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phanerozoic

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Coq-HoTT

like 2

`{Univalence} {B A : AbGroup@{u}} {E F G : AbSES B A} (p : abses_path_data_iso E F) (q : abses_path_data_iso F G) : equiv_path_abses_iso p @ equiv_path_abses_iso q = equiv_path_abses_iso (abses_path_data_compose p q). Proof. generalize p, q. equiv_intro ((equiv_path_abses_iso (E:=E) (F:=F))^-1) x. equiv_intro ((equiv_path_abses_iso (E:=F) (F:=G))^-1) y. refine ((eisretr _ _ @@ eisretr _ _) @ _). rapply abses_path_compose_beta. 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_compose_beta

`{Univalence} {X : Type} {B A : AbGroup@{u}} (f g : X -> AbSES B A) : (f $=> g) <~> f == g. Proof. srapply equiv_functor_forall_id; intro x; cbn. srapply equiv_path_abses_iso. 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_data_homotopy

{B' A' B A : AbGroup@{u}} : AbSES B A -->* AbSES B' A' := Build_BasepointPreservingFunctor (const pt) (Id pt).

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

pmap_abses_const

`{Univalence} {B' A' B A : AbGroup@{u}} : (AbSES B A -->* AbSES B' A') -> (AbSES B A ->* AbSES B' A') := fun f => Build_pMap _ _ f (equiv_path_abses_iso (bp_pointed 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

to_pointed

`{Univalence} {B' A' B A : AbGroup@{u}} : pconst ==* to_pointed (@pmap_abses_const B' A' B A). Proof. srapply Build_pHomotopy. 1: reflexivity. apply moveL_pV. refine (concat_1p _ @ _). apply equiv_path_abses_1. 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

pmap_abses_const_to_pointed

`{Univalence} {B0 B1 A0 A1 : AbGroup@{u}} (f : AbSES B0 A0 -> AbSES B1 A1) `{!Is0Functor f, !Is1Functor f} {E F : AbSES B0 A0} (p : E $== F) : ap f (equiv_path_abses_iso p) = equiv_path_abses_iso (fmap f p). Proof. revert p. apply (equiv_ind equiv_path_abses_iso^-1%equiv); intro p. induction p. refine (ap (ap f) (eisretr _ _) @ _). nrefine (_ @ ap equiv_path_abses_iso _). 2: { rapply path_hom. srefine (_ $@ fmap2 _ _). 2: exact (Id E). 2: intro x; reflexivity. exact (fmap_id f _)^$. } exact equiv_path_abses_1^. 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

abses_ap_fmap

`{Univalence} {B0 B1 B2 A0 A1 A2 : AbGroup@{u}} (f : AbSES B0 A0 -->* AbSES B1 A1) (g : AbSES B1 A1 -->* AbSES B2 A2) `{!Is1Functor f, !Is1Functor g} : to_pointed g o* to_pointed f ==* to_pointed (g $o* f). Proof. srapply Build_pHomotopy. 1: reflexivity. lazy beta. nrapply moveL_pV. nrefine (concat_1p _ @ _). unfold pmap_compose, Build_pMap, pointed_fun, point_eq, dpoint_eq. refine (_ @ ap (fun x => x @ _) _^). 2: apply (abses_ap_fmap g). nrefine (_ @ (abses_path_data_compose_beta _ _)^). nrapply (ap equiv_path_abses_iso). rapply path_hom. 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

to_pointed_compose

`{Univalence} {B' A' B A : AbGroup@{u}} {f g : AbSES B A -->* AbSES B' A'} : f $=>* g <~> to_pointed f ==* to_pointed g. Proof. refine (issig_pforall _ _ oE _). apply (equiv_functor_sigma' (equiv_path_data_homotopy f g)); intro h. refine (equiv_concat_r _ _ oE _). 1: exact ((abses_path_data_compose_beta _ _)^ @ ap (fun x => _ @ x) (abses_path_data_V _)^). refine (equiv_ap' equiv_path_abses_iso _ _ oE _). refine (equiv_path_sigma_hprop _ _ oE _). 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

equiv_ptransformation_phomotopy

`{Funext} {B A : AbGroup@{u}} : {phi : GroupHomomorphism (point (AbSES B A)) (point (AbSES B A)) & (phi o inclusion _ == inclusion _) * (projection _ == projection _ o phi)} <~> (B $-> A). Proof. srapply equiv_adjointify. - intros [phi _]. exact (ab_biprod_pr1 $o phi $o ab_biprod_inr). - intro f. snrefine (_;_). + refine (ab_biprod_rec ab_biprod_inl _). refine (ab_biprod_corec f grp_homo_id). + split; intro x; cbn. * apply path_prod; cbn. -- exact (ap _ (grp_homo_unit f) @ right_identity _). -- exact (right_identity _). * exact (left_identity _)^. - intro f. rapply equiv_path_grouphomomorphism; intro b; cbn. exact (left_identity _). - intros [phi [p q]]. apply path_sigma_hprop; cbn. rapply equiv_path_grouphomomorphism; intros [a b]; cbn. apply path_prod; cbn. + rewrite (grp_prod_decompose a b). refine (_ @ (grp_homo_op (ab_biprod_pr1 $o phi) _ _)^). apply grp_cancelR; symmetry. exact (ap fst (p a)). + rewrite (grp_prod_decompose a b). refine (_ @ (grp_homo_op (ab_biprod_pr2 $o phi) _ _)^); cbn; symmetry. exact (ap011 _ (ap snd (p a)) (q (group_unit, b))^). 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

abses_endomorphism_trivial

`{Univalence} {A B : AbGroup} : (B $-> A) <~> loops (AbSES B A) := equiv_path_abses oE abses_endomorphism_trivial^-1.

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

loops_abses

{A B : AbGroup} (E : AbSES B A) : (B $-> A) -> abses_path_data E E. Proof. intro phi. srefine (_; (_, _)). - exact (ab_homo_add grp_homo_id (inclusion E $o phi $o projection E)). - intro a; cbn. refine (ap (fun x => _ + inclusion E (phi x)) _ @ _). 1: apply iscomplex_abses. refine (ap (fun x => _ + x) (grp_homo_unit (inclusion E $o phi)) @ _). apply grp_unit_r. - intro e; symmetry. refine (grp_homo_op (projection E) _ _ @ _); cbn. refine (ap (fun x => _ + x) _ @ _). 1: apply iscomplex_abses. apply grp_unit_r. 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

hom_loops_data_abses

{A X B Y : AbGroup@{u}}

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

AbSESMorphism

{A X B Y : AbGroup@{u}} {E : AbSES B A} {F : AbSES Y X} : { f : (A $-> X) * (middle E $-> middle F) * (B $-> Y) & ((inclusion _) $o (fst (fst f)) == (snd (fst f)) $o (inclusion _)) * ((projection F) $o (snd (fst f)) == (snd f) $o (projection _)) } <~> AbSESMorphism E F := ltac:(make_equiv).

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_AbSESMorphism

{A B : AbGroup@{u}} (E : AbSES B A) : AbSESMorphism E E. Proof. snrapply (Build_AbSESMorphism grp_homo_id grp_homo_id grp_homo_id). 1,2: reflexivity. 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

abses_morphism_id

{A0 A1 A2 B0 B1 B2 : AbGroup@{u}} {E : AbSES B0 A0} {F : AbSES B1 A1} {G : AbSES B2 A2} (g : AbSESMorphism F G) (f : AbSESMorphism E F) : AbSESMorphism E G. Proof. rapply (Build_AbSESMorphism (component1 g $o component1 f) (component2 g $o component2 f) (component3 g $o component3 f)). - intro x; cbn. exact (left_square g _ @ ap _ (left_square f _)). - intro x; cbn. exact (right_square g _ @ ap _ (right_square f _)). 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

absesmorphism_compose

{B A : AbGroup} (E : AbSES B A) {s : B $-> E} (h : projection _ $o s == idmap) : (middle E) $-> (@ab_kernel E B (projection _)). Proof. snrapply (grp_kernel_corec (G:=E) (A:=E)). - refine (ab_homo_add grp_homo_id (grp_homo_compose ab_homo_negation (s $o (projection _)))). - intro x; simpl. refine (grp_homo_op (projection _) x _ @ _). refine (ap (fun y => (projection _) x + y) _ @ right_inverse ((projection _) x)). refine (grp_homo_inv _ _ @ ap negate _ ). apply h. 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

projection_split_to_kernel

{B A : AbGroup} (E : AbSES B A) {s : B $-> E} (h : (projection _) $o s == idmap) : (projection_split_to_kernel E h) $o (inclusion _) == grp_cxfib cx_isexact. Proof. intro a. apply path_sigma_hprop; cbn. apply grp_cancelL1. refine (ap (fun x => - s x) _ @ _). 1: rapply cx_isexact. exact (ap _ (grp_homo_unit _) @ negate_mon_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

projection_split_to_kernel_beta

{B A : AbGroup} (E : AbSES B A) {s : GroupHomomorphism B E} (h : (projection _) $o s == idmap) : GroupIsomorphism E (ab_biprod (@ab_kernel E B (projection _)) B). Proof. srapply Build_GroupIsomorphism. - refine (ab_biprod_corec _ (projection _)). exact (projection_split_to_kernel E h). - srapply isequiv_adjointify. + refine (ab_biprod_rec _ s). rapply subgroup_incl. + intros [a b]; simpl. apply path_prod'. * srapply path_sigma_hprop; cbn. refine ((associativity _ _ _)^ @ _). apply grp_cancelL1. refine (ap _ _ @ right_inverse _). apply (ap negate). apply (ap s). refine (grp_homo_op (projection _) a.1 (s b) @ _). exact (ap (fun y => y + _) a.2 @ left_identity _ @ h b). * refine (grp_homo_op (projection _) a.1 (s b) @ _). exact (ap (fun y => y + _) a.2 @ left_identity _ @ h b). + intro e; simpl. by apply grp_moveR_gM. 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

projection_split_iso1

{B A : AbGroup@{u}} (E : AbSES B A) {s : GroupHomomorphism B E} (h : (projection _) $o s == idmap) : GroupIsomorphism E (ab_biprod A B). Proof. etransitivity (ab_biprod (ab_kernel _) B). - exact (projection_split_iso1 E h). - srapply (equiv_functor_ab_biprod (grp_iso_inverse _) grp_iso_id). rapply grp_iso_cxfib. 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

projection_split_iso

{B A : AbGroup} (E : AbSES B A) {s : B $-> E} (h : (projection _) $o s == idmap) : projection_split_iso E h o (inclusion _) == ab_biprod_inl. Proof. intro a. nrapply path_prod'. 2: rapply cx_isexact. lhs nrapply (ap _ (projection_split_to_kernel_beta E h a)). apply eissect. 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

projection_split_beta

`{Univalence} {B A : AbGroup@{u}} (E : AbSES B A) : {s : B $-> E & (projection _) $o s == idmap} <-> (E = point (AbSES B A)). Proof. refine (iff_compose _ (iff_equiv equiv_path_abses_iso)); split. - intros [s h]. exists (projection_split_iso E h). split. + nrapply projection_split_beta. + reflexivity. - intros [phi [g h]]. exists (grp_homo_compose (grp_iso_inverse phi) ab_biprod_inr). intro x; cbn. exact (h _ @ ap snd (eisretr _ _)). 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

iff_abses_trivial_split

`{Univalence} {A E : AbGroup@{u}} (i : A $-> E) `{IsEmbedding i} : AbSES (QuotientAbGroup E (grp_image_embedding i)) A. Proof. srapply (Build_AbSES E i). 1: exact grp_quotient_map. 1: exact _. srapply Build_IsExact. - srapply phomotopy_homotopy_hset. intro x. apply qglue; cbn. exists (-x). exact (grp_homo_inv _ _ @ (grp_unit_r _)^). - snrapply (conn_map_homotopic (Tr (-1)) (B:=grp_kernel (@grp_quotient_map E _))). + exact (grp_kernel_quotient_iso _ o ab_image_in_embedding i). + intro a. by rapply (isinj_embedding (subgroup_incl _)). + rapply conn_map_isequiv. 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_from_inclusion

`{Funext} {A E B : AbGroup} (i : A $-> E) (p : E $-> B) `{IsEmbedding i, IsExact (Tr (-1)) _ _ _ i p} : GroupIsomorphism A (ab_kernel p). Proof. snrapply Build_GroupIsomorphism. - apply (grp_kernel_corec i). rapply cx_isexact. - apply isequiv_surj_emb. 2: rapply (cancelL_mapinO _ (grp_kernel_corec _ _) _). intros [y q]. assert (a : Tr (-1) (hfiber i y)). 1: by rapply isexact_preimage. strip_truncations; destruct a as [a r]. rapply contr_inhabited_hprop. refine (tr (a; _)); cbn. apply path_sigma_hprop; cbn. exact r. 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

abses_kernel_iso

`{Funext} {A E B : AbGroup} (i : A $-> E) (p : E $-> B) `{IsEmbedding i, IsExact (Tr (-1)) _ _ _ i p} : i o (abses_kernel_iso i p)^-1 == subgroup_incl _. Proof. rapply (equiv_ind (abses_kernel_iso i p)); intro a. exact (ap i (eissect (abses_kernel_iso i p) _)). 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

abses_kernel_iso_inv_beta

{E B : AbGroup@{u}} (p : E $-> B) `{IsSurjection p} : AbSES B (ab_kernel p). Proof. srapply (Build_AbSES E _ p). 1: exact (subgroup_incl _). 1: exact _. snrapply Build_IsExact. - apply phomotopy_homotopy_hset. intros [e q]; cbn. exact q. - rapply conn_map_isequiv. 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

abses_from_surjection

`{Funext} {A E B : AbGroup@{u}} (f : A $-> E) (g : GroupHomomorphism E B) `{IsSurjection g, IsExact (Tr (-1)) _ _ _ f g} : GroupIsomorphism (ab_cokernel f) B. Proof. snrapply Build_GroupIsomorphism. - snrapply (quotient_abgroup_rec _ _ g). intros e; rapply Trunc_rec; intros [a p]. refine (ap _ p^ @ _). rapply cx_isexact. - apply isequiv_surj_emb. 1: rapply cancelR_conn_map. apply isembedding_isinj_hset. srapply Quotient_ind2_hprop; intros x y. intro p. apply qglue; cbn. refine (isexact_preimage (Tr (-1)) _ _ (-x + y) _). refine (grp_homo_op _ _ _ @ _). rewrite grp_homo_inv. apply grp_moveL_M1^-1. exact p^. 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

abses_cokernel_iso

`{Funext} {A E B : AbGroup} (f : A $-> E) (g : GroupHomomorphism E B) `{IsSurjection g, IsExact (Tr (-1)) _ _ _ f g} : (abses_cokernel_iso f g)^-1 o g == grp_quotient_map. Proof. intro x; by apply moveR_equiv_V. 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_cokernel_iso_inv_beta

{A E B X F Y : AbGroup} (i : A $-> E) (p : E $-> B) `{ex0 : IsExact (Tr (-1)) _ _ _ i p} (j : X $-> F) (q : F $-> Y) `{ex1 : IsExact (Tr (-1)) _ _ _ j q} : IsExact (Tr (-1)) (functor_ab_biprod i j) (functor_ab_biprod p q). Proof. snrapply Build_IsExact. - snrapply phomotopy_homotopy_hset. 1: exact _. intro x; apply path_prod; cbn. + apply ex0. + apply ex1. - intros [[e f] u]; cbn. rapply contr_inhabited_hprop. pose (U := (equiv_path_prod _ _)^-1 u); cbn in U. pose proof (a := isexact_preimage _ i p e (fst U)). pose proof (x := isexact_preimage _ j q f (snd U)). strip_truncations; apply tr. exists (ab_biprod_inl a.1 + ab_biprod_inr x.1); cbn. pose (IS := sg_set (ab_biprod B Y)). apply path_sigma_hprop; cbn. apply path_prod; cbn. + rewrite right_identity. exact a.2. + rewrite left_identity. exact x.2. Defined.

Lemma

Require Import Basics Types Truncations.Core. Require Import Pointed.Core. Require Import WildCat.Core Homotopy.ExactSequence. Require Import AbGroups.AbelianGroup AbSES.Core AbGroups.Biproduct.

Algebra\AbSES\DirectSum.v

ab_biprod_exact

`{Funext} {B A B' A' : AbGroup} (E : AbSES B A) (F : AbSES B' A') : AbSES (ab_biprod B B') (ab_biprod A A') := Build_AbSES (ab_biprod E F) (functor_ab_biprod (inclusion E) (inclusion F)) (functor_ab_biprod (projection E) (projection F)) (functor_ab_biprod_embedding _ _) (functor_ab_biprod_surjection _ _) (ab_biprod_exact _ _ _ _).

Definition

Require Import Basics Types Truncations.Core. Require Import Pointed.Core. Require Import WildCat.Core Homotopy.ExactSequence. Require Import AbGroups.AbelianGroup AbSES.Core AbGroups.Biproduct.

Algebra\AbSES\DirectSum.v

abses_direct_sum

`{Funext} {A A' B B' C C' D D' : AbGroup} {E : AbSES B A} {E' : AbSES B' A'} {F : AbSES D C} {F' : AbSES D' C'} (f : AbSESMorphism E E') (g : AbSESMorphism F F') : AbSESMorphism (abses_direct_sum E F) (abses_direct_sum E' F'). Proof. snrapply Build_AbSESMorphism. + exact (functor_ab_biprod (component1 f) (component1 g)). + exact (functor_ab_biprod (component2 f) (component2 g)). + exact (functor_ab_biprod (component3 f) (component3 g)). + intro x. apply path_prod; apply left_square. + intro x. apply path_prod; apply right_square. Defined.

Lemma

Require Import Basics Types Truncations.Core. Require Import Pointed.Core. Require Import WildCat.Core Homotopy.ExactSequence. Require Import AbGroups.AbelianGroup AbSES.Core AbGroups.Biproduct.

Algebra\AbSES\DirectSum.v

functor_abses_directsum

`{Funext} {A B : AbGroup} (E : AbSES B A) : AbSESMorphism E (abses_direct_sum E E). Proof. snrapply Build_AbSESMorphism. 1,2,3: exact ab_diagonal. all: reflexivity. Defined.

Definition

Require Import Basics Types Truncations.Core. Require Import Pointed.Core. Require Import WildCat.Core Homotopy.ExactSequence. Require Import AbGroups.AbelianGroup AbSES.Core AbGroups.Biproduct.

Algebra\AbSES\DirectSum.v

abses_diagonal

`{Funext} {A B : AbGroup} (E : AbSES B A) : AbSESMorphism (abses_direct_sum E E) E. Proof. snrapply Build_AbSESMorphism. 1,2,3: exact ab_codiagonal. all: intro x; cbn; apply grp_homo_op. Defined.

Definition

Require Import Basics Types Truncations.Core. Require Import Pointed.Core. Require Import WildCat.Core Homotopy.ExactSequence. Require Import AbGroups.AbelianGroup AbSES.Core AbGroups.Biproduct.

Algebra\AbSES\DirectSum.v

abses_codiagonal

`{Funext} {A A' B B' : AbGroup} (E : AbSES B A) (F : AbSES B' A') : AbSESMorphism (abses_direct_sum E F) (abses_direct_sum F E). Proof. snrapply Build_AbSESMorphism. 1,2,3: exact direct_sum_swap. all: reflexivity. Defined.

Definition

Require Import Basics Types Truncations.Core. Require Import Pointed.Core. Require Import WildCat.Core Homotopy.ExactSequence. Require Import AbGroups.AbelianGroup AbSES.Core AbGroups.Biproduct.

Algebra\AbSES\DirectSum.v

abses_swap_morphism

`{Funext} {A B : AbGroup} (E F G : AbSES B A) : AbSESMorphism (abses_direct_sum (abses_direct_sum E F) G) (abses_direct_sum (abses_direct_sum G F) E). Proof. snrapply Build_AbSESMorphism. 1,2,3: exact ab_biprod_twist. all: reflexivity. Defined.

Lemma

Require Import Basics Types Truncations.Core. Require Import Pointed.Core. Require Import WildCat.Core Homotopy.ExactSequence. Require Import AbGroups.AbelianGroup AbSES.Core AbGroups.Biproduct.

Algebra\AbSES\DirectSum.v

abses_twist_directsum

(B A : AbGroup@{u}) := pTr 0 (AbSES B A).

Definition

Require Import Basics Types Truncations.Core. Require Import Pointed WildCat. Require Import Truncations.SeparatedTrunc. Require Import AbelianGroup AbHom AbProjective. Require Import AbSES.Pullback AbSES.Pushout AbSES.BaerSum AbSES.Core.

Algebra\AbSES\Ext.v

Ext

`{Univalence} {B A : AbGroup} (E : AbSES B A) : merely {s : GroupHomomorphism B E & (projection _) $o s == idmap} <~> (tr E = point (Ext B A)). Proof. refine (equiv_path_Tr _ _ oE _). srapply equiv_iff_hprop; apply Trunc_functor; apply iff_abses_trivial_split. Defined.

Proposition

Require Import Basics Types Truncations.Core. Require Import Pointed WildCat. Require Import Truncations.SeparatedTrunc. Require Import AbelianGroup AbHom AbProjective. Require Import AbSES.Pullback AbSES.Pushout AbSES.BaerSum AbSES.Core.

Algebra\AbSES\Ext.v

iff_ab_ext_trivial_split

Ext' (B A : AbGroup@{u}) := Tr 0 (AbSES' B A).

Definition

Require Import Basics Types Truncations.Core. Require Import Pointed WildCat. Require Import Truncations.SeparatedTrunc. Require Import AbelianGroup AbHom AbProjective. Require Import AbSES.Pullback AbSES.Pushout AbSES.BaerSum AbSES.Core.

Algebra\AbSES\Ext.v

Ext'

`{Univalence} (B A : AbGroup@{u}) : Group. Proof. snrapply (Build_Group (Ext B A)). - intros E F. strip_truncations. exact (tr (abses_baer_sum E F)). - exact (point (Ext B A)). - unfold Negate. exact (Trunc_functor _ (abses_pullback (- grp_homo_id))). - repeat split. 1: apply istrunc_truncation. all: intro E. 1: intros F G. all: strip_truncations; unfold mon_unit, point; apply (ap tr). + symmetry; apply baer_sum_associative. + apply baer_sum_unit_l. + apply baer_sum_unit_r. + apply baer_sum_inverse_r. + apply baer_sum_inverse_l. Defined.

Definition

Require Import Basics Types Truncations.Core. Require Import Pointed WildCat. Require Import Truncations.SeparatedTrunc. Require Import AbelianGroup AbHom AbProjective. Require Import AbSES.Pullback AbSES.Pushout AbSES.BaerSum AbSES.Core.

Algebra\AbSES\Ext.v

grp_ext

ab_ext@{u v|u < v} `{Univalence} (B : AbGroup@{u}^op) (A : AbGroup@{u}) : AbGroup@{v}. Proof. snrapply (Build_AbGroup (grp_ext@{u v} B A)). intros E F. strip_truncations; cbn. apply ap. apply baer_sum_commutative. Defined.

Definition

Require Import Basics Types Truncations.Core. Require Import Pointed WildCat. Require Import Truncations.SeparatedTrunc. Require Import AbelianGroup AbHom AbProjective. Require Import AbSES.Pullback AbSES.Pushout AbSES.BaerSum AbSES.Core.

Algebra\AbSES\Ext.v

ab_ext@

`{Univalence} {B A G : AbGroup@{u}} (E : AbSES B A) : GroupHomomorphism (ab_hom A G) (ab_ext B G). Proof. snrapply Build_GroupHomomorphism. 1: exact (fun f => fmap01 (A:=AbGroup^op) Ext' _ f (tr E)). intros f g; cbn. nrapply (ap tr). exact (baer_sum_distributive_pushouts f g). Defined.

Definition

Require Import Basics Types Truncations.Core. Require Import Pointed WildCat. Require Import Truncations.SeparatedTrunc. Require Import AbelianGroup AbHom AbProjective. Require Import AbSES.Pullback AbSES.Pushout AbSES.BaerSum AbSES.Core.

Algebra\AbSES\Ext.v

abses_pushout_ext

`{Univalence} (P : AbGroup) `{IsAbProjective P} : forall A, forall E : AbSES P A, tr E = point (Ext P A). Proof. intros A E. apply iff_ab_ext_trivial_split. exact (fst (iff_isabprojective_surjections_split P) _ _ _ _). Defined.

Proposition

Require Import Basics Types Truncations.Core. Require Import Pointed WildCat. Require Import Truncations.SeparatedTrunc. Require Import AbelianGroup AbHom AbProjective. Require Import AbSES.Pullback AbSES.Pushout AbSES.BaerSum AbSES.Core.

Algebra\AbSES\Ext.v

abext_trivial_projective

`{Univalence} (P : AbGroup) (ext_triv : forall A, forall E : AbSES P A, tr E = point (Ext P A)) : IsAbProjective P. Proof. apply iff_isabprojective_surjections_split. intros E p issurj_p. apply (iff_ab_ext_trivial_split (abses_from_surjection p))^-1. apply ext_triv. Defined.

Proposition

Require Import Basics Types Truncations.Core. Require Import Pointed WildCat. Require Import Truncations.SeparatedTrunc. Require Import AbelianGroup AbHom AbProjective. Require Import AbSES.Pullback AbSES.Pushout AbSES.BaerSum AbSES.Core.

Algebra\AbSES\Ext.v

abext_projective_trivial

{A B B' : AbGroup} (f : B' $-> B) : AbSES B A -> AbSES B' A. Proof. intro E. snrapply (Build_AbSES (ab_pullback (projection E) f) (grp_pullback_corec _ _ (inclusion _) grp_homo_const _) (grp_pullback_pr2 (projection _) f)). - intro x. nrefine (_ @ (grp_homo_unit f)^). apply isexact_inclusion_projection. - exact (cancelL_isembedding (g:= grp_pullback_pr1 _ _)). - rapply conn_map_pullback'. - snrapply Build_IsExact. + snrapply phomotopy_homotopy_hset. * exact _. * reflexivity. + nrefine (cancelL_equiv_conn_map _ _ (hfiber_pullback_along_pointed f (projection _) (grp_homo_unit _))). nrefine (conn_map_homotopic _ _ _ _ (conn_map_isexact (IsExact:=isexact_inclusion_projection _))). intro a. by apply path_sigma_hprop. Defined.

Definition

Require Import Basics Types. Require Import HSet Limits.Pullback. Require Import WildCat Pointed.Core Homotopy.ExactSequence. Require Import Modalities.ReflectiveSubuniverse. Require Import AbGroups.AbelianGroup AbGroups.AbPullback AbGroups.Biproduct. Require Import AbSES.Core AbSES.DirectSum.

Algebra\AbSES\Pullback.v

abses_pullback

{A B B' : AbGroup@{u}} (E : AbSES B A) (f : B' $-> B) : AbSESMorphism (abses_pullback f E) E. Proof. snrapply (Build_AbSESMorphism grp_homo_id _ f). - apply grp_pullback_pr1. - reflexivity. - apply pullback_commsq. Defined.

Definition

Require Import Basics Types. Require Import HSet Limits.Pullback. Require Import WildCat Pointed.Core Homotopy.ExactSequence. Require Import Modalities.ReflectiveSubuniverse. Require Import AbGroups.AbelianGroup AbGroups.AbPullback AbGroups.Biproduct. Require Import AbSES.Core AbSES.DirectSum.

Algebra\AbSES\Pullback.v

abses_pullback_morphism

{A B X Y : AbGroup@{u}} {E : AbSES B A} {F : AbSES Y X} (f : AbSESMorphism E F) : AbSESMorphism E (abses_pullback (component3 f) F). Proof. snrapply (Build_AbSESMorphism (component1 f) _ grp_homo_id). - apply (grp_pullback_corec (projection F) (component3 f) (component2 f) (projection E)). apply right_square. - intro x; cbn. apply equiv_path_pullback_hset; cbn; split. + apply left_square. + symmetry; apply iscomplex_abses. - reflexivity. Defined.

Definition

Require Import Basics Types. Require Import HSet Limits.Pullback. Require Import WildCat Pointed.Core Homotopy.ExactSequence. Require Import Modalities.ReflectiveSubuniverse. Require Import AbGroups.AbelianGroup AbGroups.AbPullback AbGroups.Biproduct. Require Import AbSES.Core AbSES.DirectSum.

Algebra\AbSES\Pullback.v

abses_pullback_morphism_corec

`{Funext} {A B X Y : AbGroup@{u}} {E : AbSES B A} {F : AbSES Y X} (f : AbSESMorphism E F) : f = absesmorphism_compose (abses_pullback_morphism F (component3 f)) (abses_pullback_morphism_corec f). Proof. apply (equiv_ap issig_AbSESMorphism^-1 _ _). srapply path_sigma_hprop. apply path_prod. 1: apply path_prod. all: by apply equiv_path_grouphomomorphism. Defined.

Definition

Require Import Basics Types. Require Import HSet Limits.Pullback. Require Import WildCat Pointed.Core Homotopy.ExactSequence. Require Import Modalities.ReflectiveSubuniverse. Require Import AbGroups.AbelianGroup AbGroups.AbPullback AbGroups.Biproduct. Require Import AbSES.Core AbSES.DirectSum.

Algebra\AbSES\Pullback.v

abses_pullback_morphism_corec_beta

abses_pullback_component1_id' {A B B' : AbGroup@{u}} {E : AbSES B A} {F : AbSES B' A} (f : AbSESMorphism E F) (h : component1 f == grp_homo_id) : E $== abses_pullback (component3 f) F. Proof. pose (g := abses_pullback_morphism_corec f). nrapply abses_path_data_to_iso. exists (component2 g); split. - exact (fun a => (left_square g a)^ @ ap _ (h a)). - reflexivity. Defined.

Definition

Require Import Basics Types. Require Import HSet Limits.Pullback. Require Import WildCat Pointed.Core Homotopy.ExactSequence. Require Import Modalities.ReflectiveSubuniverse. Require Import AbGroups.AbelianGroup AbGroups.AbPullback AbGroups.Biproduct. Require Import AbSES.Core AbSES.DirectSum.

Algebra\AbSES\Pullback.v

abses_pullback_component1_id'

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

Definition

Require Import Basics Types. Require Import HSet Limits.Pullback. Require Import WildCat Pointed.Core Homotopy.ExactSequence. Require Import Modalities.ReflectiveSubuniverse. Require Import AbGroups.AbelianGroup AbGroups.AbPullback AbGroups.Biproduct. Require Import AbSES.Core AbSES.DirectSum.

Algebra\AbSES\Pullback.v

abses_pullback_component1_id

`{Funext} {A B B' C D D' : AbGroup@{u}} {E : AbSES B A} {F : AbSES D C} (f : B' $-> B) (g : D' $-> D) : AbSESMorphism (abses_direct_sum (abses_pullback f E) (abses_pullback g F)) (abses_direct_sum E F) := functor_abses_directsum (abses_pullback_morphism E f) (abses_pullback_morphism F g).

Definition

Require Import Basics Types. Require Import HSet Limits.Pullback. Require Import WildCat Pointed.Core Homotopy.ExactSequence. Require Import Modalities.ReflectiveSubuniverse. Require Import AbGroups.AbelianGroup AbGroups.AbPullback AbGroups.Biproduct. Require Import AbSES.Core AbSES.DirectSum.

Algebra\AbSES\Pullback.v

abses_directsum_pullback_morphism

`{Univalence} {A B B' C D D' : AbGroup@{u}} {E : AbSES B A} {F : AbSES D C} (f : B' $-> B) (g : D' $-> D) : abses_pullback (functor_ab_biprod f g) (abses_direct_sum E F) = abses_direct_sum (abses_pullback f E) (abses_pullback g F) := (abses_pullback_component1_id (abses_directsum_pullback_morphism f g) (fun _ => idpath))^.

Definition

Require Import Basics Types. Require Import HSet Limits.Pullback. Require Import WildCat Pointed.Core Homotopy.ExactSequence. Require Import Modalities.ReflectiveSubuniverse. Require Import AbGroups.AbelianGroup AbGroups.AbPullback AbGroups.Biproduct. Require Import AbSES.Core AbSES.DirectSum.

Algebra\AbSES\Pullback.v

abses_directsum_distributive_pullbacks

{A B B' : AbGroup@{u}} (bt : B' $-> B) (E : AbSES B A) (F : AbSES B' A) (p : abses_pullback bt E = F) : exists phi : middle F $-> E, projection E o phi == bt o projection F. Proof. induction p. exists (grp_pullback_pr1 _ _); intro x. nrapply pullback_commsq. Defined.

Definition

Require Import Basics Types. Require Import HSet Limits.Pullback. Require Import WildCat Pointed.Core Homotopy.ExactSequence. Require Import Modalities.ReflectiveSubuniverse. Require Import AbGroups.AbelianGroup AbGroups.AbPullback AbGroups.Biproduct. Require Import AbSES.Core AbSES.DirectSum.

Algebra\AbSES\Pullback.v

abses_path_pullback_projection_commsq

`{Univalence} {A B B' : AbGroup} (f : B' $-> B) {E F : AbSES B A} (p : E = F) : ap (abses_pullback f) p = equiv_path_abses_iso (fmap (abses_pullback f) (equiv_path_abses_iso^-1 p)). Proof. induction p. nrefine (_ @ ap equiv_path_abses_iso _). 2: refine ((fmap_id_strong _ _)^ @ ap _ equiv_path_absesV_1^). exact equiv_path_abses_1^. Defined.

Lemma

Require Import Basics Types. Require Import HSet Limits.Pullback. Require Import WildCat Pointed.Core Homotopy.ExactSequence. Require Import Modalities.ReflectiveSubuniverse. Require Import AbGroups.AbelianGroup AbGroups.AbPullback AbGroups.Biproduct. Require Import AbSES.Core AbSES.DirectSum.

Algebra\AbSES\Pullback.v

ap_abses_pullback

`{Univalence} {A B B' : AbGroup} (f : B' $-> B) {E F : AbSES B A} (p : abses_path_data_iso E F) : ap (abses_pullback f) (equiv_path_abses_iso p) = equiv_path_abses_iso (fmap (abses_pullback f) p). Proof. refine (ap_abses_pullback _ _ @ _). apply (ap (equiv_path_abses_iso o _)). apply eissect. Defined.

Lemma

Require Import Basics Types. Require Import HSet Limits.Pullback. Require Import WildCat Pointed.Core Homotopy.ExactSequence. Require Import Modalities.ReflectiveSubuniverse. Require Import AbGroups.AbelianGroup AbGroups.AbPullback AbGroups.Biproduct. Require Import AbSES.Core AbSES.DirectSum.

Algebra\AbSES\Pullback.v

ap_abses_pullback_data

abses_pullback_point' {A B B' : AbGroup} (f : B' $-> B) : (abses_pullback f pt) $== (point (AbSES B' A)). Proof. snrefine (_; (_, _)). - snrapply Build_GroupIsomorphism. + srapply ab_biprod_corec. * refine (ab_biprod_pr1 $o _). apply grp_pullback_pr1. * apply projection. + srapply isequiv_adjointify. * snrapply grp_pullback_corec. -- exact (functor_ab_biprod grp_homo_id f). -- exact ab_biprod_pr2. -- reflexivity. * reflexivity. * intros [[a b] [b' c]]. srapply equiv_path_pullback_hset; split; cbn. 2: reflexivity. exact (path_prod' idpath c^). - reflexivity. - reflexivity. Defined.

Definition

Require Import Basics Types. Require Import HSet Limits.Pullback. Require Import WildCat Pointed.Core Homotopy.ExactSequence. Require Import Modalities.ReflectiveSubuniverse. Require Import AbGroups.AbelianGroup AbGroups.AbPullback AbGroups.Biproduct. Require Import AbSES.Core AbSES.DirectSum.

Algebra\AbSES\Pullback.v

abses_pullback_point'

`{Univalence} {A B B' : AbGroup} (f : B' $-> B) : abses_pullback f pt = pt :> AbSES B' A := equiv_path_abses_iso (' f).

Definition

Require Import Basics Types. Require Import HSet Limits.Pullback. Require Import WildCat Pointed.Core Homotopy.ExactSequence. Require Import Modalities.ReflectiveSubuniverse. Require Import AbGroups.AbelianGroup AbGroups.AbPullback AbGroups.Biproduct. Require Import AbSES.Core AbSES.DirectSum.

Algebra\AbSES\Pullback.v

abses_pullback_point

abses_pullback' {A B B' : AbGroup} (f : B' $-> B) : AbSES B A -->* AbSES B' A := Build_BasepointPreservingFunctor (abses_pullback f) (abses_pullback_point' f).

Definition

Require Import Basics Types. Require Import HSet Limits.Pullback. Require Import WildCat Pointed.Core Homotopy.ExactSequence. Require Import Modalities.ReflectiveSubuniverse. Require Import AbGroups.AbelianGroup AbGroups.AbPullback AbGroups.Biproduct. Require Import AbSES.Core AbSES.DirectSum.

Algebra\AbSES\Pullback.v

abses_pullback'

`{Univalence} {A B B' : AbGroup} (f : B' $-> B) : AbSES B A ->* AbSES B' A := to_pointed (abses_pullback' f).

Definition

Require Import Basics Types. Require Import HSet Limits.Pullback. Require Import WildCat Pointed.Core Homotopy.ExactSequence. Require Import Modalities.ReflectiveSubuniverse. Require Import AbGroups.AbelianGroup AbGroups.AbPullback AbGroups.Biproduct. Require Import AbSES.Core AbSES.DirectSum.

Algebra\AbSES\Pullback.v

abses_pullback_pmap

`{Univalence} {A B : AbGroup} : abses_pullback (A:=A) (@grp_homo_id B) == idmap. Proof. intro E. apply equiv_path_abses_iso; srefine (_; (_, _)). 1: rapply (Build_GroupIsomorphism _ _ (grp_pullback_pr1 _ _)). 1: reflexivity. intros [a [p q]]; cbn. exact q^. Defined.

Definition

Require Import Basics Types. Require Import HSet Limits.Pullback. Require Import WildCat Pointed.Core Homotopy.ExactSequence. Require Import Modalities.ReflectiveSubuniverse. Require Import AbGroups.AbelianGroup AbGroups.AbPullback AbGroups.Biproduct. Require Import AbSES.Core AbSES.DirectSum.

Algebra\AbSES\Pullback.v

abses_pullback_id

`{Univalence} {A B : AbGroup} : abses_pullback_pmap (A:=A) (@grp_homo_id B) ==* pmap_idmap. Proof. srapply Build_pHomotopy. 1: apply abses_pullback_id. refine (_ @ (concat_p1 _)^). nrapply (ap equiv_path_abses_iso). apply path_sigma_hprop. apply equiv_path_groupisomorphism. intros [[a b] [b' p]]; cbn; cbn in p. by apply path_prod'. Defined.

Definition

Require Import Basics Types. Require Import HSet Limits.Pullback. Require Import WildCat Pointed.Core Homotopy.ExactSequence. Require Import Modalities.ReflectiveSubuniverse. Require Import AbGroups.AbelianGroup AbGroups.AbPullback AbGroups.Biproduct. Require Import AbSES.Core AbSES.DirectSum.

Algebra\AbSES\Pullback.v

abses_pullback_pmap_id

abses_pullback_compose' {A B0 B1 B2 : AbGroup@{u}} (f : B0 $-> B1) (g : B1 $-> B2) : abses_pullback (A:=A) f o abses_pullback g $=> abses_pullback (g $o f). Proof. intro E; srefine (_; (_,_)). - apply equiv_grp_pullback_compose_r. - intro a. by srapply equiv_path_pullback_hset. - reflexivity. Defined.

Definition

Require Import Basics Types. Require Import HSet Limits.Pullback. Require Import WildCat Pointed.Core Homotopy.ExactSequence. Require Import Modalities.ReflectiveSubuniverse. Require Import AbGroups.AbelianGroup AbGroups.AbPullback AbGroups.Biproduct. Require Import AbSES.Core AbSES.DirectSum.

Algebra\AbSES\Pullback.v

abses_pullback_compose'

`{Univalence} {A B0 B1 B2 : AbGroup@{u}} (f : B0 $-> B1) (g : B1 $-> B2) : abses_pullback (A:=A) f o abses_pullback g == abses_pullback (g $o f) := fun x => equiv_path_abses_iso (' f g x).

Definition

Require Import Basics Types. Require Import HSet Limits.Pullback. Require Import WildCat Pointed.Core Homotopy.ExactSequence. Require Import Modalities.ReflectiveSubuniverse. Require Import AbGroups.AbelianGroup AbGroups.AbPullback AbGroups.Biproduct. Require Import AbSES.Core AbSES.DirectSum.

Algebra\AbSES\Pullback.v

abses_pullback_compose

abses_pullback_pcompose' {B0 B1 B2 A : AbGroup} (f : B0 $-> B1) (g : B1 $-> B2) : abses_pullback' f $o* abses_pullback' g $=>* abses_pullback' (A:=A) (g $o f). Proof. exists (abses_pullback_compose' f g). intros [[[a b2] [b1 c]] [b0 c']]; cbn in c, c'. srapply equiv_path_pullback_hset; split; cbn. 2: reflexivity. exact (path_prod' idpath (c @ ap g c')). Defined.

Definition

Require Import Basics Types. Require Import HSet Limits.Pullback. Require Import WildCat Pointed.Core Homotopy.ExactSequence. Require Import Modalities.ReflectiveSubuniverse. Require Import AbGroups.AbelianGroup AbGroups.AbPullback AbGroups.Biproduct. Require Import AbSES.Core AbSES.DirectSum.

Algebra\AbSES\Pullback.v

abses_pullback_pcompose'

`{Univalence} {A B0 B1 B2 : AbGroup} (f : B0 $-> B1) (g : B1 $-> B2) : abses_pullback_pmap (A:=A) f o* abses_pullback_pmap g ==* abses_pullback_pmap (g $o f). Proof. refine (to_pointed_compose _ _ @* _). apply equiv_ptransformation_phomotopy. apply '. Defined.

Definition

Require Import Basics Types. Require Import HSet Limits.Pullback. Require Import WildCat Pointed.Core Homotopy.ExactSequence. Require Import Modalities.ReflectiveSubuniverse. Require Import AbGroups.AbelianGroup AbGroups.AbPullback AbGroups.Biproduct. Require Import AbSES.Core AbSES.DirectSum.

Algebra\AbSES\Pullback.v

abses_pullback_pcompose

abses_pullback_const' {A B B' : AbGroup} : const pt $=> (@abses_pullback A B B' grp_homo_const). Proof. intro E. simpl. nrapply abses_path_data_to_iso. srefine (_;(_,_)); cbn. - srapply grp_pullback_corec. + exact (inclusion _ $o ab_biprod_pr1). + exact ab_biprod_pr2. + intro x; cbn. apply iscomplex_abses. - intro a; cbn. by srapply equiv_path_pullback_hset; split. - reflexivity. Defined.

Lemma

Require Import Basics Types. Require Import HSet Limits.Pullback. Require Import WildCat Pointed.Core Homotopy.ExactSequence. Require Import Modalities.ReflectiveSubuniverse. Require Import AbGroups.AbelianGroup AbGroups.AbPullback AbGroups.Biproduct. Require Import AbSES.Core AbSES.DirectSum.

Algebra\AbSES\Pullback.v

abses_pullback_const'

`{Univalence} {A B B' : AbGroup} : const pt == @abses_pullback A B B' grp_homo_const := fun x => (equiv_path_abses_iso (' x)).

Definition

Require Import Basics Types. Require Import HSet Limits.Pullback. Require Import WildCat Pointed.Core Homotopy.ExactSequence. Require Import Modalities.ReflectiveSubuniverse. Require Import AbGroups.AbelianGroup AbGroups.AbPullback AbGroups.Biproduct. Require Import AbSES.Core AbSES.DirectSum.

Algebra\AbSES\Pullback.v

abses_pullback_const

abses_pullback_pconst' {A B B' : AbGroup} : @pmap_abses_const B' A B A $=>* abses_pullback' grp_homo_const. Proof. srefine (_; _). 1: rapply abses_pullback_const'. lazy beta. intro x; reflexivity. Defined.

Lemma

Require Import Basics Types. Require Import HSet Limits.Pullback. Require Import WildCat Pointed.Core Homotopy.ExactSequence. Require Import Modalities.ReflectiveSubuniverse. Require Import AbGroups.AbelianGroup AbGroups.AbPullback AbGroups.Biproduct. Require Import AbSES.Core AbSES.DirectSum.

Algebra\AbSES\Pullback.v

abses_pullback_pconst'

`{Univalence} {A B B' : AbGroup} : pconst ==* @abses_pullback_pmap _ A B B' grp_homo_const. Proof. refine (pmap_abses_const_to_pointed @* _). rapply equiv_ptransformation_phomotopy. exact '. Defined.

Definition

Require Import Basics Types. Require Import HSet Limits.Pullback. Require Import WildCat Pointed.Core Homotopy.ExactSequence. Require Import Modalities.ReflectiveSubuniverse. Require Import AbGroups.AbelianGroup AbGroups.AbPullback AbGroups.Biproduct. Require Import AbSES.Core AbSES.DirectSum.

Algebra\AbSES\Pullback.v

abses_pullback_pconst

{B A : AbGroup} (E : AbSES B A) : AbSESMorphism (pt : AbSES E A) E. Proof. srapply (Build_AbSESMorphism grp_homo_id _ (projection E)). - cbn. exact (ab_biprod_rec (inclusion E) grp_homo_id). - intro x; cbn. exact (right_identity _)^. - intros [a e]; cbn. refine (grp_homo_op _ _ _ @ _). refine (ap (fun x => sg_op x _) _ @ _). 1: apply isexact_inclusion_projection. apply left_identity. Defined.

Definition

Require Import Basics Types. Require Import HSet Limits.Pullback. Require Import WildCat Pointed.Core Homotopy.ExactSequence. Require Import Modalities.ReflectiveSubuniverse. Require Import AbGroups.AbelianGroup AbGroups.AbPullback AbGroups.Biproduct. Require Import AbSES.Core AbSES.DirectSum.

Algebra\AbSES\Pullback.v

abses_pullback_projection_morphism

`{Univalence} {B A : AbGroup} (E : AbSES B A) : pt = abses_pullback (projection E) E := abses_pullback_component1_id (abses_pullback_projection_morphism E) (fun _ => idpath).

Definition

Require Import Basics Types. Require Import HSet Limits.Pullback. Require Import WildCat Pointed.Core Homotopy.ExactSequence. Require Import Modalities.ReflectiveSubuniverse. Require Import AbGroups.AbelianGroup AbGroups.AbPullback AbGroups.Biproduct. Require Import AbSES.Core AbSES.DirectSum.

Algebra\AbSES\Pullback.v

abses_pullback_projection

abses_pullback_homotopic' {A B B' : AbGroup} (f f' : B $-> B') (h : f == f') : abses_pullback (A:=A) f $=> abses_pullback f'. Proof. intro E. srefine (_; (_, _)). - srapply equiv_functor_grp_pullback. 1-3: exact grp_iso_id. 1: reflexivity. apply h. - intro a; cbn. by srapply equiv_path_pullback_hset; split. - reflexivity. Defined.

Lemma

Require Import Basics Types. Require Import HSet Limits.Pullback. Require Import WildCat Pointed.Core Homotopy.ExactSequence. Require Import Modalities.ReflectiveSubuniverse. Require Import AbGroups.AbelianGroup AbGroups.AbPullback AbGroups.Biproduct. Require Import AbSES.Core AbSES.DirectSum.

Algebra\AbSES\Pullback.v

abses_pullback_homotopic'

`{Univalence} {A B B' : AbGroup} (f f' : B $-> B') (h : f == f') : abses_pullback (A:=A) f == abses_pullback f'. Proof. intro E. apply equiv_path_abses_iso. exact (' _ _ h _). Defined.

Lemma

Require Import Basics Types. Require Import HSet Limits.Pullback. Require Import WildCat Pointed.Core Homotopy.ExactSequence. Require Import Modalities.ReflectiveSubuniverse. Require Import AbGroups.AbelianGroup AbGroups.AbPullback AbGroups.Biproduct. Require Import AbSES.Core AbSES.DirectSum.

Algebra\AbSES\Pullback.v

abses_pullback_homotopic

abses_pullback_phomotopic' {A B B' : AbGroup} (f f' : B $-> B') (h : f == f') : abses_pullback' (A:=A) f $=>* abses_pullback' f'. Proof. exists (abses_pullback_homotopic' f f' h); cbn. intros [[a b'] [b c]]; cbn in c. srapply equiv_path_pullback_hset; split; cbn. 2: reflexivity. exact (path_prod' idpath (c @ h b)). Defined.

Lemma

Require Import Basics Types. Require Import HSet Limits.Pullback. Require Import WildCat Pointed.Core Homotopy.ExactSequence. Require Import Modalities.ReflectiveSubuniverse. Require Import AbGroups.AbelianGroup AbGroups.AbPullback AbGroups.Biproduct. Require Import AbSES.Core AbSES.DirectSum.

Algebra\AbSES\Pullback.v

abses_pullback_phomotopic'

`{Univalence} {A B B' : AbGroup} (f f' : B $-> B') (h : f == f') : abses_pullback_pmap (A:=A) f ==* abses_pullback_pmap f' := equiv_ptransformation_phomotopy (' f f' h).

Definition

Require Import Basics Types. Require Import HSet Limits.Pullback. Require Import WildCat Pointed.Core Homotopy.ExactSequence. Require Import Modalities.ReflectiveSubuniverse. Require Import AbGroups.AbelianGroup AbGroups.AbPullback AbGroups.Biproduct. Require Import AbSES.Core AbSES.DirectSum.

Algebra\AbSES\Pullback.v

abses_pullback_phomotopic

iscomplex_abses_pullback' {A B0 B1 B2 : AbGroup} (f : B0 $-> B1) (g : B1 $-> B2) (h : g $o f == grp_homo_const) : abses_pullback' f $o* abses_pullback' g $=>* @pmap_abses_const _ _ _ A. Proof. refine (abses_pullback_pcompose' _ _ $@* _). refine (abses_pullback_phomotopic' _ _ h $@* _). exact abses_pullback_pconst'^*$. Defined.

Definition

Require Import Basics Types. Require Import HSet Limits.Pullback. Require Import WildCat Pointed.Core Homotopy.ExactSequence. Require Import Modalities.ReflectiveSubuniverse. Require Import AbGroups.AbelianGroup AbGroups.AbPullback AbGroups.Biproduct. Require Import AbSES.Core AbSES.DirectSum.

Algebra\AbSES\Pullback.v

iscomplex_abses_pullback'

`{Univalence} {A B0 B1 B2 : AbGroup} (f : B0 $-> B1) (g : B1 $-> B2) (h : g $o f == grp_homo_const) : IsComplex (abses_pullback_pmap (A:=A) g) (abses_pullback_pmap f). Proof. refine (_ @* _). 2: symmetry; exact pmap_abses_const_to_pointed. refine (to_pointed_compose _ _ @* _). apply equiv_ptransformation_phomotopy. by rapply '. Defined.

Definition

Require Import Basics Types. Require Import HSet Limits.Pullback. Require Import WildCat Pointed.Core Homotopy.ExactSequence. Require Import Modalities.ReflectiveSubuniverse. Require Import AbGroups.AbelianGroup AbGroups.AbPullback AbGroups.Biproduct. Require Import AbSES.Core AbSES.DirectSum.

Algebra\AbSES\Pullback.v

iscomplex_abses_pullback

`{Univalence} {A B C : AbGroup} (E : AbSES C B) : IsComplex (abses_pullback_pmap (A:=A) (projection E)) (abses_pullback_pmap (inclusion E)). Proof. rapply iscomplex_abses_pullback. rapply iscomplex_abses. Defined.

Definition

Require Import Basics Types. Require Import HSet Limits.Pullback. Require Import WildCat Pointed.Core Homotopy.ExactSequence. Require Import Modalities.ReflectiveSubuniverse. Require Import AbGroups.AbelianGroup AbGroups.AbPullback AbGroups.Biproduct. Require Import AbSES.Core AbSES.DirectSum.

Algebra\AbSES\Pullback.v

iscomplex_pullback_abses

`{Univalence} {X : Type} {A B : AbGroup} (f : X -> AbSES B A) (E : AbSES B A) : graph_hfiber f E <~> hfiber f E := equiv_functor_sigma_id (fun _ => equiv_path_abses_iso).

Definition

Require Import Basics Types. Require Import HSet Limits.Pullback. Require Import WildCat Pointed.Core Homotopy.ExactSequence. Require Import Modalities.ReflectiveSubuniverse. Require Import AbGroups.AbelianGroup AbGroups.AbPullback AbGroups.Biproduct. Require Import AbSES.Core AbSES.DirectSum.

Algebra\AbSES\Pullback.v

equiv_hfiber_abses

{A B B' : AbGroup} {f : B' $-> B} {X : AbSES B' A} (E F : graph_hfiber (abses_pullback f) X) := {p : E.

Definition

Require Import Basics Types. Require Import HSet Limits.Pullback. Require Import WildCat Pointed.Core Homotopy.ExactSequence. Require Import Modalities.ReflectiveSubuniverse. Require Import AbGroups.AbelianGroup AbGroups.AbPullback AbGroups.Biproduct. Require Import AbSES.Core AbSES.DirectSum.

Algebra\AbSES\Pullback.v

hfiber_abses_path

`{Univalence} {A B B' : AbGroup} {f : B' $-> B} {Y : AbSES B' A} {X0 : graph_hfiber (abses_pullback f) Y} {X1 : AbSES B A} (p : X0.1 = X1) : transport (fun x : AbSES B A => abses_pullback f x $== Y) p X0.2 = fmap (abses_pullback f) (equiv_path_abses_iso^-1 p^) $@ X0.2. Proof. induction p. refine (transport_1 _ _ @ _). nrefine (_ @ (ap (fun x => x $@ _)) _). 2: { refine (_ @ ap _ equiv_path_absesV_1^). exact (fmap_id_strong _ _)^. } exact (cat_idr_strong _)^. Defined.

Definition

Require Import Basics Types. Require Import HSet Limits.Pullback. Require Import WildCat Pointed.Core Homotopy.ExactSequence. Require Import Modalities.ReflectiveSubuniverse. Require Import AbGroups.AbelianGroup AbGroups.AbPullback AbGroups.Biproduct. Require Import AbSES.Core AbSES.DirectSum.

Algebra\AbSES\Pullback.v

transport_path_data_hfiber_abses_pullback_l

`{Univalence} {A B B' : AbGroup} {f : B' $-> B} (Y : AbSES B' A) (U V : graph_hfiber (abses_pullback f) Y) : hfiber_abses_path U V <~> U = V. Proof. refine (equiv_path_sigma _ _ _ oE _). srapply (equiv_functor_sigma' equiv_path_abses_iso); intro p; lazy beta. refine (equiv_concat_l _ _ oE _). { refine (transport_path_data_hfiber_abses_pullback_l _ @ _). refine (ap (fun x => (fmap (abses_pullback f) x) $@ _) _ @ _). { refine (ap _ (abses_path_data_V p) @ _). apply eissect. } refine (ap (fun x => x $@ _) _). rapply gpd_strong_1functor_V. } refine (equiv_path_sigma_hprop _ _ oE _). apply equiv_path_groupisomorphism. Defined.

Definition

Require Import Basics Types. Require Import HSet Limits.Pullback. Require Import WildCat Pointed.Core Homotopy.ExactSequence. Require Import Modalities.ReflectiveSubuniverse. Require Import AbGroups.AbelianGroup AbGroups.AbPullback AbGroups.Biproduct. Require Import AbSES.Core AbSES.DirectSum.

Algebra\AbSES\Pullback.v

equiv_hfiber_abses_pullback

{A B E : AbGroup} (i : B $-> E) `{IsEmbedding i} (F : AbSES E A) (p : abses_pullback i F $== pt) : B $-> F := grp_pullback_pr1 _ _ $o p^$.1 $o ab_biprod_inr. Local Instance abses_pullback_inclusion_lemma {A B E : AbGroup} (i : B $-> E) `{IsEmbedding i} (F : AbSES E A) (p : abses_pullback i F $== pt) : IsEmbedding (grp_pullback_pr1 _ _ $o p^$.1). Proof. nrapply (istruncmap_compose (-1) p^$.1 (grp_pullback_pr1 (projection F) i)). all: rapply istruncmap_mapinO_tr. Defined.

Definition

Require Import Basics Types HSet HFiber Limits.Pullback. Require Import WildCat Pointed.Core Homotopy.ExactSequence. Require Import Groups.QuotientGroup. Require Import AbGroups.AbelianGroup AbGroups.AbPullback AbGroups.Biproduct. Require Import AbSES.Core AbSES.Pullback. Require Import Modalities.Identity Modalities.Modality Truncations.Core.

Algebra\AbSES\PullbackFiberSequence.v

abses_pullback_inclusion_transpose_map

abses_pullback_inclusion_transpose_endpoint' {A B E : AbGroup} (i : B $-> E) `{IsEmbedding i} (F : AbSES E A) (p : abses_pullback i F $== pt) : AbGroup := ab_cokernel_embedding (abses_pullback_inclusion_transpose_map i F p).

Definition

Require Import Basics Types HSet HFiber Limits.Pullback. Require Import WildCat Pointed.Core Homotopy.ExactSequence. Require Import Groups.QuotientGroup. Require Import AbGroups.AbelianGroup AbGroups.AbPullback AbGroups.Biproduct. Require Import AbSES.Core AbSES.Pullback. Require Import Modalities.Identity Modalities.Modality Truncations.Core.

Algebra\AbSES\PullbackFiberSequence.v

abses_pullback_inclusion_transpose_endpoint'

{A B E : AbGroup} (i : B $-> E) `{IsEmbedding i} (F : AbSES E A) (p : abses_pullback i F $== pt) : projection F $o (abses_pullback_inclusion_transpose_map i F p) == i. Proof. intro b. change b with (ab_biprod_pr2 (A:=A) (mon_unit, b)). refine (pullback_commsq _ _ _ @ ap i _). exact (snd p^$.2 _)^. Defined.

Lemma

Require Import Basics Types HSet HFiber Limits.Pullback. Require Import WildCat Pointed.Core Homotopy.ExactSequence. Require Import Groups.QuotientGroup. Require Import AbGroups.AbelianGroup AbGroups.AbPullback AbGroups.Biproduct. Require Import AbSES.Core AbSES.Pullback. Require Import Modalities.Identity Modalities.Modality Truncations.Core.

Algebra\AbSES\PullbackFiberSequence.v

abses_pullback_inclusion_transpose_beta

`{Univalence} {A B C : AbGroup} (E : AbSES C B) (F : AbSES (middle E) A) (p : abses_pullback (inclusion E) F $== pt) : AbSES C A. Proof. snrapply Build_AbSES. - exact (abses_pullback_inclusion_transpose_endpoint' (inclusion E) F p). - exact (grp_quotient_map $o inclusion F). - srapply (ab_cokernel_embedding_rec _ (projection E $o projection F)). intro b. refine (ap (projection E) (abses_pullback_inclusion_transpose_beta (inclusion E) F p b) @ _). apply iscomplex_abses. - apply isembedding_grouphomomorphism. intros a q0. pose proof (in_coset := related_quotient_paths _ _ _ q0). destruct in_coset as [b q1]; rewrite grp_unit_r in q1. assert (q2 : ab_biprod_inr b = ab_biprod_inl (-a)). 1: { apply (isinj_embedding (grp_pullback_pr1 _ _ $o p^$.1)). - apply abses_pullback_inclusion_lemma. exact _. - nrefine (q1 @ _); symmetry. refine (ap (grp_pullback_pr1 _ _) (fst p^$.2 (-a)) @ _). exact (grp_homo_inv _ _). } pose proof (q3 := ap negate (fst ((equiv_path_prod _ _)^-1 q2))); cbn in q3. exact ((negate_involutive _)^ @ q3^ @ negate_mon_unit). - apply (cancelR_conn_map (Tr (-1)) grp_quotient_map). 1: exact _. simpl. exact _. - snrapply Build_IsExact. + srapply phomotopy_homotopy_hset. intro a; simpl. refine (ap (projection E) _ @ _). 1: apply iscomplex_abses. apply grp_homo_unit. + intros [y q]. apply (@contr_inhabited_hprop _ _). assert (f : merely (hfiber grp_quotient_map y)). 1: apply center, issurj_class_of. revert_opaque f; apply Trunc_rec; intros [f q0]. assert (b : merely (hfiber (inclusion E) (projection F f))). 1: { rapply isexact_preimage. exact (ap _ q0 @ q). } revert_opaque b; apply Trunc_rec; intros [b q1]. assert (a : merely (hfiber (inclusion F) (sg_op f (-(grp_pullback_pr1 _ _ (p^$.1 (ab_biprod_inr b))))))). 1: { rapply isexact_preimage. refine (grp_homo_op _ _ _ @ _). refine (ap (fun x => _ + x) (grp_homo_inv _ _) @ _). refine (ap (fun x => _ - x) (abses_pullback_inclusion_transpose_beta (inclusion E) F p b @ q1) @ _). apply right_inverse. } revert_opaque a; apply Trunc_rec; intros [a q2]. refine (tr (a; _)). let T := type of y in apply (@path_sigma_hprop T). 1: intros ?; apply istrunc_paths; apply group_isgroup. refine (ap grp_quotient_map q2 @ _ @ q0). refine (grp_homo_op _ _ _ @ _). apply grp_moveR_Mg. refine (_ @ (left_inverse _)^). apply qglue. exists b. refine (_ @ (grp_unit_r _)^). exact (negate_involutive _)^. Defined.

Definition

Require Import Basics Types HSet HFiber Limits.Pullback. Require Import WildCat Pointed.Core Homotopy.ExactSequence. Require Import Groups.QuotientGroup. Require Import AbGroups.AbelianGroup AbGroups.AbPullback AbGroups.Biproduct. Require Import AbSES.Core AbSES.Pullback. Require Import Modalities.Identity Modalities.Modality Truncations.Core.

Algebra\AbSES\PullbackFiberSequence.v

abses_pullback_trivial_preimage

abses_pullback_inclusion0_map' `{Univalence} {A B C : AbGroup} (E : AbSES C B) (F : AbSES (middle E) A) (p : abses_pullback (inclusion E) F $== pt) : AbSESMorphism F (abses_pullback_trivial_preimage E F p). Proof. srapply Build_AbSESMorphism. - exact grp_homo_id. - exact grp_quotient_map. - exact (projection E). - reflexivity. - reflexivity. Defined.

Definition

Require Import Basics Types HSet HFiber Limits.Pullback. Require Import WildCat Pointed.Core Homotopy.ExactSequence. Require Import Groups.QuotientGroup. Require Import AbGroups.AbelianGroup AbGroups.AbPullback AbGroups.Biproduct. Require Import AbSES.Core AbSES.Pullback. Require Import Modalities.Identity Modalities.Modality Truncations.Core.

Algebra\AbSES\PullbackFiberSequence.v

abses_pullback_inclusion0_map'

cxfib' {A B C : AbGroup} (E : AbSES C B) : AbSES C A -> graph_hfiber (abses_pullback (A:=A) (inclusion E)) pt. Proof. intro Y. exists (abses_pullback (projection E) Y). refine (abses_pullback_compose' _ _ Y $@ _). refine (abses_pullback_homotopic' _ grp_homo_const _ Y $@ _). 1: rapply iscomplex_abses. symmetry; apply abses_pullback_const'. Defined.

Definition

Require Import Basics Types HSet HFiber Limits.Pullback. Require Import WildCat Pointed.Core Homotopy.ExactSequence. Require Import Groups.QuotientGroup. Require Import AbGroups.AbelianGroup AbGroups.AbPullback AbGroups.Biproduct. Require Import AbSES.Core AbSES.Pullback. Require Import Modalities.Identity Modalities.Modality Truncations.Core.

Algebra\AbSES\PullbackFiberSequence.v

cxfib'

hfiber_cxfib' {A B C : AbGroup} (E : AbSES C B) (F : AbSES (middle E) A) (p : abses_pullback (inclusion E) F $== pt) := {Y : AbSES C A & hfiber_abses_path (cxfib' E Y) (F; p)}. Local pr2_cxfib' `{Univalence} {A B C : AbGroup} {E : AbSES C B} (U : AbSES C A) : equiv_ptransformation_phomotopy (iscomplex_abses_pullback' _ _ (iscomplex_abses E)) U = equiv_path_abses_iso (cxfib' E U).2. Proof. change (equiv_ptransformation_phomotopy (iscomplex_abses_pullback' _ _ (iscomplex_abses E)) U) with (equiv_path_abses_iso ((iscomplex_abses_pullback' _ _ (iscomplex_abses E)).1 U)). apply (ap equiv_path_abses_iso). rapply path_hom. refine (_ $@R abses_pullback_compose' (inclusion E) (projection E) U); unfold trans_comp. refine (_ $@R abses_pullback_homotopic' (projection E $o inclusion E) grp_homo_const (iscomplex_abses E) U). reflexivity. Defined.

Definition

Require Import Basics Types HSet HFiber Limits.Pullback. Require Import WildCat Pointed.Core Homotopy.ExactSequence. Require Import Groups.QuotientGroup. Require Import AbGroups.AbelianGroup AbGroups.AbPullback AbGroups.Biproduct. Require Import AbSES.Core AbSES.Pullback. Require Import Modalities.Identity Modalities.Modality Truncations.Core.

Algebra\AbSES\PullbackFiberSequence.v

hfiber_cxfib'

equiv_hfiber_cxfib' `{Univalence} {A B C : AbGroup} {E : AbSES C B} (F : AbSES (middle E) A) (p : abses_pullback (inclusion E) F $== pt) : hfiber_cxfib' E F p <~> hfiber (cxfib (iscomplex_pullback_abses E)) (equiv_hfiber_abses _ pt (F;p)). Proof. srapply equiv_functor_sigma_id; intro U; lazy beta. refine (_ oE equiv_hfiber_abses_pullback _ _ _). refine (_ oE equiv_ap' (equiv_hfiber_abses _ pt) _ _). apply equiv_concat_l. apply eq_cxfib_cxfib'. Defined.

Definition

Require Import Basics Types HSet HFiber Limits.Pullback. Require Import WildCat Pointed.Core Homotopy.ExactSequence. Require Import Groups.QuotientGroup. Require Import AbGroups.AbelianGroup AbGroups.AbPullback AbGroups.Biproduct. Require Import AbSES.Core AbSES.Pullback. Require Import Modalities.Identity Modalities.Modality Truncations.Core.

Algebra\AbSES\PullbackFiberSequence.v

equiv_hfiber_cxfib'

path_hfiber_cxfib' {A B C : AbGroup} {E : AbSES C B} {F : AbSES (middle E) A} {p : abses_pullback (inclusion E) F $== pt} (X Y : hfiber_cxfib' (B:=B) E F p) : Type. Proof. refine (sig (fun q0 : X.1 $== Y.1 => _)). exact ((fmap (abses_pullback (projection E)) q0)^$ $@ X.2.1 $== Y.2.1). Defined.

Definition

Require Import Basics Types HSet HFiber Limits.Pullback. Require Import WildCat Pointed.Core Homotopy.ExactSequence. Require Import Groups.QuotientGroup. Require Import AbGroups.AbelianGroup AbGroups.AbPullback AbGroups.Biproduct. Require Import AbSES.Core AbSES.Pullback. Require Import Modalities.Identity Modalities.Modality Truncations.Core.

Algebra\AbSES\PullbackFiberSequence.v

path_hfiber_cxfib'

`{Univalence} {A B C : AbGroup} {E : AbSES C B} (F : AbSES (middle E) A) (p : abses_pullback (inclusion E) F $== pt) (U V : hfiber_cxfib' E F p) (q : U.1 = V.1) : (transport (fun Y : AbSES C A => hfiber_abses_path (cxfib' E Y) (F; p)) q U.2).1 = fmap (abses_pullback (projection E)) (equiv_path_abses_iso^-1 q^) $@ U.2.1. Proof. induction q. refine (ap pr1 (transport_1 _ _) @ _). refine (_ @ ap (fun x => fmap (abses_pullback (projection E)) x $@ _) equiv_path_absesV_1^). refine (_ @ ap (fun x => x $@ _) (fmap_id_strong _ _)^). exact (cat_idr_strong _)^. Defined.

Definition

Require Import Basics Types HSet HFiber Limits.Pullback. Require Import WildCat Pointed.Core Homotopy.ExactSequence. Require Import Groups.QuotientGroup. Require Import AbGroups.AbelianGroup AbGroups.AbPullback AbGroups.Biproduct. Require Import AbSES.Core AbSES.Pullback. Require Import Modalities.Identity Modalities.Modality Truncations.Core.

Algebra\AbSES\PullbackFiberSequence.v

transport_hfiber_abses_path_cxfib'_l

equiv_path_hfiber_cxfib' `{Univalence} {A B C : AbGroup} {E : AbSES C B} (F : AbSES (middle E) A) (p : abses_pullback (inclusion E) F $== pt) (U V : hfiber_cxfib' E F p) : path_hfiber_cxfib' U V <~> U = V. Proof. refine (equiv_path_sigma _ _ _ oE _). srapply (equiv_functor_sigma' equiv_path_abses_iso); intro q; lazy beta. refine (equiv_path_sigma_hprop _ _ oE _). refine (equiv_concat_l _ _ oE _). 1: apply transport_hfiber_abses_path_cxfib'_l. refine (equiv_path_sigma_hprop _ _ oE equiv_concat_l _ _ oE _). 1: { refine (ap (fun x => (fmap (abses_pullback _) x $@ _).1) _). nrefine (ap _ (abses_path_data_V q) @ _). apply eissect. } refine (equiv_concat_l _ _ oE _). 1: { refine (ap (fun x => (x $@ _).1) _). rapply gpd_strong_1functor_V. } apply equiv_path_groupisomorphism. Defined.

Definition

Require Import Basics Types HSet HFiber Limits.Pullback. Require Import WildCat Pointed.Core Homotopy.ExactSequence. Require Import Groups.QuotientGroup. Require Import AbGroups.AbelianGroup AbGroups.AbPullback AbGroups.Biproduct. Require Import AbSES.Core AbSES.Pullback. Require Import Modalities.Identity Modalities.Modality Truncations.Core.

Algebra\AbSES\PullbackFiberSequence.v

equiv_path_hfiber_cxfib'

`{Univalence} {A B C : AbGroup} (E : AbSES C B) (F : AbSES (middle E) A) (p : abses_pullback (inclusion E) F $== pt) : hfiber_cxfib' E F p. Proof. exists (abses_pullback_trivial_preimage E F p). srefine (_^$; _). 1: by rapply (abses_pullback_component1_id' (abses_pullback_inclusion0_map' E F p)). lazy beta; unfold pr2. refine (cat_assoc _ _ _ $@ _). refine (cat_assoc _ _ _ $@ _). apply gpd_moveR_Vh. apply gpd_moveL_hM. apply equiv_ab_biprod_ind_homotopy. split; apply equiv_path_pullback_rec_hset; split; cbn. - intro a. exact (ap (class_of _ o pullback_pr1) (fst p^$.2 a)). - intro a. exact ((snd p^$.2 _)^). - intro b; apply qglue. exists (-b). apply grp_moveL_Vg. refine ((grp_homo_op (grp_pullback_pr1 _ _ $o p^$.1 $o ab_biprod_inr) _ _)^ @ _). exact (ap _ (right_inverse _) @ grp_homo_unit _ @ (grp_homo_unit _)^). - intro b. exact (snd p^$.2 _)^. Defined.

Definition

Require Import Basics Types HSet HFiber Limits.Pullback. Require Import WildCat Pointed.Core Homotopy.ExactSequence. Require Import Groups.QuotientGroup. Require Import AbGroups.AbelianGroup AbGroups.AbPullback AbGroups.Biproduct. Require Import AbSES.Core AbSES.Pullback. Require Import Modalities.Identity Modalities.Modality Truncations.Core.

Algebra\AbSES\PullbackFiberSequence.v

hfiber_cxfib'_inhabited

abses_pullback_splits_induced_map' {A B C : AbGroup} (E : AbSES C B) (Y : AbSES C A) : ab_biprod A B $-> abses_pullback (projection E) Y. Proof. srapply (ab_biprod_rec (inclusion _)). srapply grp_pullback_corec. - exact grp_homo_const. - exact (inclusion E). - intro x. refine (grp_homo_unit _ @ _). symmetry; apply iscomplex_abses. Defined.

Definition

Require Import Basics Types HSet HFiber Limits.Pullback. Require Import WildCat Pointed.Core Homotopy.ExactSequence. Require Import Groups.QuotientGroup. Require Import AbGroups.AbelianGroup AbGroups.AbPullback AbGroups.Biproduct. Require Import AbSES.Core AbSES.Pullback. Require Import Modalities.Identity Modalities.Modality Truncations.Core.

Algebra\AbSES\PullbackFiberSequence.v

abses_pullback_splits_induced_map'

`{Univalence} {A B B' : AbGroup} (f : B' $-> B) (X Y : graph_hfiber (abses_pullback (A:=A) f) pt) (Q : hfiber_abses_path X Y) : fmap (abses_pullback f) Q.1^$ $o Y.2^$ $== X.2^$. Proof. generalize Q. equiv_intro (equiv_hfiber_abses_pullback _ X Y)^-1%equiv p; induction p. refine ((_ $@R _) $@ _). { Unshelve. 2: exact (Id _). refine (fmap2 _ _ $@ fmap_id _ _). intro x; reflexivity. } exact (cat_idl _). Defined.

Lemma

Require Import Basics Types HSet HFiber Limits.Pullback. Require Import WildCat Pointed.Core Homotopy.ExactSequence. Require Import Groups.QuotientGroup. Require Import AbGroups.AbelianGroup AbGroups.AbPullback AbGroups.Biproduct. Require Import AbSES.Core AbSES.Pullback. Require Import Modalities.Identity Modalities.Modality Truncations.Core.

Algebra\AbSES\PullbackFiberSequence.v

fmap_hfiber_abses_lemma

`{Univalence} {A B C : AbGroup} (E : AbSES C B) (F : AbSES (middle E) A) (p : abses_pullback (inclusion E) F $== pt) (Y : hfiber_cxfib' E F p) : hfiber_cxfib'_induced_map E F p Y == abses_pullback_splits_induced_map' E Y.1. Proof. intros [a b]; cbn. refine (ap pullback_pr1 (fmap_hfiber_abses_lemma _ _ (F;p) Y.2 _) @ _). srapply equiv_path_pullback_hset; split; cbn. - exact (grp_unit_r _)^. - exact (grp_unit_l _)^. Defined.

Lemma

Require Import Basics Types HSet HFiber Limits.Pullback. Require Import WildCat Pointed.Core Homotopy.ExactSequence. Require Import Groups.QuotientGroup. Require Import AbGroups.AbelianGroup AbGroups.AbPullback AbGroups.Biproduct. Require Import AbSES.Core AbSES.Pullback. Require Import Modalities.Identity Modalities.Modality Truncations.Core.

Algebra\AbSES\PullbackFiberSequence.v

induced_map_eq

`{Univalence} {A B C : AbGroup} (E : AbSES C B) (F : AbSES (middle E) A) (p : abses_pullback (inclusion E) F $== pt) (Y : hfiber_cxfib' E F p) : abses_pullback_trivial_preimage E F p $== Y.1. Proof. destruct Y as [Y Q]. apply abses_path_data_to_iso; srefine (_; (_,_)). - snrapply (ab_cokernel_embedding_rec _ (grp_pullback_pr1 _ _$o (Q.1^$).1)). 1-3: exact _. intro f. nrefine (ap _ (induced_map_eq E F p (Y;Q) _) @ _); cbn. exact (grp_unit_r _ @ grp_homo_unit _). - intro a. refine (_ @ ap (grp_pullback_pr1 _ _) (fst (Q.1^$).2 a)). exact (grp_quotient_rec_beta' _ F _ _ (inclusion F a)). - nrapply (conn_map_elim _ grp_quotient_map). 1: apply issurj_class_of. 1: intros ?; apply istrunc_paths; apply group_isgroup. intro f. refine (ap (projection E) (snd (Q.1^$).2 f) @ _); unfold pr1. exact (pullback_commsq _ _ ((Q.1^$).1 f))^. Defined.

Lemma

Require Import Basics Types HSet HFiber Limits.Pullback. Require Import WildCat Pointed.Core Homotopy.ExactSequence. Require Import Groups.QuotientGroup. Require Import AbGroups.AbelianGroup AbGroups.AbPullback AbGroups.Biproduct. Require Import AbSES.Core AbSES.Pullback. Require Import Modalities.Identity Modalities.Modality Truncations.Core.

Algebra\AbSES\PullbackFiberSequence.v

hfiber_cxfib'_induced_path'0

hfiber_cxfib'_induced_path' `{Univalence} {A B C : AbGroup} (E : AbSES C B) (F : AbSES (middle E) A) (p : abses_pullback (inclusion E) F $== pt) (Y : hfiber_cxfib' E F p) : path_hfiber_cxfib' (hfiber_cxfib'_inhabited E F p) Y. Proof. exists (0 E F p Y). rapply gpd_moveR_Vh. rapply gpd_moveL_hM. rapply gpd_moveR_Vh. intro x. srapply equiv_path_pullback_hset; split. 2: exact (snd Y.2.1^$.2 x)^. reflexivity. Defined.

Lemma

Require Import Basics Types HSet HFiber Limits.Pullback. Require Import WildCat Pointed.Core Homotopy.ExactSequence. Require Import Groups.QuotientGroup. Require Import AbGroups.AbelianGroup AbGroups.AbPullback AbGroups.Biproduct. Require Import AbSES.Core AbSES.Pullback. Require Import Modalities.Identity Modalities.Modality Truncations.Core.

Algebra\AbSES\PullbackFiberSequence.v

hfiber_cxfib'_induced_path'

contr_hfiber_cxfib' `{Univalence} {A B C : AbGroup} (E : AbSES C B) (F : AbSES (middle E) A) (p : abses_pullback (inclusion E) F $== pt) : Contr (hfiber_cxfib' E F p). Proof. srapply Build_Contr. 1: apply hfiber_cxfib'_inhabited. intros [Y q]. apply equiv_path_hfiber_cxfib'. apply hfiber_cxfib'_induced_path'. Defined.

Lemma

Require Import Basics Types HSet HFiber Limits.Pullback. Require Import WildCat Pointed.Core Homotopy.ExactSequence. Require Import Groups.QuotientGroup. Require Import AbGroups.AbelianGroup AbGroups.AbPullback AbGroups.Biproduct. Require Import AbSES.Core AbSES.Pullback. Require Import Modalities.Identity Modalities.Modality Truncations.Core.

Algebra\AbSES\PullbackFiberSequence.v

contr_hfiber_cxfib'

`{Univalence} {A A' B : AbGroup} (f : A $-> A') : AbSES B A -> AbSES B A'. Proof. intro E. snrapply (Build_AbSES (ab_pushout f (inclusion E)) ab_pushout_inl (ab_pushout_rec grp_homo_const (projection E) _)). - symmetry; rapply iscomplex_abses. - rapply ab_pushout_embedding_inl. - nrapply (cancelR_issurjection ab_pushout_inr _). rapply (conn_map_homotopic _ (projection E)); symmetry. nrapply ab_pushout_rec_beta_right. - snrapply Build_IsExact. + srapply phomotopy_homotopy_hset. nrapply ab_pushout_rec_beta_left. + intros [bc' p]. rapply contr_inhabited_hprop. assert (bc : merely (hfiber grp_quotient_map bc')). 1: apply center, issurj_class_of. strip_truncations. destruct bc as [[b c] q]. assert (c_in_kernel : (projection E) c = mon_unit). 1: { refine (_ @ p); symmetry. rewrite <- q; simpl. apply left_identity. } pose proof (a := isexact_preimage _ _ _ c c_in_kernel). strip_truncations. destruct a as [a s]. apply tr. exists (b + - f (- a)); cbn. apply path_sigma_hprop; cbn. change (ab_pushout_inl (b + - f (- a)) = bc'). refine (_ @ q). symmetry. apply path_ab_pushout; cbn. refine (tr (-a; _)). apply path_prod; cbn. * apply grp_moveL_Mg. by rewrite negate_involutive. * exact ((preserves_negate a) @ ap _ s @ (right_identity _)^). Defined.

Definition

Require Import Basics Types Truncations.Core. Require Import WildCat Pointed.Core Homotopy.ExactSequence HIT.epi. Require Import Modalities.ReflectiveSubuniverse. Require Import AbelianGroup AbPushout AbHom AbGroups.Biproduct. Require Import AbSES.Core AbSES.DirectSum.

Algebra\AbSES\Pushout.v

abses_pushout

`{Univalence} {A A' B : AbGroup} (E : AbSES B A) (f : A $-> A') : AbSESMorphism E (abses_pushout f E). Proof. snrapply (Build_AbSESMorphism f _ grp_homo_id). - exact ab_pushout_inr. - exact ab_pushout_commsq. - rapply ab_pushout_rec_beta_right. Defined.

Definition

Require Import Basics Types Truncations.Core. Require Import WildCat Pointed.Core Homotopy.ExactSequence HIT.epi. Require Import Modalities.ReflectiveSubuniverse. Require Import AbelianGroup AbPushout AbHom AbGroups.Biproduct. Require Import AbSES.Core AbSES.DirectSum.

Algebra\AbSES\Pushout.v

abses_pushout_morphism