Library HoTT.Algebra.Groups.FreeGroup
Require Import Basics Types Group Subgroup
WildCat.Core WildCat.Universe Colimits.Coeq
Truncations.Core Truncations.SeparatedTrunc
Spaces.List.Core Spaces.List.Theory.
Local Open Scope mc_scope.
Local Open Scope mc_mult_scope.
WildCat.Core WildCat.Universe Colimits.Coeq
Truncations.Core Truncations.SeparatedTrunc
Spaces.List.Core Spaces.List.Theory.
Local Open Scope mc_scope.
Local Open Scope mc_mult_scope.
IsFreeGroup is defined in Group.v. In this file we construct free groups and and prove properties about them.
We construct the free group on a type A as a higher inductive type. This construction is due to Kraus-Altenkirch 2018 arXiv:1805.02069. Their construction is actually more general, but we set-truncate it to suit our needs which is the free group as a set. This is a very simple HIT in a similar manner to the abelianization HIT used in Algebra.AbGroup.Abelianization.
We define words (with inverses) on A to be lists of marked elements of A
Given a marked element of A we can change its mark
We introduce a local notation for change_sign. It is only defined in this section however.
Changing sign is an involution
Now we wish to define the free group on A as the following HIT:
HIT N(A) : hSet
| eta : Words -> N(A)
| tau (x : Words) (a : A + A) (y : Words)
: eta (x ++ a ++ a^ ++ y) = eta (x ++ y).
Since we cannot write our HITs directly like this (without resorting to private inductive types), we will construct this HIT out of HITs we know. In fact, we can define N(A) as a coequalizer.
Local Definition map1 : Words × (A + A) × Words → Words.
Proof.
intros [[x a] y].
exact (x ++ [a] ++ [a^'] ++ y).
Defined.
Arguments map1 _ /.
Local Definition map2 : Words × (A + A) × Words → Words.
Proof.
intros [[x a] y].
exact (x ++ y).
Defined.
Arguments map2 _ /.
Now we can define the underlying type of the free group as the 0-truncated coequalizer of these two maps
This is the point constructor
This is the path constructor
Definition freegroup_tau (x : Words) (a : A + A) (y : Words)
: freegroup_eta (x ++ [a] ++ [a^'] ++ y) = freegroup_eta (x ++ y).
Proof.
apply path_Tr, tr.
exact ((cglue (x, a, y))).
Defined.
: freegroup_eta (x ++ [a] ++ [a^'] ++ y) = freegroup_eta (x ++ y).
Proof.
apply path_Tr, tr.
exact ((cglue (x, a, y))).
Defined.
The group operation
#[export] Instance sgop_freegroup : SgOp freegroup_type.
Proof.
intros x y.
strip_truncations.
revert x; snapply Coeq_rec.
{ intros x; revert y.
snapply Coeq_rec.
{ intros y.
exact (freegroup_eta (x ++ y)). }
intros [[y a] z]; simpl.
change (freegroup_eta (x ++ y ++ ([a] ++ [a^'] ++ z))
= freegroup_eta (x ++ y ++ z)).
rhs napply ap.
2: napply app_assoc.
lhs napply ap.
1: napply app_assoc.
napply (freegroup_tau _ a). }
intros [[c b] d].
revert y.
srapply Coeq_ind_hprop.
intro a.
change (freegroup_eta ((c ++ [b] ++ [b^'] ++ d) ++ a)
= freegroup_eta ((c ++ d) ++ a)).
lhs_V napply ap.
1: napply app_assoc.
lhs_V napply (ap (fun x ⇒ freegroup_eta (c ++ x))).
1: napply app_assoc.
lhs_V napply (ap (fun x ⇒ freegroup_eta (c ++ _ ++ x))).
1: napply app_assoc.
rhs_V napply ap.
2: napply app_assoc.
napply freegroup_tau.
Defined.
Proof.
intros x y.
strip_truncations.
revert x; snapply Coeq_rec.
{ intros x; revert y.
snapply Coeq_rec.
{ intros y.
exact (freegroup_eta (x ++ y)). }
intros [[y a] z]; simpl.
change (freegroup_eta (x ++ y ++ ([a] ++ [a^'] ++ z))
= freegroup_eta (x ++ y ++ z)).
rhs napply ap.
2: napply app_assoc.
lhs napply ap.
1: napply app_assoc.
napply (freegroup_tau _ a). }
intros [[c b] d].
revert y.
srapply Coeq_ind_hprop.
intro a.
change (freegroup_eta ((c ++ [b] ++ [b^'] ++ d) ++ a)
= freegroup_eta ((c ++ d) ++ a)).
lhs_V napply ap.
1: napply app_assoc.
lhs_V napply (ap (fun x ⇒ freegroup_eta (c ++ x))).
1: napply app_assoc.
lhs_V napply (ap (fun x ⇒ freegroup_eta (c ++ _ ++ x))).
1: napply app_assoc.
rhs_V napply ap.
2: napply app_assoc.
napply freegroup_tau.
Defined.
The unit of the free group is the empty word
We can change the sign of all the elements in a word and reverse the order. This will be the inversion in the group
Changing the sign changes the order of word concatenation
Definition word_change_sign_ww (x y : Words)
: word_change_sign (x ++ y) = word_change_sign y ++ word_change_sign x.
Proof.
unfold word_change_sign.
lhs napply (ap reverse).
1: napply list_map_app.
napply reverse_app.
Defined.
: word_change_sign (x ++ y) = word_change_sign y ++ word_change_sign x.
Proof.
unfold word_change_sign.
lhs napply (ap reverse).
1: napply list_map_app.
napply reverse_app.
Defined.
This is also involutive
Lemma word_change_sign_inv x : word_change_sign (word_change_sign x) = x.
Proof.
unfold word_change_sign.
lhs_V napply list_map_reverse.
lhs napply ap.
1: napply reverse_reverse.
lhs_V napply list_map_compose.
snapply list_map_id.
intros a ?.
apply change_sign_inv.
Defined.
Proof.
unfold word_change_sign.
lhs_V napply list_map_reverse.
lhs napply ap.
1: napply reverse_reverse.
lhs_V napply list_map_compose.
snapply list_map_id.
intros a ?.
apply change_sign_inv.
Defined.
Changing the sign gives us left inverses
Lemma word_concat_Vw x : freegroup_eta (word_change_sign x ++ x) = mon_unit.
Proof.
induction x.
1: reflexivity.
lhs napply (ap (fun x ⇒ freegroup_eta (x ++ _))).
1: napply reverse_cons.
change (freegroup_eta ((word_change_sign x ++ [a^']) ++ [a] ++ x)
= mon_unit).
lhs_V napply ap.
1: napply app_assoc.
set (a' := a^').
rewrite <- (change_sign_inv a).
lhs napply freegroup_tau.
exact IHx.
Defined.
Proof.
induction x.
1: reflexivity.
lhs napply (ap (fun x ⇒ freegroup_eta (x ++ _))).
1: napply reverse_cons.
change (freegroup_eta ((word_change_sign x ++ [a^']) ++ [a] ++ x)
= mon_unit).
lhs_V napply ap.
1: napply app_assoc.
set (a' := a^').
rewrite <- (change_sign_inv a).
lhs napply freegroup_tau.
exact IHx.
Defined.
And since changing the sign is involutive we get right inverses from left inverses
Lemma word_concat_wV x : freegroup_eta (x ++ word_change_sign x) = mon_unit.
Proof.
set (x' := word_change_sign x).
rewrite <- (word_change_sign_inv x).
change (freegroup_eta (word_change_sign x' ++ x') = mon_unit).
apply word_concat_Vw.
Defined.
Proof.
set (x' := word_change_sign x).
rewrite <- (word_change_sign_inv x).
change (freegroup_eta (word_change_sign x' ++ x') = mon_unit).
apply word_concat_Vw.
Defined.
Inverses are defined by changing the order of a word that appears in eta. Most of the work here is checking that it is agreeable with the path constructor.
#[export] Instance inverse_freegroup_type : Inverse freegroup_type.
Proof.
intro x.
strip_truncations.
revert x; srapply Coeq_rec.
{ intro x.
apply freegroup_eta.
exact (word_change_sign x). }
intros [[b a] c].
unfold map1, map2.
lhs napply ap.
{ lhs napply word_change_sign_ww.
napply (ap (fun x ⇒ x ++ _)).
lhs napply word_change_sign_ww.
napply (ap (fun x ⇒ x ++ _)).
lhs napply word_change_sign_ww.
napply (ap (fun x ⇒ _ ++ x)).
exact (word_change_sign_inv [a]). }
lhs_V napply ap.
1: rhs_V napply app_assoc.
1: napply app_assoc.
rhs napply ap.
2: napply word_change_sign_ww.
napply freegroup_tau.
Defined.
Proof.
intro x.
strip_truncations.
revert x; srapply Coeq_rec.
{ intro x.
apply freegroup_eta.
exact (word_change_sign x). }
intros [[b a] c].
unfold map1, map2.
lhs napply ap.
{ lhs napply word_change_sign_ww.
napply (ap (fun x ⇒ x ++ _)).
lhs napply word_change_sign_ww.
napply (ap (fun x ⇒ x ++ _)).
lhs napply word_change_sign_ww.
napply (ap (fun x ⇒ _ ++ x)).
exact (word_change_sign_inv [a]). }
lhs_V napply ap.
1: rhs_V napply app_assoc.
1: napply app_assoc.
rhs napply ap.
2: napply word_change_sign_ww.
napply freegroup_tau.
Defined.
Now we can start to prove the group laws. Since these are hprops we can ignore what happens with the path constructor.
Our operation is associative
#[export] Instance associative_freegroup_type : Associative sg_op.
Proof.
intros x y z.
strip_truncations.
revert x; srapply Coeq_ind_hprop; intro x.
revert y; srapply Coeq_ind_hprop; intro y.
revert z; srapply Coeq_ind_hprop; intro z.
napply (ap (tr o coeq)).
napply app_assoc.
Defined.
Proof.
intros x y z.
strip_truncations.
revert x; srapply Coeq_ind_hprop; intro x.
revert y; srapply Coeq_ind_hprop; intro y.
revert z; srapply Coeq_ind_hprop; intro z.
napply (ap (tr o coeq)).
napply app_assoc.
Defined.
Left identity
#[export] Instance leftidentity_freegroup_type : LeftIdentity sg_op mon_unit.
Proof.
rapply Trunc_ind.
srapply Coeq_ind_hprop; intros x.
reflexivity.
Defined.
Proof.
rapply Trunc_ind.
srapply Coeq_ind_hprop; intros x.
reflexivity.
Defined.
Right identity
#[export] Instance rightidentity_freegroup_type : RightIdentity sg_op mon_unit.
Proof.
rapply Trunc_ind.
srapply Coeq_ind_hprop; intros x.
apply (ap tr), ap.
napply app_nil.
Defined.
Proof.
rapply Trunc_ind.
srapply Coeq_ind_hprop; intros x.
apply (ap tr), ap.
napply app_nil.
Defined.
Left inverse
#[export] Instance leftinverse_freegroup_type : LeftInverse (.*.) (^) mon_unit.
Proof.
rapply Trunc_ind.
srapply Coeq_ind_hprop; intro x.
apply word_concat_Vw.
Defined.
Proof.
rapply Trunc_ind.
srapply Coeq_ind_hprop; intro x.
apply word_concat_Vw.
Defined.
Right inverse
#[export] Instance rightinverse_freegroup_type : RightInverse (.*.) (^) mon_unit.
Proof.
rapply Trunc_ind.
srapply Coeq_ind_hprop; intro x.
apply word_concat_wV.
Defined.
Proof.
rapply Trunc_ind.
srapply Coeq_ind_hprop; intro x.
apply word_concat_wV.
Defined.
Finally we have defined the free group on A
Definition FreeGroup : Group.
Proof.
snapply (Build_Group freegroup_type); repeat split; exact _.
Defined.
Definition word_rec (G : Group) (s : A → G) : A + A → G.
Proof.
intros [x|x].
- exact (s x).
- exact (s x)^.
Defined.
Proof.
snapply (Build_Group freegroup_type); repeat split; exact _.
Defined.
Definition word_rec (G : Group) (s : A → G) : A + A → G.
Proof.
intros [x|x].
- exact (s x).
- exact (s x)^.
Defined.
When we have a list of words we can recursively define a group element. The obvious choice would be to map nil to the identity and x :: xs to x × words_rec xs. This has the disadvantage that a single generating element gets mapped to x × 1 instead of x. To fix this issue, we map nil to the identity, the singleton to the element we want, and do the rest recursively.
Definition words_rec (G : Group) (s : A → G) : Words → G.
Proof.
intro xs.
induction xs as [|x [|y xs] IHxs].
- exact mon_unit.
- exact (word_rec G s x).
- exact (word_rec G s x × IHxs).
Defined.
Definition words_rec_cons (G : Group) (s : A → G) (x : A + A) (xs : Words)
: words_rec G s (x :: xs)%list = word_rec G s x × words_rec G s xs.
Proof.
induction xs in x |- ×.
- symmetry; napply grp_unit_r.
- reflexivity.
Defined.
Lemma words_rec_pp (G : Group) (s : A → G) (x y : Words)
: words_rec G s (x ++ y) = words_rec G s x × words_rec G s y.
Proof.
induction x as [|x xs IHxs] in y |- ×.
- symmetry; napply grp_unit_l.
- change ((?x :: ?xs) ++ y) with (x :: xs ++ y).
lhs napply words_rec_cons.
lhs napply ap.
1: napply IHxs.
lhs napply grp_assoc.
napply (ap (.* _)).
symmetry.
apply words_rec_cons.
Defined.
Lemma words_rec_coh (G : Group) (s : A → G) (a : A + A) (b c : Words)
: words_rec G s (map1 (b, a, c)) = words_rec G s (map2 (b, a, c)).
Proof.
unfold map1, map2.
rhs napply (words_rec_pp G s).
lhs napply words_rec_pp.
napply (ap (_ *.)).
lhs napply words_rec_pp.
lhs napply ap.
1: napply words_rec_pp.
lhs napply grp_assoc.
rhs_V napply grp_unit_l.
napply (ap (.* _)).
destruct a; simpl.
- napply grp_inv_r.
- napply grp_inv_l.
Defined.
Proof.
intro xs.
induction xs as [|x [|y xs] IHxs].
- exact mon_unit.
- exact (word_rec G s x).
- exact (word_rec G s x × IHxs).
Defined.
Definition words_rec_cons (G : Group) (s : A → G) (x : A + A) (xs : Words)
: words_rec G s (x :: xs)%list = word_rec G s x × words_rec G s xs.
Proof.
induction xs in x |- ×.
- symmetry; napply grp_unit_r.
- reflexivity.
Defined.
Lemma words_rec_pp (G : Group) (s : A → G) (x y : Words)
: words_rec G s (x ++ y) = words_rec G s x × words_rec G s y.
Proof.
induction x as [|x xs IHxs] in y |- ×.
- symmetry; napply grp_unit_l.
- change ((?x :: ?xs) ++ y) with (x :: xs ++ y).
lhs napply words_rec_cons.
lhs napply ap.
1: napply IHxs.
lhs napply grp_assoc.
napply (ap (.* _)).
symmetry.
apply words_rec_cons.
Defined.
Lemma words_rec_coh (G : Group) (s : A → G) (a : A + A) (b c : Words)
: words_rec G s (map1 (b, a, c)) = words_rec G s (map2 (b, a, c)).
Proof.
unfold map1, map2.
rhs napply (words_rec_pp G s).
lhs napply words_rec_pp.
napply (ap (_ *.)).
lhs napply words_rec_pp.
lhs napply ap.
1: napply words_rec_pp.
lhs napply grp_assoc.
rhs_V napply grp_unit_l.
napply (ap (.* _)).
destruct a; simpl.
- napply grp_inv_r.
- napply grp_inv_l.
Defined.
Definition FreeGroup_rec {G : Group} (s : A → G) : FreeGroup $-> G.
Proof.
snapply Build_GroupHomomorphism.
{ rapply Trunc_rec.
srapply Coeq_rec.
1: apply words_rec, s.
intros [[b a] c].
apply words_rec_coh. }
intros x y; strip_truncations.
revert x; srapply Coeq_ind_hprop; intro x.
revert y; srapply Coeq_ind_hprop; intro y.
simpl.
apply words_rec_pp.
Defined.
Definition freegroup_in : A → FreeGroup
:= freegroup_eta o (fun x ⇒ [ x ]) o inl.
Definition FreeGroup_rec_beta {G : Group} (f : A → G)
: FreeGroup_rec f o freegroup_in == f
:= fun _ ⇒ idpath.
Coercion freegroup_in : A >-> group_type.
Definition FreeGroup_ind_hprop' (P : FreeGroup → Type)
`{∀ x, IsHProp (P x)}
(H1 : ∀ w, P (freegroup_eta w))
: ∀ x, P x.
Proof.
rapply Trunc_ind.
srapply Coeq_ind_hprop.
exact H1.
Defined.
Definition FreeGroup_ind_hprop (P : FreeGroup → Type)
`{∀ x, IsHProp (P x)}
(H1 : P mon_unit)
(Hin : ∀ x, P (freegroup_in x))
(Hop : ∀ x y, P x → P y → P (x^ × y))
: ∀ x, P x.
Proof.
rapply FreeGroup_ind_hprop'.
intros w.
induction w as [|a w IHw].
- exact H1.
- destruct a as [a|a].
+ change (P ((freegroup_in a) × freegroup_eta w)).
rewrite <- (grp_inv_inv a).
apply Hop.
× rewrite <- grp_unit_r.
by apply Hop.
× assumption.
+ change (P ((freegroup_in a)^ × freegroup_eta w)).
by apply Hop.
Defined.
Definition FreeGroup_ind_homotopy {G : Group} {f f' : FreeGroup $-> G}
(H : ∀ x, f (freegroup_in x) = f' (freegroup_in x))
: f $== f'.
Proof.
rapply FreeGroup_ind_hprop.
- exact (concat (grp_homo_unit f) (grp_homo_unit f')^).
- exact H.
- intros x y p q. refine (grp_homo_op_agree f f' _ q).
lhs napply grp_homo_inv.
rhs napply grp_homo_inv.
exact (ap _ p).
Defined.
Proof.
snapply Build_GroupHomomorphism.
{ rapply Trunc_rec.
srapply Coeq_rec.
1: apply words_rec, s.
intros [[b a] c].
apply words_rec_coh. }
intros x y; strip_truncations.
revert x; srapply Coeq_ind_hprop; intro x.
revert y; srapply Coeq_ind_hprop; intro y.
simpl.
apply words_rec_pp.
Defined.
Definition freegroup_in : A → FreeGroup
:= freegroup_eta o (fun x ⇒ [ x ]) o inl.
Definition FreeGroup_rec_beta {G : Group} (f : A → G)
: FreeGroup_rec f o freegroup_in == f
:= fun _ ⇒ idpath.
Coercion freegroup_in : A >-> group_type.
Definition FreeGroup_ind_hprop' (P : FreeGroup → Type)
`{∀ x, IsHProp (P x)}
(H1 : ∀ w, P (freegroup_eta w))
: ∀ x, P x.
Proof.
rapply Trunc_ind.
srapply Coeq_ind_hprop.
exact H1.
Defined.
Definition FreeGroup_ind_hprop (P : FreeGroup → Type)
`{∀ x, IsHProp (P x)}
(H1 : P mon_unit)
(Hin : ∀ x, P (freegroup_in x))
(Hop : ∀ x y, P x → P y → P (x^ × y))
: ∀ x, P x.
Proof.
rapply FreeGroup_ind_hprop'.
intros w.
induction w as [|a w IHw].
- exact H1.
- destruct a as [a|a].
+ change (P ((freegroup_in a) × freegroup_eta w)).
rewrite <- (grp_inv_inv a).
apply Hop.
× rewrite <- grp_unit_r.
by apply Hop.
× assumption.
+ change (P ((freegroup_in a)^ × freegroup_eta w)).
by apply Hop.
Defined.
Definition FreeGroup_ind_homotopy {G : Group} {f f' : FreeGroup $-> G}
(H : ∀ x, f (freegroup_in x) = f' (freegroup_in x))
: f $== f'.
Proof.
rapply FreeGroup_ind_hprop.
- exact (concat (grp_homo_unit f) (grp_homo_unit f')^).
- exact H.
- intros x y p q. refine (grp_homo_op_agree f f' _ q).
lhs napply grp_homo_inv.
rhs napply grp_homo_inv.
exact (ap _ p).
Defined.
Now we need to prove that the free group satisfies the universal property of the free group. TODO: remove funext from here and universal property of free group
#[export] Instance isfreegroupon_freegroup `{Funext}
: IsFreeGroupOn A FreeGroup freegroup_in.
Proof.
intros G f.
snapply Build_Contr.
{ srefine (_;_); simpl.
1: apply FreeGroup_rec, f.
intro x; reflexivity. }
intros [g h].
napply path_sigma_hprop; [ exact _ |].
simpl.
apply equiv_path_grouphomomorphism.
symmetry.
snapply FreeGroup_ind_homotopy.
exact h.
Defined.
: IsFreeGroupOn A FreeGroup freegroup_in.
Proof.
intros G f.
snapply Build_Contr.
{ srefine (_;_); simpl.
1: apply FreeGroup_rec, f.
intro x; reflexivity. }
intros [g h].
napply path_sigma_hprop; [ exact _ |].
simpl.
apply equiv_path_grouphomomorphism.
symmetry.
snapply FreeGroup_ind_homotopy.
exact h.
Defined.
Typeclass search can already find this but we leave it here as a definition for reference.
Definition isfreegroup_freegroup `{Funext} : IsFreeGroup FreeGroup := _.
End Reduction.
Arguments FreeGroup_rec {A G}.
Arguments FreeGroup_ind_homotopy {A G f f'}.
Arguments freegroup_eta {A}.
Arguments freegroup_in {A}.
End Reduction.
Arguments FreeGroup_rec {A G}.
Arguments FreeGroup_ind_homotopy {A G f f'}.
Arguments freegroup_eta {A}.
Arguments freegroup_in {A}.
Properties of free groups
(* Given a function on the generators, there is an induced group homomorphism from the free group. *)
Definition isfreegroupon_rec {S : Type} {F_S : Group}
{i : S → F_S} `{IsFreeGroupOn S F_S i}
{G : Group} (f : S → G) : F_S $-> G
:= (center (FactorsThroughFreeGroup S F_S i G f)).1.
(* The propositional computation rule for the recursor. *)
Definition isfreegroupon_rec_beta
{S : Type} {F_S : Group} {i : S → F_S} `{IsFreeGroupOn S F_S i}
{G : Group} (f : S → G)
: isfreegroupon_rec f o i == f
:= (center (FactorsThroughFreeGroup S F_S i G f)).2.
(* Two homomorphisms from a free group are equal if they agree on the generators. *)
Definition path_homomorphism_from_free_group {S : Type}
{F_S : Group} {i : S → F_S} `{IsFreeGroupOn S F_S i}
{G : Group} (f g : F_S $-> G) (K : f o i == g o i)
: f = g.
Proof.
(* By assumption, the type FactorsThroughFreeGroup S F_S i G (g o i) of factorizations of g o i through i is contractible. Therefore the two elements we have are equal. Therefore, their first components are equal. *)
exact (path_contr (f; K) (g; fun x ⇒ idpath))..1.
Defined.
Instance isequiv_isfreegroupon_rec `{Funext} {S : Type}
{F_S : Group} {i : S → F_S} `{IsFreeGroupOn S F_S i} {G : Group}
: IsEquiv (@isfreegroupon_rec S F_S i _ G).
Proof.
apply (isequiv_adjointify isfreegroupon_rec (fun f ⇒ f o i)).
- intro f.
apply path_homomorphism_from_free_group.
apply isfreegroupon_rec_beta.
- intro f.
(* here we need Funext: *)
apply path_arrow, isfreegroupon_rec_beta.
Defined.
The universal property of a free group.
Definition equiv_isfreegroupon_rec `{Funext} {G F : Group} {A : Type}
{i : A → F} `{IsFreeGroupOn A F i}
: (A → G) <~> (F $-> G) := Build_Equiv _ _ isfreegroupon_rec _.
{i : A → F} `{IsFreeGroupOn A F i}
: (A → G) <~> (F $-> G) := Build_Equiv _ _ isfreegroupon_rec _.
The above theorem is true regardless of the implementation of free groups. This lets us state the more specific theorem about the canonical free groups. This can be read as FreeGroup is left adjoint to the forgetful functor group_type.
Definition equiv_freegroup_rec `{Funext} (G : Group) (A : Type)
: (A → G) <~> (FreeGroup A $-> G)
:= equiv_isfreegroupon_rec.
Instance ishprop_isfreegroupon `{Funext} (F : Group) (A : Type) (i : A → F)
: IsHProp (IsFreeGroupOn A F i).
Proof.
unfold IsFreeGroupOn.
exact istrunc_forall.
Defined.
: (A → G) <~> (FreeGroup A $-> G)
:= equiv_isfreegroupon_rec.
Instance ishprop_isfreegroupon `{Funext} (F : Group) (A : Type) (i : A → F)
: IsHProp (IsFreeGroupOn A F i).
Proof.
unfold IsFreeGroupOn.
exact istrunc_forall.
Defined.
Both ways of stating the universal property are equivalent.
Definition equiv_isfreegroupon_isequiv_precomp `{Funext}
(F : Group) (A : Type) (i : A → F)
: IsFreeGroupOn A F i <~> ∀ G, IsEquiv (fun f : F $-> G ⇒ f o i).
Proof.
srapply equiv_iff_hprop.
1: intros ? ?; exact (equiv_isequiv (equiv_isfreegroupon_rec)^-1).
intros k G g.
specialize (k G).
snapply contr_equiv'.
1: exact (hfiber (fun f x ⇒ grp_homo_map f (i x)) g).
{ rapply equiv_functor_sigma_id.
intro y; symmetry.
apply equiv_path_forall. }
exact _.
Defined.
(F : Group) (A : Type) (i : A → F)
: IsFreeGroupOn A F i <~> ∀ G, IsEquiv (fun f : F $-> G ⇒ f o i).
Proof.
srapply equiv_iff_hprop.
1: intros ? ?; exact (equiv_isequiv (equiv_isfreegroupon_rec)^-1).
intros k G g.
specialize (k G).
snapply contr_equiv'.
1: exact (hfiber (fun f x ⇒ grp_homo_map f (i x)) g).
{ rapply equiv_functor_sigma_id.
intro y; symmetry.
apply equiv_path_forall. }
exact _.
Defined.
(* We say that a group G is generated by a subtype X if the natural map from the subgroup generated by X to G is a surjection. One could equivalently say IsEquiv (subgroup_incl (subgroup_generated X)), ∀ g, subgroup_generated X g, or subgroup_generated X = maximal_subgroup, but the definition using surjectivity is convenient later. *)
Definition isgeneratedby (G : Group) (X : G → Type)
:= IsSurjection (subgroup_incl (subgroup_generated X)).
Section FreeGroupGenerated.
(* In this Section, we prove that the free group F_S on a type S is generated in the above sense by the image of S. We conclude that the inclusion map is an equivalence, and that the free group is isomorphic as a group to the subgroup. We show that the inclusion is a surjection by showing that it is split epi in the category of groups. *)
Context {S : Type} {F_S : Group} {i : S → F_S} `{IsFreeGroupOn S F_S i}.
(* We define a group homomorphism from F_S to the subgroup G generated by S by sending a generator s to "itself". This map will be a section of the inclusion map. *)
Local Definition to_subgroup_generated
: F_S $-> subgroup_generated (hfiber i).
Proof.
apply isfreegroupon_rec.
intro s.
snapply subgroup_generated_gen_incl.
- exact (i s).
- exact (s; idpath).
Defined.
(* We record the computation rule that to_subgroup_generated satisfies. *)
Local Definition to_subgroup_generated_beta (s : S)
: to_subgroup_generated (i s) = subgroup_generated_gen_incl (i s) (s; idpath)
:= isfreegroupon_rec_beta _ _.
(* It follows that to_subgroup_generated is a section of the inclusion map from G to F_S. *)
Local Definition is_retraction
: (subgroup_incl _) $o to_subgroup_generated = grp_homo_id.
Proof.
apply path_homomorphism_from_free_group; cbn.
intro s.
exact (ap pr1 (to_subgroup_generated_beta s)).
Defined.
(* It follows that the inclusion map is a surjection, i.e., that F_S is generated by the image of S. *)
Definition isgenerated_isfreegroupon
: isgeneratedby F_S (hfiber i).
Proof.
snapply issurj_retr.
- exact to_subgroup_generated.
- apply ap10; cbn.
exact (ap grp_homo_map is_retraction).
Defined.
(* Therefore, the inclusion map is an equivalence, since it is known to be an embedding. *)
Definition isequiv_subgroup_incl_freegroupon
: IsEquiv (subgroup_incl (subgroup_generated (hfiber i))).
Proof.
apply isequiv_surj_emb.
- exact isgenerated_isfreegroupon.
- exact _.
Defined.
(* Therefore, the subgroup is isomorphic to the free group. *)
Definition iso_subgroup_incl_freegroupon
: GroupIsomorphism (subgroup_generated (hfiber i)) F_S.
Proof.
napply Build_GroupIsomorphism.
exact isequiv_subgroup_incl_freegroupon.
Defined.
End FreeGroupGenerated.
Definition freegroup_rec_in {A : Type}
: FreeGroup_rec freegroup_in $== Id (FreeGroup A).
Proof.
by snapply FreeGroup_ind_homotopy.
Defined.
Definition freegroup_rec_compose {A : Type} {G H : Group}
(f : A → G) (k : G $-> H)
: FreeGroup_rec (k o f) $== k $o FreeGroup_rec f.
Proof.
by snapply FreeGroup_ind_homotopy; intros x.
Defined.
Definition freegroup_const {A : Type} {G : Group}
: FreeGroup_rec (fun _ : A ⇒ 1) $== @grp_homo_const _ G.
Proof.
by snapply FreeGroup_ind_homotopy.
Defined.
Instance is0functor_freegroup : Is0Functor FreeGroup.
Proof.
snapply Build_Is0Functor.
intros X Y f.
snapply FreeGroup_rec.
exact (freegroup_in o f).
Defined.
Instance is1functor_freegroup : Is1Functor FreeGroup.
Proof.
snapply Build_Is1Functor.
- intros X Y f g p.
snapply FreeGroup_ind_homotopy.
intros x.
exact (ap freegroup_in (p x)).
- intro; exact freegroup_rec_in.
- intros X Y Z f g.
by snapply FreeGroup_ind_homotopy.
Defined.