_ind_homotopy. intros a b; simpl. reflexivity. - intros [f [l r]]. snrapply (equiv_ap_inv' issig_Biadditive). rapply path_sigma_hprop; simpl. reflexivity. Defined.

Definition

Require Import Basics.Overture Basics.Tactics. Require Import Types.Forall Types.Sigma Types.Prod. Require Import WildCat.Core WildCat.Equiv WildCat.Monoidal WildCat.Bifunctor. Require Import WildCat.NatTrans WildCat.MonoidalTwistConstruction. Require Import Algebra.Groups.Group Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Algebra.AbGroups.AbHom Algebra.AbGroups.FreeAbelianGroup. Require Import Algebra.AbGroups.Abelianization Algebra Algebra.Groups.FreeGroup. Require Import Colimits.Quotient. Require Import Spaces.List.Core Spaces.Int. Require Import AbGroups.Z. Require Import Truncations.

Algebra\AbGroups\TensorProduct.v

equiv_ab_tensor_prod_rec

{A B A' B' : AbGroup} (f : A $-> A') (g : B $-> B') : ab_tensor_prod A B $-> ab_tensor_prod A' B'. Proof. snrapply ab_tensor_prod_rec'. - intro a. exact (grp_homo_tensor_l (f a) $o g). - intros a a' b; hnf. rewrite grp_homo_op. nrapply tensor_dist_r. Defined.

Definition

Require Import Basics.Overture Basics.Tactics. Require Import Types.Forall Types.Sigma Types.Prod. Require Import WildCat.Core WildCat.Equiv WildCat.Monoidal WildCat.Bifunctor. Require Import WildCat.NatTrans WildCat.MonoidalTwistConstruction. Require Import Algebra.Groups.Group Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Algebra.AbGroups.AbHom Algebra.AbGroups.FreeAbelianGroup. Require Import Algebra.AbGroups.Abelianization Algebra Algebra.Groups.FreeGroup. Require Import Colimits.Quotient. Require Import Spaces.List.Core Spaces.Int. Require Import AbGroups.Z. Require Import Truncations.

Algebra\AbGroups\TensorProduct.v

functor_ab_tensor_prod

{A B A' B' : AbGroup} {f f' : A $-> A'} (p : f $== f') {g g' : B $-> B'} (q : g $== g') : functor_ab_tensor_prod f g $== functor_ab_tensor_prod f' g'. Proof. snrapply ab_tensor_prod_ind_homotopy. intros a b; simpl. exact (ap011 tensor (p _) (q _)). Defined.

Definition

Require Import Basics.Overture Basics.Tactics. Require Import Types.Forall Types.Sigma Types.Prod. Require Import WildCat.Core WildCat.Equiv WildCat.Monoidal WildCat.Bifunctor. Require Import WildCat.NatTrans WildCat.MonoidalTwistConstruction. Require Import Algebra.Groups.Group Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Algebra.AbGroups.AbHom Algebra.AbGroups.FreeAbelianGroup. Require Import Algebra.AbGroups.Abelianization Algebra Algebra.Groups.FreeGroup. Require Import Colimits.Quotient. Require Import Spaces.List.Core Spaces.Int. Require Import AbGroups.Z. Require Import Truncations.

Algebra\AbGroups\TensorProduct.v

functor2_ab_tensor_prod

(A B : AbGroup) : functor_ab_tensor_prod (Id A) (Id B) $== Id (ab_tensor_prod A B). Proof. snrapply ab_tensor_prod_ind_homotopy. intros a b; simpl. reflexivity. Defined.

Definition

Require Import Basics.Overture Basics.Tactics. Require Import Types.Forall Types.Sigma Types.Prod. Require Import WildCat.Core WildCat.Equiv WildCat.Monoidal WildCat.Bifunctor. Require Import WildCat.NatTrans WildCat.MonoidalTwistConstruction. Require Import Algebra.Groups.Group Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Algebra.AbGroups.AbHom Algebra.AbGroups.FreeAbelianGroup. Require Import Algebra.AbGroups.Abelianization Algebra Algebra.Groups.FreeGroup. Require Import Colimits.Quotient. Require Import Spaces.List.Core Spaces.Int. Require Import AbGroups.Z. Require Import Truncations.

Algebra\AbGroups\TensorProduct.v

functor_ab_tensor_prod_id

{A B C A' B' C' : AbGroup} (f : A $-> B) (g : B $-> C) (f' : A' $-> B') (g' : B' $-> C') : functor_ab_tensor_prod (g $o f) (g' $o f') $== functor_ab_tensor_prod g g' $o functor_ab_tensor_prod f f'. Proof. snrapply ab_tensor_prod_ind_homotopy. intros a b; simpl. reflexivity. Defined.

Definition

Require Import Basics.Overture Basics.Tactics. Require Import Types.Forall Types.Sigma Types.Prod. Require Import WildCat.Core WildCat.Equiv WildCat.Monoidal WildCat.Bifunctor. Require Import WildCat.NatTrans WildCat.MonoidalTwistConstruction. Require Import Algebra.Groups.Group Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Algebra.AbGroups.AbHom Algebra.AbGroups.FreeAbelianGroup. Require Import Algebra.AbGroups.Abelianization Algebra Algebra.Groups.FreeGroup. Require Import Colimits.Quotient. Require Import Spaces.List.Core Spaces.Int. Require Import AbGroups.Z. Require Import Truncations.

Algebra\AbGroups\TensorProduct.v

functor_ab_tensor_prod_compose

{A B} : ab_tensor_prod A B $-> ab_tensor_prod B A. Proof. snrapply ab_tensor_prod_rec. - exact (flip tensor). - intros a b b'. apply tensor_dist_r. - intros a a' b. apply tensor_dist_l. Defined.

Definition

Require Import Basics.Overture Basics.Tactics. Require Import Types.Forall Types.Sigma Types.Prod. Require Import WildCat.Core WildCat.Equiv WildCat.Monoidal WildCat.Bifunctor. Require Import WildCat.NatTrans WildCat.MonoidalTwistConstruction. Require Import Algebra.Groups.Group Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Algebra.AbGroups.AbHom Algebra.AbGroups.FreeAbelianGroup. Require Import Algebra.AbGroups.Abelianization Algebra Algebra.Groups.FreeGroup. Require Import Colimits.Quotient. Require Import Spaces.List.Core Spaces.Int. Require Import AbGroups.Z. Require Import Truncations.

Algebra\AbGroups\TensorProduct.v

ab_tensor_swap

{A B} : ab_tensor_swap $o @ab_tensor_swap A B $== Id _. Proof. snrapply ab_tensor_prod_ind_homotopy. reflexivity. Defined.

Definition

Require Import Basics.Overture Basics.Tactics. Require Import Types.Forall Types.Sigma Types.Prod. Require Import WildCat.Core WildCat.Equiv WildCat.Monoidal WildCat.Bifunctor. Require Import WildCat.NatTrans WildCat.MonoidalTwistConstruction. Require Import Algebra.Groups.Group Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Algebra.AbGroups.AbHom Algebra.AbGroups.FreeAbelianGroup. Require Import Algebra.AbGroups.Abelianization Algebra Algebra.Groups.FreeGroup. Require Import Colimits.Quotient. Require Import Spaces.List.Core Spaces.Int. Require Import AbGroups.Z. Require Import Truncations.

Algebra\AbGroups\TensorProduct.v

ab_tensor_swap_swap

{A B A' B'} (f : A $-> A') (g : B $-> B') : ab_tensor_swap $o functor_ab_tensor_prod f g $== functor_ab_tensor_prod g f $o ab_tensor_swap. Proof. snrapply ab_tensor_prod_ind_homotopy. simpl. reflexivity. Defined.

Definition

Require Import Basics.Overture Basics.Tactics. Require Import Types.Forall Types.Sigma Types.Prod. Require Import WildCat.Core WildCat.Equiv WildCat.Monoidal WildCat.Bifunctor. Require Import WildCat.NatTrans WildCat.MonoidalTwistConstruction. Require Import Algebra.Groups.Group Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Algebra.AbGroups.AbHom Algebra.AbGroups.FreeAbelianGroup. Require Import Algebra.AbGroups.Abelianization Algebra Algebra.Groups.FreeGroup. Require Import Colimits.Quotient. Require Import Spaces.List.Core Spaces.Int. Require Import AbGroups.Z. Require Import Truncations.

Algebra\AbGroups\TensorProduct.v

ab_tensor_swap_natural

{A B C} : ab_tensor_prod A (ab_tensor_prod B C) $-> ab_tensor_prod B (ab_tensor_prod A C). Proof. snrapply ab_tensor_prod_rec'. - exact ab_tensor_prod_twist_map. - exact ab_tensor_prod_twist_map_additive_l. Defined.

Definition

Require Import Basics.Overture Basics.Tactics. Require Import Types.Forall Types.Sigma Types.Prod. Require Import WildCat.Core WildCat.Equiv WildCat.Monoidal WildCat.Bifunctor. Require Import WildCat.NatTrans WildCat.MonoidalTwistConstruction. Require Import Algebra.Groups.Group Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Algebra.AbGroups.AbHom Algebra.AbGroups.FreeAbelianGroup. Require Import Algebra.AbGroups.Abelianization Algebra Algebra.Groups.FreeGroup. Require Import Colimits.Quotient. Require Import Spaces.List.Core Spaces.Int. Require Import AbGroups.Z. Require Import Truncations.

Algebra\AbGroups\TensorProduct.v

ab_tensor_prod_twist

{A B C} : ab_tensor_prod_twist $o @ab_tensor_prod_twist A B C $== Id _. Proof. snrapply ab_tensor_prod_ind_homotopy_triple. reflexivity. Defined.

Definition

Require Import Basics.Overture Basics.Tactics. Require Import Types.Forall Types.Sigma Types.Prod. Require Import WildCat.Core WildCat.Equiv WildCat.Monoidal WildCat.Bifunctor. Require Import WildCat.NatTrans WildCat.MonoidalTwistConstruction. Require Import Algebra.Groups.Group Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Algebra.AbGroups.AbHom Algebra.AbGroups.FreeAbelianGroup. Require Import Algebra.AbGroups.Abelianization Algebra Algebra.Groups.FreeGroup. Require Import Colimits.Quotient. Require Import Spaces.List.Core Spaces.Int. Require Import AbGroups.Z. Require Import Truncations.

Algebra\AbGroups\TensorProduct.v

ab_tensor_prod_twist_twist

{A B C A' B' C'} (f : A $-> A') (g : B $-> B') (h : C $-> C') : ab_tensor_prod_twist $o fmap11 ab_tensor_prod f (fmap11 ab_tensor_prod g h) $== fmap11 ab_tensor_prod g (fmap11 ab_tensor_prod f h) $o ab_tensor_prod_twist. Proof. snrapply ab_tensor_prod_ind_homotopy_triple. intros a b c. change (tensor (g b) (tensor (f a) (h c)) = tensor (g b) (tensor (f a) (h c))). reflexivity. Defined.

Definition

Require Import Basics.Overture Basics.Tactics. Require Import Types.Forall Types.Sigma Types.Prod. Require Import WildCat.Core WildCat.Equiv WildCat.Monoidal WildCat.Bifunctor. Require Import WildCat.NatTrans WildCat.MonoidalTwistConstruction. Require Import Algebra.Groups.Group Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Algebra.AbGroups.AbHom Algebra.AbGroups.FreeAbelianGroup. Require Import Algebra.AbGroups.Abelianization Algebra Algebra.Groups.FreeGroup. Require Import Colimits.Quotient. Require Import Spaces.List.Core Spaces.Int. Require Import AbGroups.Z. Require Import Truncations.

Algebra\AbGroups\TensorProduct.v

ab_tensor_prod_twist_natural

{A B : AbGroup} (z : Int) (a : A) (b : B) : tensor (ab_mul z a) b = ab_mul z (tensor a b) := ab_mul_natural (grp_homo_tensor_r b) z a.

Definition

Require Import Basics.Overture Basics.Tactics. Require Import Types.Forall Types.Sigma Types.Prod. Require Import WildCat.Core WildCat.Equiv WildCat.Monoidal WildCat.Bifunctor. Require Import WildCat.NatTrans WildCat.MonoidalTwistConstruction. Require Import Algebra.Groups.Group Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Algebra.AbGroups.AbHom Algebra.AbGroups.FreeAbelianGroup. Require Import Algebra.AbGroups.Abelianization Algebra Algebra.Groups.FreeGroup. Require Import Colimits.Quotient. Require Import Spaces.List.Core Spaces.Int. Require Import AbGroups.Z. Require Import Truncations.

Algebra\AbGroups\TensorProduct.v

tensor_ab_mul_l

{A B : AbGroup} (z : Int) (a : A) (b : B) : tensor a (ab_mul z b) = ab_mul z (tensor a b) := ab_mul_natural (grp_homo_tensor_l a) z b.

Definition

Require Import Basics.Overture Basics.Tactics. Require Import Types.Forall Types.Sigma Types.Prod. Require Import WildCat.Core WildCat.Equiv WildCat.Monoidal WildCat.Bifunctor. Require Import WildCat.NatTrans WildCat.MonoidalTwistConstruction. Require Import Algebra.Groups.Group Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Algebra.AbGroups.AbHom Algebra.AbGroups.FreeAbelianGroup. Require Import Algebra.AbGroups.Abelianization Algebra Algebra.Groups.FreeGroup. Require Import Colimits.Quotient. Require Import Spaces.List.Core Spaces.Int. Require Import AbGroups.Z. Require Import Truncations.

Algebra\AbGroups\TensorProduct.v

tensor_ab_mul_r

{A B : AbGroup} (z : Int) (a : A) (b : B) : tensor (ab_mul z a) b = tensor a (ab_mul z b). Proof. rhs nrapply tensor_ab_mul_r. nrapply tensor_ab_mul_l. Defined.

Definition

Require Import Basics.Overture Basics.Tactics. Require Import Types.Forall Types.Sigma Types.Prod. Require Import WildCat.Core WildCat.Equiv WildCat.Monoidal WildCat.Bifunctor. Require Import WildCat.NatTrans WildCat.MonoidalTwistConstruction. Require Import Algebra.Groups.Group Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Algebra.AbGroups.AbHom Algebra.AbGroups.FreeAbelianGroup. Require Import Algebra.AbGroups.Abelianization Algebra Algebra.Groups.FreeGroup. Require Import Colimits.Quotient. Require Import Spaces.List.Core Spaces.Int. Require Import AbGroups.Z. Require Import Truncations.

Algebra\AbGroups\TensorProduct.v

tensor_ab_mul

{A} : ab_tensor_prod A abgroup_Z $<~> A. Proof. symmetry. snrapply Build_GroupIsomorphism. - nrapply grp_homo_tensor_r. exact 1%int. - snrapply isequiv_adjointify. + snrapply ab_tensor_prod_rec'. * exact grp_pow_homo. * intros a a' z; cbn beta. nrapply (grp_homo_op (ab_mul z)). + hnf. change (forall x : ?A, (grp_homo_map ?f) ((grp_homo_map ?g) x) = x) with (f $o g $== Id _). snrapply ab_tensor_prod_ind_homotopy. intros a z. change (tensor (B:=abgroup_Z) (grp_pow a z) 1%int = tensor a z). lhs nrapply tensor_ab_mul. nrapply ap. lhs nrapply abgroup_Z_ab_mul. apply int_mul_1_r. + exact grp_unit_r. Defined.

Definition

Require Import Basics.Overture Basics.Tactics. Require Import Types.Forall Types.Sigma Types.Prod. Require Import WildCat.Core WildCat.Equiv WildCat.Monoidal WildCat.Bifunctor. Require Import WildCat.NatTrans WildCat.MonoidalTwistConstruction. Require Import Algebra.Groups.Group Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Algebra.AbGroups.AbHom Algebra.AbGroups.FreeAbelianGroup. Require Import Algebra.AbGroups.Abelianization Algebra Algebra.Groups.FreeGroup. Require Import Colimits.Quotient. Require Import Spaces.List.Core Spaces.Int. Require Import AbGroups.Z. Require Import Truncations.

Algebra\AbGroups\TensorProduct.v

ab_tensor_prod_Z_r

A {B C} (f g : B $-> C) : ab_coeq (fmap01 ab_tensor_prod A f) (fmap01 ab_tensor_prod A g) $<~> ab_tensor_prod A (ab_coeq f g). Proof. snrapply cate_adjointify. - snrapply ab_coeq_rec. + rapply (fmap01 ab_tensor_prod A). nrapply ab_coeq_in. + refine (_^$ $@ fmap02 ab_tensor_prod _ _ $@ _). 1,3: rapply fmap01_comp. nrapply ab_coeq_glue. - snrapply ab_tensor_prod_rec'. + intros a. snrapply functor_ab_coeq. 1,2: snrapply (grp_homo_tensor_l a). 1,2: hnf; reflexivity. + intros a a'; cbn beta. srapply ab_coeq_ind_hprop. intros x. exact (ap (ab_coeq_in (f:=fmap01 ab_tensor_prod A f) (g:=fmap01 ab_tensor_prod A g)) (tensor_dist_r a a' x)). - snrapply ab_tensor_prod_ind_homotopy. intros a. srapply ab_coeq_ind_hprop. intros c. reflexivity. - snrapply ab_coeq_ind_homotopy. snrapply ab_tensor_prod_ind_homotopy. reflexivity. Defined.

Definition

Require Import Basics.Overture Basics.Tactics. Require Import Types.Forall Types.Sigma Types.Prod. Require Import WildCat.Core WildCat.Equiv WildCat.Monoidal WildCat.Bifunctor. Require Import WildCat.NatTrans WildCat.MonoidalTwistConstruction. Require Import Algebra.Groups.Group Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Algebra.AbGroups.AbHom Algebra.AbGroups.FreeAbelianGroup. Require Import Algebra.AbGroups.Abelianization Algebra Algebra.Groups.FreeGroup. Require Import Colimits.Quotient. Require Import Spaces.List.Core Spaces.Int. Require Import AbGroups.Z. Require Import Truncations.

Algebra\AbGroups\TensorProduct.v

grp_iso_ab_tensor_prod_coeq_l

A {B C} (f g : B $-> C) : grp_iso_ab_tensor_prod_coeq_l A f g $o ab_coeq_in $== fmap01 ab_tensor_prod A ab_coeq_in. Proof. snrapply ab_tensor_prod_ind_homotopy. reflexivity. Defined.

Definition

Require Import Basics.Overture Basics.Tactics. Require Import Types.Forall Types.Sigma Types.Prod. Require Import WildCat.Core WildCat.Equiv WildCat.Monoidal WildCat.Bifunctor. Require Import WildCat.NatTrans WildCat.MonoidalTwistConstruction. Require Import Algebra.Groups.Group Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Algebra.AbGroups.AbHom Algebra.AbGroups.FreeAbelianGroup. Require Import Algebra.AbGroups.Abelianization Algebra Algebra.Groups.FreeGroup. Require Import Colimits.Quotient. Require Import Spaces.List.Core Spaces.Int. Require Import AbGroups.Z. Require Import Truncations.

Algebra\AbGroups\TensorProduct.v

ab_tensor_prod_coeq_l_triangle

{A B} (f g : A $-> B) C : ab_coeq (fmap10 ab_tensor_prod f C) (fmap10 ab_tensor_prod g C) $<~> ab_tensor_prod (ab_coeq f g) C. Proof. refine (braide _ _ $oE _). nrefine (grp_iso_ab_tensor_prod_coeq_l _ f g $oE _). snrapply grp_iso_ab_coeq. 1,2: rapply braide. 1,2: symmetry; nrapply ab_tensor_swap_natural. Defined.

Definition

Require Import Basics.Overture Basics.Tactics. Require Import Types.Forall Types.Sigma Types.Prod. Require Import WildCat.Core WildCat.Equiv WildCat.Monoidal WildCat.Bifunctor. Require Import WildCat.NatTrans WildCat.MonoidalTwistConstruction. Require Import Algebra.Groups.Group Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Algebra.AbGroups.AbHom Algebra.AbGroups.FreeAbelianGroup. Require Import Algebra.AbGroups.Abelianization Algebra Algebra.Groups.FreeGroup. Require Import Colimits.Quotient. Require Import Spaces.List.Core Spaces.Int. Require Import AbGroups.Z. Require Import Truncations.

Algebra\AbGroups\TensorProduct.v

grp_iso_ab_tensor_prod_coeq_r

{A B} (f g : A $-> B) C : grp_iso_ab_tensor_prod_coeq_r f g C $o ab_coeq_in $== fmap10 ab_tensor_prod ab_coeq_in C. Proof. snrapply ab_tensor_prod_ind_homotopy. reflexivity. Defined.

Definition

Require Import Basics.Overture Basics.Tactics. Require Import Types.Forall Types.Sigma Types.Prod. Require Import WildCat.Core WildCat.Equiv WildCat.Monoidal WildCat.Bifunctor. Require Import WildCat.NatTrans WildCat.MonoidalTwistConstruction. Require Import Algebra.Groups.Group Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Algebra.AbGroups.AbHom Algebra.AbGroups.FreeAbelianGroup. Require Import Algebra.AbGroups.Abelianization Algebra Algebra.Groups.FreeGroup. Require Import Colimits.Quotient. Require Import Spaces.List.Core Spaces.Int. Require Import AbGroups.Z. Require Import Truncations.

Algebra\AbGroups\TensorProduct.v

ab_tensor_prod_coeq_r_triangle

X Y : FreeAbGroup (X * Y) $<~> ab_tensor_prod (FreeAbGroup X) (FreeAbGroup Y). Proof. srefine (let f:=_ in let g:=_ in cate_adjointify f g _ _). - snrapply FreeAbGroup_rec. intros [x y]. exact (tensor (freeabgroup_in x) (freeabgroup_in y)). - snrapply ab_tensor_prod_rec. + intros x. snrapply FreeAbGroup_rec. intros y; revert x. unfold FreeAbGroup. snrapply FreeAbGroup_rec. intros x. apply abel_unit. apply freegroup_in. exact (x, y). + intros x y y'. snrapply grp_homo_op. + intros x x'. rapply Abel_ind_hprop. snrapply (FreeGroup_ind_homotopy _ (f' := ab_homo_add _ _)). intros y. lhs nrapply FreeGroup_rec_beta. lhs nrapply grp_homo_op. snrapply (ap011 (+) _^ _^). 1,2: nrapply FreeGroup_rec_beta. - snrapply ab_tensor_prod_ind_homotopy. intros x. change (f $o g $o grp_homo_tensor_l x $== grp_homo_tensor_l x). rapply Abel_ind_hprop. change (@abel_in ?G) with (grp_homo_map (@abel_unit G)). repeat change (cat_comp (A:=AbGroup) ?f ?g) with (cat_comp (A:=Group) f g). change (forall y, grp_homo_map ?f (abel_unit y) = grp_homo_map ?g (abel_unit y)) with (cat_comp (A:=Group) f abel_unit $== cat_comp (A:=Group) g abel_unit). rapply FreeGroup_ind_homotopy. intros y; revert x. change (f $o g $o grp_homo_tensor_r (freeabgroup_in y) $== grp_homo_tensor_r (freeabgroup_in y)). rapply Abel_ind_hprop. change (@abel_in ?G) with (grp_homo_map (@abel_unit G)). repeat change (cat_comp (A:=AbGroup) ?f ?g) with (cat_comp (A:=Group) f g). change (forall y, grp_homo_map ?f (abel_unit y) = grp_homo_map ?g (abel_unit y)) with (cat_comp (A:=Group) f abel_unit $== cat_comp (A:=Group) g abel_unit). rapply FreeGroup_ind_homotopy. intros x. reflexivity. - rapply Abel_ind_hprop. change (GpdHom (A:=Hom(A:=Group) (FreeGroup (X * Y)) _) (cat_comp (A:=Group) (g $o f) (@abel_unit (FreeGroup (X * Y)))) (@abel_unit (FreeGroup (X * Y)))). snrapply FreeGroup_ind_homotopy. reflexivity. Defined.

Definition

Require Import Basics.Overture Basics.Tactics. Require Import Types.Forall Types.Sigma Types.Prod. Require Import WildCat.Core WildCat.Equiv WildCat.Monoidal WildCat.Bifunctor. Require Import WildCat.NatTrans WildCat.MonoidalTwistConstruction. Require Import Algebra.Groups.Group Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Algebra.AbGroups.AbHom Algebra.AbGroups.FreeAbelianGroup. Require Import Algebra.AbGroups.Abelianization Algebra Algebra.Groups.FreeGroup. Require Import Colimits.Quotient. Require Import Spaces.List.Core Spaces.Int. Require Import AbGroups.Z. Require Import Truncations.

Algebra\AbGroups\TensorProduct.v

equiv_ab_tensor_prod_freeabgroup

{A B C : AbGroup} : ab_tensor_prod A (ab_biprod B C) $<~> ab_biprod (ab_tensor_prod A B) (ab_tensor_prod A C). Proof. srapply (let f := _ in let g := _ in cate_adjointify f g _ _). - snrapply ab_tensor_prod_rec. + intros a bc. exact (tensor a (fst bc), tensor a (snd bc)). + intros a bc bc'; cbn beta. snrapply path_prod'; snrapply tensor_dist_l. + intros a a' bc; cbn beta. snrapply path_prod; snrapply tensor_dist_r. - snrapply ab_biprod_rec. + exact (fmap01 ab_tensor_prod A ab_biprod_inl). + exact (fmap01 ab_tensor_prod A ab_biprod_inr). - snrapply ab_biprod_ind_homotopy. + refine (cat_assoc _ _ _ $@ (_ $@L _) $@ _). 1: snrapply ab_biprod_rec_beta_inl. snrapply ab_tensor_prod_ind_homotopy. intros a b. snrapply path_prod; simpl. * reflexivity. * snrapply tensor_zero_r. + refine (cat_assoc _ _ _ $@ (_ $@L _) $@ _). 1: snrapply ab_biprod_rec_beta_inr. snrapply ab_tensor_prod_ind_homotopy. intros a b. snrapply path_prod; simpl. * snrapply tensor_zero_r. * reflexivity. - snrapply ab_tensor_prod_ind_homotopy. intros a [b c]. lhs_V nrapply tensor_dist_l; simpl. snrapply ap. symmetry; apply grp_prod_decompose. Defined.

Definition

Require Import Basics.Overture Basics.Tactics. Require Import Types.Forall Types.Sigma Types.Prod. Require Import WildCat.Core WildCat.Equiv WildCat.Monoidal WildCat.Bifunctor. Require Import WildCat.NatTrans WildCat.MonoidalTwistConstruction. Require Import Algebra.Groups.Group Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Algebra.AbGroups.AbHom Algebra.AbGroups.FreeAbelianGroup. Require Import Algebra.AbGroups.Abelianization Algebra Algebra.Groups.FreeGroup. Require Import Colimits.Quotient. Require Import Spaces.List.Core Spaces.Int. Require Import AbGroups.Z. Require Import Truncations.

Algebra\AbGroups\TensorProduct.v

ab_tensor_prod_dist_l

{A B C : AbGroup} : ab_tensor_prod (ab_biprod A B) C $<~> ab_biprod (ab_tensor_prod A C) (ab_tensor_prod B C). Proof. refine (emap11 ab_biprod (braide _ _) (braide _ _) $oE _ $oE braide _ _). snrapply ab_tensor_prod_dist_l. Defined.

Definition

Require Import Basics.Overture Basics.Tactics. Require Import Types.Forall Types.Sigma Types.Prod. Require Import WildCat.Core WildCat.Equiv WildCat.Monoidal WildCat.Bifunctor. Require Import WildCat.NatTrans WildCat.MonoidalTwistConstruction. Require Import Algebra.Groups.Group Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Algebra.AbGroups.AbHom Algebra.AbGroups.FreeAbelianGroup. Require Import Algebra.AbGroups.Abelianization Algebra Algebra.Groups.FreeGroup. Require Import Colimits.Quotient. Require Import Spaces.List.Core Spaces.Int. Require Import AbGroups.Z. Require Import Truncations.

Algebra\AbGroups\TensorProduct.v

ab_tensor_prod_dist_r

abgroup_Z@{} : AbGroup@{Set}. Proof. snrapply Build_AbGroup'. - exact Int. - exact 0. - exact int_neg. - exact int_add. - exact _. - exact int_add_comm. - exact int_add_assoc. - exact int_add_0_l. - exact int_add_neg_l. Defined.

Definition

Require Import Basics. Require Import Spaces.Pos.Core Spaces.Int. Require Import Algebra.AbGroups.AbelianGroup.

Algebra\AbGroups\Z.v

abgroup_Z@

{G : Group} (g : G) : GroupHomomorphism abgroup_Z G. Proof. snrapply Build_GroupHomomorphism. 1: exact (grp_pow g). intros m n; apply grp_pow_add. Defined.

Definition

Require Import Basics. Require Import Spaces.Pos.Core Spaces.Int. Require Import Algebra.AbGroups.AbelianGroup.

Algebra\AbGroups\Z.v

grp_pow_homo

(z z' : Int) : ab_mul (A:=abgroup_Z) z z' = z * z'. Proof. induction z. - reflexivity. - cbn. lhs nrapply (grp_pow_succ (G:=abgroup_Z)). rhs nrapply int_mul_succ_l. f_ap. - cbn. lhs nrapply (grp_pow_pred (G:=abgroup_Z)). rhs nrapply int_mul_pred_l. f_ap. Defined.

Definition

Require Import Basics. Require Import Spaces.Pos.Core Spaces.Int. Require Import Algebra.AbGroups.AbelianGroup.

Algebra\AbGroups\Z.v

abgroup_Z_ab_mul

`{Univalence} {B A : AbGroup@{u}} (E F : AbSES B A) : AbSES B A := abses_pullback ab_diagonal (abses_pushout ab_codiagonal (abses_direct_sum E F)).

Definition

Require Import Basics Types. Require Import WildCat Pointed.Core. Require Import AbGroups.AbelianGroup AbGroups.Biproduct AbGroups.AbHom. Require Import AbSES.Core AbSES.Pullback AbSES.Pushout AbSES.DirectSum. Require Import Homotopy.HSpace.Core.

Algebra\AbSES\BaerSum.v

abses_baer_sum

abses_pushout_is_pullback' `{Univalence} {A A' B B' : AbGroup@{u}} {E : AbSES B A} {E' : AbSES B' A'} (f : AbSESMorphism E E') : abses_pushout (component1 f) E $== abses_pullback (component3 f) E'. Proof. exact (abses_pullback_component1_id' (abses_pushout_morphism_rec f) (fun _ => idpath)). Defined.

Lemma

Require Import Basics Types. Require Import WildCat Pointed.Core. Require Import AbGroups.AbelianGroup AbGroups.Biproduct AbGroups.AbHom. Require Import AbSES.Core AbSES.Pullback AbSES.Pushout AbSES.DirectSum. Require Import Homotopy.HSpace.Core.

Algebra\AbSES\BaerSum.v

abses_pushout_is_pullback'

`{Univalence} {A A' B B' : AbGroup} {E : AbSES B A} {E' : AbSES B' A'} (f : AbSESMorphism E E') : abses_pushout (component1 f) E = abses_pullback (component3 f) E' := equiv_path_abses_iso (' f).

Definition

Require Import Basics Types. Require Import WildCat Pointed.Core. Require Import AbGroups.AbelianGroup AbGroups.Biproduct AbGroups.AbHom. Require Import AbSES.Core AbSES.Pullback AbSES.Pushout AbSES.DirectSum. Require Import Homotopy.HSpace.Core.

Algebra\AbSES\BaerSum.v

abses_pushout_is_pullback

abses_pushout_pullback_reorder' `{Univalence} {A A' B B' : AbGroup} (E : AbSES B A) (f : A $-> A') (g : B' $-> B) : abses_pushout f (abses_pullback g E) $== abses_pullback g (abses_pushout f E). Proof. pose (F := absesmorphism_compose (abses_pushout_morphism E f) (abses_pullback_morphism E g)). refine (abses_pushout_is_pullback' (Build_AbSESMorphism f (component2 F) g _ _)); apply F. Defined.

Definition

Require Import Basics Types. Require Import WildCat Pointed.Core. Require Import AbGroups.AbelianGroup AbGroups.Biproduct AbGroups.AbHom. Require Import AbSES.Core AbSES.Pullback AbSES.Pushout AbSES.DirectSum. Require Import Homotopy.HSpace.Core.

Algebra\AbSES\BaerSum.v

abses_pushout_pullback_reorder'

`{Univalence} {A A' B B' : AbGroup} (E : AbSES B A) (f : A $-> A') (g : B' $-> B) : abses_pushout f (abses_pullback g E) = abses_pullback g (abses_pushout f E). Proof. apply equiv_path_abses_iso. apply '. Defined.

Definition

Require Import Basics Types. Require Import WildCat Pointed.Core. Require Import AbGroups.AbelianGroup AbGroups.Biproduct AbGroups.AbHom. Require Import AbSES.Core AbSES.Pullback AbSES.Pushout AbSES.DirectSum. Require Import Homotopy.HSpace.Core.

Algebra\AbSES\BaerSum.v

abses_pushout_pullback_reorder

`{Univalence} {A A' B B' : AbGroup} (E : AbSES B A) (f : A $-> A') (g : B' $-> B) : abses_pushout f (abses_pullback g E) = abses_pullback g (abses_pushout f E). Proof. pose (F := absesmorphism_compose (abses_pushout_morphism E f) (abses_pullback_morphism E g)). refine (abses_pushout_is_pullback (Build_AbSESMorphism f (component2 F) g _ _)); apply F. Defined.

Lemma

Require Import Basics Types. Require Import WildCat Pointed.Core. Require Import AbGroups.AbelianGroup AbGroups.Biproduct AbGroups.AbHom. Require Import AbSES.Core AbSES.Pullback AbSES.Pushout AbSES.DirectSum. Require Import Homotopy.HSpace.Core.

Algebra\AbSES\BaerSum.v

abses_reorder_pullback_pushout

`{Univalence} {A B B' : AbGroup} {E : AbSES B A} (f g : ab_hom B' B) : abses_pullback (f + g) E = abses_baer_sum (abses_pullback f E) (abses_pullback g E). Proof. unfold abses_baer_sum. refine ((abses_pullback_compose (B1:=ab_biprod B B) _ _ E)^ @ _). refine (ap (abses_pullback _) (abses_pushout_is_pullback (abses_codiagonal E))^ @ _). unfold abses_codiagonal, component1. refine (_^ @ _ @ _). 1,3: apply abses_reorder_pullback_pushout. refine (ap (abses_pushout _) _). refine (ap (fun h => abses_pullback h _) (ab_biprod_corec_diagonal _ _) @ _). refine ((abses_pullback_compose _ _ (abses_direct_sum E E))^ @ _). exact (ap (abses_pullback _) (abses_directsum_distributive_pullbacks f g)). Defined.

Lemma

Require Import Basics Types. Require Import WildCat Pointed.Core. Require Import AbGroups.AbelianGroup AbGroups.Biproduct AbGroups.AbHom. Require Import AbSES.Core AbSES.Pullback AbSES.Pushout AbSES.DirectSum. Require Import Homotopy.HSpace.Core.

Algebra\AbSES\BaerSum.v

baer_sum_distributive_pullbacks

`{Univalence} {A B : AbGroup} (E F : AbSES B A) : abses_baer_sum E F = abses_baer_sum F E. Proof. unfold abses_baer_sum. refine (_ @ abses_pullback_compose ab_diagonal direct_sum_swap _). refine (ap (abses_pullback ab_diagonal) _). refine (ap (fun f => abses_pushout f _) ab_codiagonal_swap^ @ _). refine ((abses_pushout_compose _ _ _) @ _). refine (ap _ (abses_pushout_is_pullback (abses_swap_morphism E F)) @ _). unfold abses_swap_morphism, component3. apply abses_pushout_pullback_reorder. Defined.

Lemma

Require Import Basics Types. Require Import WildCat Pointed.Core. Require Import AbGroups.AbelianGroup AbGroups.Biproduct AbGroups.AbHom. Require Import AbSES.Core AbSES.Pullback AbSES.Pushout AbSES.DirectSum. Require Import Homotopy.HSpace.Core.

Algebra\AbSES\BaerSum.v

baer_sum_commutative

`{Univalence} {A B : AbGroup} (E : AbSES B A) : abses_baer_sum E (point (AbSES B A)) = E. Proof. refine (ap (abses_baer_sum E) _ @ _). - exact (abses_pullback_const E). - refine (ap (fun F => abses_baer_sum F (abses_pullback grp_homo_const E)) (abses_pullback_id E)^ @ _). refine ((baer_sum_distributive_pullbacks grp_homo_id grp_homo_const)^ @ _). refine (ap (fun f => abses_pullback f E) (grp_unit_r (G := ab_hom _ _) _) @ _). apply abses_pullback_id. Defined.

Lemma

Require Import Basics Types. Require Import WildCat Pointed.Core. Require Import AbGroups.AbelianGroup AbGroups.Biproduct AbGroups.AbHom. Require Import AbSES.Core AbSES.Pullback AbSES.Pushout AbSES.DirectSum. Require Import Homotopy.HSpace.Core.

Algebra\AbSES\BaerSum.v

baer_sum_unit_r

`{Univalence} {A B : AbGroup} (E : AbSES B A) : abses_baer_sum (point (AbSES B A)) E = E := baer_sum_commutative _ _ @ baer_sum_unit_r _.

Definition

Require Import Basics Types. Require Import WildCat Pointed.Core. Require Import AbGroups.AbelianGroup AbGroups.Biproduct AbGroups.AbHom. Require Import AbSES.Core AbSES.Pullback AbSES.Pushout AbSES.DirectSum. Require Import Homotopy.HSpace.Core.

Algebra\AbSES\BaerSum.v

baer_sum_unit_l

`{Univalence} {A B : AbGroup} (E : AbSES B A) : abses_baer_sum E (abses_pullback (- grp_homo_id) E) = point (AbSES B A). Proof. refine (ap (fun F => abses_baer_sum F (abses_pullback _ E)) (abses_pullback_id E)^ @ _). refine ((baer_sum_distributive_pullbacks grp_homo_id (-grp_homo_id))^ @ _). refine (ap (fun f => abses_pullback f _) (grp_inv_r (G := ab_hom _ _) _) @ _). symmetry; apply abses_pullback_const. Defined.

Lemma

Require Import Basics Types. Require Import WildCat Pointed.Core. Require Import AbGroups.AbelianGroup AbGroups.Biproduct AbGroups.AbHom. Require Import AbSES.Core AbSES.Pullback AbSES.Pushout AbSES.DirectSum. Require Import Homotopy.HSpace.Core.

Algebra\AbSES\BaerSum.v

baer_sum_inverse_l

`{Univalence} {A B : AbGroup} (E : AbSES B A) : abses_baer_sum (abses_pullback (-grp_homo_id) E) E = point (AbSES B A) := baer_sum_commutative _ _ @ baer_sum_inverse_l _.

Definition

Require Import Basics Types. Require Import WildCat Pointed.Core. Require Import AbGroups.AbelianGroup AbGroups.Biproduct AbGroups.AbHom. Require Import AbSES.Core AbSES.Pullback AbSES.Pushout AbSES.DirectSum. Require Import Homotopy.HSpace.Core.

Algebra\AbSES\BaerSum.v

baer_sum_inverse_r

`{Univalence} {A A' B : AbGroup} {E : AbSES B A'} (f g : ab_hom A' A) : abses_pushout (f + g) E = abses_baer_sum (abses_pushout f E) (abses_pushout g E). Proof. unfold abses_baer_sum. refine (abses_pushout_compose (A1 := ab_biprod A A) _ _ E @ _). refine (_ @ abses_pushout_pullback_reorder _ _ _). refine (ap (abses_pushout ab_codiagonal) _). refine (ap (fun f => abses_pushout f E) (ab_biprod_corec_diagonal f g) @ _). refine (abses_pushout_compose _ _ E @ _). refine (ap (abses_pushout _) (abses_pushout_is_pullback (abses_diagonal E)) @ _). refine (abses_pushout_pullback_reorder _ _ _ @ _). exact (ap (abses_pullback _) (abses_directsum_distributive_pushouts f g)). Defined.

Lemma

Require Import Basics Types. Require Import WildCat Pointed.Core. Require Import AbGroups.AbelianGroup AbGroups.Biproduct AbGroups.AbHom. Require Import AbSES.Core AbSES.Pullback AbSES.Pushout AbSES.DirectSum. Require Import Homotopy.HSpace.Core.

Algebra\AbSES\BaerSum.v

baer_sum_distributive_pushouts

`{Univalence} {A B : AbGroup@{u}} (E F G : AbSES B A) : AbSES B A := abses_pullback ab_triagonal (abses_pushout ab_cotriagonal (abses_direct_sum (abses_direct_sum E F) G)).

Definition

Require Import Basics Types. Require Import WildCat Pointed.Core. Require Import AbGroups.AbelianGroup AbGroups.Biproduct AbGroups.AbHom. Require Import AbSES.Core AbSES.Pullback AbSES.Pushout AbSES.DirectSum. Require Import Homotopy.HSpace.Core.

Algebra\AbSES\BaerSum.v

abses_trinary_baer_sum

`{Univalence} {A B : AbGroup@{u}} (E F G : AbSES B A) : abses_baer_sum (abses_baer_sum E F) G = abses_trinary_baer_sum E F G. Proof. unfold abses_baer_sum, abses_trinary_baer_sum, ab_triagonal, ab_cotriagonal. refine (ap (abses_pullback _ o abses_pushout _) _^ @ _). - refine (_ @ ap (abses_direct_sum _) (abses_pullback_id G)). refine (_ @ abses_directsum_distributive_pullbacks _ _). refine (ap (abses_pullback _) _). refine (_ @ ap (abses_direct_sum _) (abses_pushout_id G)). apply abses_directsum_distributive_pushouts. - refine (ap (abses_pullback _) (abses_pushout_pullback_reorder _ _ _) @ _). refine (abses_pullback_compose _ _ _ @ _). refine (ap (abses_pullback _) _^). apply abses_pushout_compose. Defined.

Lemma

Require Import Basics Types. Require Import WildCat Pointed.Core. Require Import AbGroups.AbelianGroup AbGroups.Biproduct AbGroups.AbHom. Require Import AbSES.Core AbSES.Pullback AbSES.Pushout AbSES.DirectSum. Require Import Homotopy.HSpace.Core.

Algebra\AbSES\BaerSum.v

baer_sum_is_trinary

`{Univalence} {A B : AbGroup@{u}} (E F G : AbSES B A) : abses_trinary_baer_sum E F G = abses_trinary_baer_sum G F E. Proof. unfold abses_trinary_baer_sum. refine (_ @ abses_pullback_compose ab_triagonal ab_biprod_twist _). refine (ap (abses_pullback _) _). refine (ap (fun f => abses_pushout f _) ab_cotriagonal_twist^ @ _). refine (abses_pushout_compose _ _ _ @ _). refine (ap _ (abses_pushout_is_pullback (abses_twist_directsum E F G)) @ _). unfold abses_twist_directsum, component3. exact (abses_pushout_pullback_reorder _ _ _). Defined.

Lemma

Require Import Basics Types. Require Import WildCat Pointed.Core. Require Import AbGroups.AbelianGroup AbGroups.Biproduct AbGroups.AbHom. Require Import AbSES.Core AbSES.Pullback AbSES.Pushout AbSES.DirectSum. Require Import Homotopy.HSpace.Core.

Algebra\AbSES\BaerSum.v

twist_trinary_baer_sum

`{Univalence} {A B : AbGroup@{u}} (E F G : AbSES B A) : abses_baer_sum (abses_baer_sum E F) G = abses_baer_sum (abses_baer_sum G F) E. Proof. refine ((baer_sum_is_trinary E F G) @ _ @ (baer_sum_is_trinary G F E)^). apply twist_trinary_baer_sum. Defined.

Lemma

Require Import Basics Types. Require Import WildCat Pointed.Core. Require Import AbGroups.AbelianGroup AbGroups.Biproduct AbGroups.AbHom. Require Import AbSES.Core AbSES.Pullback AbSES.Pushout AbSES.DirectSum. Require Import Homotopy.HSpace.Core.

Algebra\AbSES\BaerSum.v

baer_sum_twist

`{Univalence} {A B : AbGroup@{u}} (E F G : AbSES B A) : abses_baer_sum (abses_baer_sum E F) G = abses_baer_sum E (abses_baer_sum F G). Proof. refine ((baer_sum_twist _ _ _)^ @ _). refine (baer_sum_commutative _ _ @ _). apply ap. apply baer_sum_commutative. Defined.

Lemma

Require Import Basics Types. Require Import WildCat Pointed.Core. Require Import AbGroups.AbelianGroup AbGroups.Biproduct AbGroups.AbHom. Require Import AbSES.Core AbSES.Pullback AbSES.Pushout AbSES.DirectSum. Require Import Homotopy.HSpace.Core.

Algebra\AbSES\BaerSum.v

baer_sum_associative

`{Univalence} {A A' B : AbGroup} (f : A $-> A') (E F : AbSES B A) : abses_pushout f (abses_baer_sum E F) = abses_baer_sum (abses_pushout f E) (abses_pushout f F). Proof. unfold abses_baer_sum. refine (abses_pushout_pullback_reorder _ _ _ @ ap _ _). refine ((abses_pushout_compose _ _ _)^ @ _). refine (abses_pushout_homotopic _ _ _ _ @ _). 1: apply ab_codiagonal_natural. refine (abses_pushout_compose _ _ _ @ ap _ _). apply abses_directsum_distributive_pushouts. Defined.

Definition

Require Import Basics Types. Require Import WildCat Pointed.Core. Require Import AbGroups.AbelianGroup AbGroups.Biproduct AbGroups.AbHom. Require Import AbSES.Core AbSES.Pullback AbSES.Pushout AbSES.DirectSum. Require Import Homotopy.HSpace.Core.

Algebra\AbSES\BaerSum.v

baer_sum_pushout

`{Univalence} {A B B' : AbGroup} (f : B' $-> B) (E F : AbSES B A) : abses_pullback f (abses_baer_sum E F) = abses_baer_sum (abses_pullback f E) (abses_pullback f F). Proof. unfold abses_baer_sum. refine (abses_pullback_compose _ _ _ @ _). refine ((abses_pushout_pullback_reorder _ _ _)^ @ ap _ _ @ abses_pushout_pullback_reorder _ _ _). refine (abses_pullback_homotopic _ (functor_ab_biprod f f $o ab_diagonal) _ _ @ _). 1: reflexivity. refine ((abses_pullback_compose _ _ _)^ @ ap _ _). apply abses_directsum_distributive_pullbacks. Defined.

Definition

Require Import Basics Types. Require Import WildCat Pointed.Core. Require Import AbGroups.AbelianGroup AbGroups.Biproduct AbGroups.AbHom. Require Import AbSES.Core AbSES.Pullback AbSES.Pushout AbSES.DirectSum. Require Import Homotopy.HSpace.Core.

Algebra\AbSES\BaerSum.v

baer_sum_pullback

AbSES' {B A : AbGroup@{u}} := Build_AbSES { middle : AbGroup@{u}; inclusion : A $-> middle; projection : middle $-> B; isembedding_inclusion : IsEmbedding inclusion; issurjection_projection : IsSurjection projection; isexact_inclusion_projection : IsExact (Tr (-1)) inclusion projection; }.

Record

Require Import Basics Types Truncations.Core. Require Import HSet WildCat. Require Import Groups.QuotientGroup Groups.ShortExactSequence. Require Import AbelianGroup AbGroups.Biproduct AbHom. Require Import Homotopy.ExactSequence Pointed. Require Import Modalities.ReflectiveSubuniverse.

Algebra\AbSES\Core.v

AbSES'

{B A : AbGroup} : {X : {E : AbGroup & (A $-> E) * (E $-> B)} & (IsEmbedding (fst X.

Definition

Require Import Basics Types Truncations.Core. Require Import HSet WildCat. Require Import Groups.QuotientGroup Groups.ShortExactSequence. Require Import AbelianGroup AbGroups.Biproduct AbHom. Require Import Homotopy.ExactSequence Pointed. Require Import Modalities.ReflectiveSubuniverse.

Algebra\AbSES\Core.v

issig_abses

{A B : AbGroup} (E : AbSES' B A) : IsComplex (inclusion E) (projection E) := cx_isexact. Global Instance ispointed_abses {B A : AbGroup@{u}} : IsPointed (AbSES' B A). Proof. rapply (Build_AbSES (ab_biprod A B) ab_biprod_inl ab_biprod_pr2). snrapply Build_IsExact. - srapply phomotopy_homotopy_hset; reflexivity. - intros [[a b] p]; cbn; cbn in p. rapply contr_inhabited_hprop. apply tr. exists a. rapply path_sigma_hprop; cbn. exact (path_prod' idpath p^). Defined.

Definition

Require Import Basics Types Truncations.Core. Require Import HSet WildCat. Require Import Groups.QuotientGroup Groups.ShortExactSequence. Require Import AbelianGroup AbGroups.Biproduct AbHom. Require Import Homotopy.ExactSequence Pointed. Require Import Modalities.ReflectiveSubuniverse.

Algebra\AbSES\Core.v

iscomplex_abses

(B A : AbGroup@{u}) : pType := [' B A, _].

Definition

Require Import Basics Types Truncations.Core. Require Import HSet WildCat. Require Import Groups.QuotientGroup Groups.ShortExactSequence. Require Import AbelianGroup AbGroups.Biproduct AbHom. Require Import Homotopy.ExactSequence Pointed. Require Import Modalities.ReflectiveSubuniverse.

Algebra\AbSES\Core.v

AbSES

{B A : AbGroup@{u}} (E F : AbSES B A) := {phi : GroupIsomorphism E F & (phi $o inclusion _ == inclusion _) * (projection _ == projection _ $o phi)}. Local Lemma shuffle_abses_path_data_iso `{Funext} {B A : AbGroup@{u}} (E F : AbSES B A) : ( E F) <~> {phi : GroupIsomorphism E F & (phi $o inclusion _ == inclusion _) * (projection _ $o grp_iso_inverse phi == projection _)}. Proof. srapply equiv_functor_sigma_id; intro phi. srapply equiv_functor_prod'. 1: exact equiv_idmap. srapply (equiv_functor_forall' phi^-1); intro e; cbn. apply equiv_concat_r. exact (ap _ (eisretr _ _)). Defined.

Definition

Require Import Basics Types Truncations.Core. Require Import HSet WildCat. Require Import Groups.QuotientGroup Groups.ShortExactSequence. Require Import AbelianGroup AbGroups.Biproduct AbHom. Require Import Homotopy.ExactSequence Pointed. Require Import Modalities.ReflectiveSubuniverse.

Algebra\AbSES\Core.v

abses_path_data_iso

`{Univalence} {B A : AbGroup@{u}} {E F : AbSES' B A} : abses_path_data_iso E F <~> E = F. Proof. refine (_ oE shuffle_abses_path_data_iso E F). refine (equiv_ap_inv issig_abses _ _ oE _). refine (equiv_path_sigma_hprop _ _ oE _). refine (equiv_path_sigma _ _ _ oE _). srapply equiv_functor_sigma'. 1: exact equiv_path_abgroup. intro q; lazy beta. snrefine (equiv_concat_l _ _ oE _). 1: exact (q $o inclusion _, projection _ $o grp_iso_inverse q). 2: { refine (equiv_path_prod _ _ oE _). exact (equiv_functor_prod' equiv_path_grouphomomorphism equiv_path_grouphomomorphism). } refine (transport_prod _ _ @ _). apply path_prod'. - apply transport_iso_abgrouphomomorphism_from_const. - apply transport_iso_abgrouphomomorphism_to_const. Defined.

Proposition

Require Import Basics Types Truncations.Core. Require Import HSet WildCat. Require Import Groups.QuotientGroup Groups.ShortExactSequence. Require Import AbelianGroup AbGroups.Biproduct AbHom. Require Import Homotopy.ExactSequence Pointed. Require Import Modalities.ReflectiveSubuniverse.

Algebra\AbSES\Core.v

equiv_path_abses_iso

`{Univalence} {B A : AbGroup@{u}} {E F : AbSES B A} (phi : GroupIsomorphism E F) (p : phi $o inclusion _ == inclusion _) (q : projection _ == projection _ $o phi) : E = F := equiv_path_abses_iso (phi; (p,q)).

Definition

Require Import Basics Types Truncations.Core. Require Import HSet WildCat. Require Import Groups.QuotientGroup Groups.ShortExactSequence. Require Import AbelianGroup AbGroups.Biproduct AbHom. Require Import Homotopy.ExactSequence Pointed. Require Import Modalities.ReflectiveSubuniverse.

Algebra\AbSES\Core.v

path_abses_iso

{B A : AbGroup@{u}} {E F : AbSES B A} (phi : GroupHomomorphism E F) (p0 : phi $o inclusion E == inclusion F) (p1 : projection E == projection F $o phi) : IsEquiv phi. Proof. apply isequiv_surj_emb. - intro f. rapply contr_inhabited_hprop. assert (e0 : Tr (-1) (hfiber (projection E) (projection F f))). 1: apply center, issurjection_projection. strip_truncations. assert (a : Tr (-1) (hfiber (inclusion F) (f + (- phi e0.1)))). 1: { refine (isexact_preimage (Tr (-1)) (inclusion F) (projection F) _ _). refine (grp_homo_op _ _ _ @ _). refine (ap _ (grp_homo_inv _ _) @ _). apply (grp_moveL_1M)^-1. exact (e0.2^ @ p1 e0.1). } strip_truncations. refine (tr (inclusion E a.1 + e0.1; _)). refine (grp_homo_op _ _ _ @ _). refine (ap (fun x => x + phi e0.1) (p0 a.1 @ a.2) @ _). refine ((grp_assoc _ _ _)^ @ _). refine (ap _ (left_inverse (phi e0.1)) @ _). apply grp_unit_r. - apply isembedding_grouphomomorphism. intros e p. assert (a : Tr (-1) (hfiber (inclusion E) e)). 1: { refine (isexact_preimage _ (inclusion E) (projection E) _ _). exact (p1 e @ ap (projection F) p @ grp_homo_unit _). } strip_truncations. refine (a.2^ @ ap (inclusion E ) _ @ grp_homo_unit (inclusion E)). rapply (isinj_embedding (inclusion F) _ _). refine ((p0 a.1)^ @ (ap phi a.2) @ p @ (grp_homo_unit _)^). Defined.

Lemma

Require Import Basics Types Truncations.Core. Require Import HSet WildCat. Require Import Groups.QuotientGroup Groups.ShortExactSequence. Require Import AbelianGroup AbGroups.Biproduct AbHom. Require Import Homotopy.ExactSequence Pointed. Require Import Modalities.ReflectiveSubuniverse.

Algebra\AbSES\Core.v

short_five_lemma

{B A : AbGroup@{u}} (E F : AbSES B A) := {phi : GroupHomomorphism E F & (phi $o inclusion _ == inclusion _) * (projection _ == projection _ $o phi)}.

Definition

Require Import Basics Types Truncations.Core. Require Import HSet WildCat. Require Import Groups.QuotientGroup Groups.ShortExactSequence. Require Import AbelianGroup AbGroups.Biproduct AbHom. Require Import Homotopy.ExactSequence Pointed. Require Import Modalities.ReflectiveSubuniverse.

Algebra\AbSES\Core.v

abses_path_data

{B A : AbGroup@{u}} (E F: AbSES B A) : abses_path_data E F -> abses_path_data_iso E F. Proof. - intros [phi [p q]]. exact ({| grp_iso_homo := phi; isequiv_group_iso := short_five_lemma phi p q |}; (p, q)). Defined.

Definition

Require Import Basics Types Truncations.Core. Require Import HSet WildCat. Require Import Groups.QuotientGroup Groups.ShortExactSequence. Require Import AbelianGroup AbGroups.Biproduct AbHom. Require Import Homotopy.ExactSequence Pointed. Require Import Modalities.ReflectiveSubuniverse.

Algebra\AbSES\Core.v

abses_path_data_to_iso

`{Funext} {B A : AbGroup@{u}} (E F: AbSES B A) : abses_path_data E F <~> abses_path_data_iso E F. Proof. srapply equiv_adjointify. - apply abses_path_data_to_iso. - srapply (functor_sigma (grp_iso_homo _ _)). exact (fun _ => idmap). - intros [phi [p q]]. apply path_sigma_hprop. by apply equiv_path_groupisomorphism. - reflexivity. Defined.

Proposition

Require Import Basics Types Truncations.Core. Require Import HSet WildCat. Require Import Groups.QuotientGroup Groups.ShortExactSequence. Require Import AbelianGroup AbGroups.Biproduct AbHom. Require Import Homotopy.ExactSequence Pointed. Require Import Modalities.ReflectiveSubuniverse.

Algebra\AbSES\Core.v

equiv_path_abses_data

`{Univalence} {B A : AbGroup@{u}} {E F : AbSES B A} : abses_path_data E F <~> E = F := equiv_path_abses_iso oE equiv_path_abses_data E F.

Definition

Require Import Basics Types Truncations.Core. Require Import HSet WildCat. Require Import Groups.QuotientGroup Groups.ShortExactSequence. Require Import AbelianGroup AbGroups.Biproduct AbHom. Require Import Homotopy.ExactSequence Pointed. Require Import Modalities.ReflectiveSubuniverse.

Algebra\AbSES\Core.v

equiv_path_abses

`{Univalence} {B A : AbGroup@{u}} {E F : AbSES B A} (phi : middle E $-> F) (p : phi $o inclusion _ == inclusion _) (q : projection _ == projection _ $o phi) : E = F := equiv_path_abses (phi; (p,q)).

Definition

Require Import Basics Types Truncations.Core. Require Import HSet WildCat. Require Import Groups.QuotientGroup Groups.ShortExactSequence. Require Import AbelianGroup AbGroups.Biproduct AbHom. Require Import Homotopy.ExactSequence Pointed. Require Import Modalities.ReflectiveSubuniverse.

Algebra\AbSES\Core.v

path_abses

{B A : AbGroup@{u}} (E : AbSES B A) : E $-> E := (grp_iso_id; (fun _ => idpath, fun _ => idpath)).

Definition

Require Import Basics Types Truncations.Core. Require Import HSet WildCat. Require Import Groups.QuotientGroup Groups.ShortExactSequence. Require Import AbelianGroup AbGroups.Biproduct AbHom. Require Import Homotopy.ExactSequence Pointed. Require Import Modalities.ReflectiveSubuniverse.

Algebra\AbSES\Core.v

abses_path_data_1

{B A : AbGroup@{u}} {E F G : AbSES B A} (p : E $-> F) (q : F $-> G) : E $-> G := (q.

Definition

Require Import Basics Types Truncations.Core. Require Import HSet WildCat. Require Import Groups.QuotientGroup Groups.ShortExactSequence. Require Import AbelianGroup AbGroups.Biproduct AbHom. Require Import Homotopy.ExactSequence Pointed. Require Import Modalities.ReflectiveSubuniverse.

Algebra\AbSES\Core.v

abses_path_data_compose

{B A : AbGroup@{u}} {E F : AbSES B A} : (E $-> F) -> (F $-> E). Proof. intros [phi [p q]]. srefine (_; (_,_)). - exact (grp_iso_inverse phi). - intro a. exact (ap _ (p a)^ @ eissect _ (inclusion E a)). - intro a; simpl. exact (ap (projection F) (eisretr _ _)^ @ (q _)^). Defined.

Definition

Require Import Basics Types Truncations.Core. Require Import HSet WildCat. Require Import Groups.QuotientGroup Groups.ShortExactSequence. Require Import AbelianGroup AbGroups.Biproduct AbHom. Require Import Homotopy.ExactSequence Pointed. Require Import Modalities.ReflectiveSubuniverse.

Algebra\AbSES\Core.v

abses_path_data_inverse

`{Univalence} {B A : AbGroup@{u}} {E : AbSES B A} : equiv_path_abses_iso (abses_path_data_1 E) = idpath. Proof. apply (equiv_ap_inv' equiv_path_abses_iso). refine (eissect _ _ @ _). srapply path_sigma_hprop; simpl. srapply equiv_path_groupisomorphism. reflexivity. Defined.

Definition

Require Import Basics Types Truncations.Core. Require Import HSet WildCat. Require Import Groups.QuotientGroup Groups.ShortExactSequence. Require Import AbelianGroup AbGroups.Biproduct AbHom. Require Import Homotopy.ExactSequence Pointed. Require Import Modalities.ReflectiveSubuniverse.

Algebra\AbSES\Core.v

equiv_path_abses_1

`{Univalence} {B A : AbGroup@{u}} {E : AbSES B A} : (@equiv_path_abses_iso _ B A E E)^-1 idpath = Id E. Proof. apply moveR_equiv_M; symmetry. apply equiv_path_abses_1. Defined.

Definition

Require Import Basics Types Truncations.Core. Require Import HSet WildCat. Require Import Groups.QuotientGroup Groups.ShortExactSequence. Require Import AbelianGroup AbGroups.Biproduct AbHom. Require Import Homotopy.ExactSequence Pointed. Require Import Modalities.ReflectiveSubuniverse.

Algebra\AbSES\Core.v

equiv_path_absesV_1

`{Univalence} {B A : AbGroup@{u}} {E F : AbSES B A} (p : abses_path_data_iso E F) : (equiv_path_abses_iso p)^ = equiv_path_abses_iso (abses_path_data_inverse p). Proof. revert p. equiv_intro (equiv_path_abses_iso (E:=E) (F:=F))^-1 p; induction p. refine (ap _ (eisretr _ _) @ _); symmetry. nrefine (ap (equiv_path_abses_iso o abses_path_data_inverse) equiv_path_absesV_1 @ _). refine (ap equiv_path_abses_iso gpd_strong_rev_1 @ _). exact equiv_path_abses_1. Defined.

Definition

Require Import Basics Types Truncations.Core. Require Import HSet WildCat. Require Import Groups.QuotientGroup Groups.ShortExactSequence. Require Import AbelianGroup AbGroups.Biproduct AbHom. Require Import Homotopy.ExactSequence Pointed. Require Import Modalities.ReflectiveSubuniverse.

Algebra\AbSES\Core.v

abses_path_data_V

`{Univalence} {B A : AbGroup@{u}} {E F G : AbSES B A} (p : E = F) (q : F = G) : p @ q = equiv_path_abses_iso (abses_path_data_compose (equiv_path_abses_iso^-1 p) (equiv_path_abses_iso^-1 q)). Proof. induction p, q. refine (equiv_path_abses_1^ @ _). apply (ap equiv_path_abses_iso). apply path_sigma_hprop. by apply equiv_path_groupisomorphism. Defined.

Definition

Require Import Basics Types Truncations.Core. Require Import HSet WildCat. Require Import Groups.QuotientGroup Groups.ShortExactSequence. Require Import AbelianGroup AbGroups.Biproduct AbHom. Require Import Homotopy.ExactSequence Pointed. Require Import Modalities.ReflectiveSubuniverse.

Algebra\AbSES\Core.v

abses_path_compose_beta