Library HoTT.Spaces.BAut

(* -*- mode: coq; mode: visual-line -*- *)
Require Import HoTT.Basics HoTT.Types.
Require Import Constant.
Require Import HoTT.Truncations.
Require Import ObjectClassifier Homotopy.ExactSequence Pointed.

Local Open Scope type_scope.
Local Open Scope path_scope.

BAut(X)

Basics

BAut X is the type of types that are merely equal to X. It is connected, by is0connected_component and any two points are merely equal by merely_path_component.

Definition BAut (X : Type@{u}) := { Z : Type@{u} & merely (Z = X) }.

Coercion BAut_pr1 X : BAut X → Type := pr1.

Global Instance ispointed_baut {X : Type} : IsPointed (BAut X) := (X; tr 1).

We also define a pointed version pBAut X, since the coercion BAut_pr1 doesn't work if BAut X is a pType.

Definition pBAut (X : Type) : pType
  := [BAut X , _].

Definition path_baut `{Univalence} {X} (Z Z' : BAut X)
: (Z <~> Z') <~> (Z = Z' :> BAut X )
  := equiv_path_sigma_hprop _ _ oE equiv_path_universe _ _.

Definition ap_pr1_path_baut `{Univalence} {X}
           {Z Z' : BAut X} (f : Z <~> Z')
: ap (BAut_pr1 X) (path_baut Z Z' f) = path_universe f.
Proof.
  unfold path_baut, BAut_pr1; simpl.
  apply ap_pr1_path_sigma_hprop.
Defined.

Definition transport_path_baut `{Univalence} {X}
           {Z Z' : BAut X} (f : Z <~> Z') (z : Z)
: transport (fun (W:BAut X) ⇒ W) (path_baut Z Z' f) z = f z.
Proof.
  refine (transport_compose idmap (BAut_pr1 X) _ _ @ _).
  refine (_ @ transport_path_universe f z).
  apply ap10, ap, ap_pr1_path_baut.
Defined.

The following tactic, which applies when trying to prove an hprop, replaces all assumed elements of BAut X by X itself. With Univalence, this would work for any 0-connected type, but using merely_path_component we can avoid univalence.

Ltac baut_reduce :=
  progress repeat
    match goal with
      | [ Z : BAut ?X |- _ ]
        ⇒ let Zispoint := fresh "Zispoint" in
           assert (Zispoint := merely_path_component (point (BAut X)) Z);
           revert Zispoint;
           refine (@Trunc_ind _ _ _ _ _);
           intro Zispoint;
           destruct Zispoint
    end.

Truncation

If X is an n.+1-type, then BAut X is an n.+2-type.

Global Instance trunc_baut `{Univalence} {n X} `{IsTrunc n .+1 X}
: IsTrunc n .+2 (BAut X).
Proof.
  apply istrunc_S.
  intros Z W.
  baut_reduce.
  exact (@istrunc_equiv_istrunc _ _ (path_baut _ _) n.+1 _).
Defined.

If X is truncated, then so is every element of BAut X.

Global Instance trunc_el_baut {n X} `{Funext} `{IsTrunc n X} (Z : BAut X)
: IsTrunc n Z
:= ltac:(by baut_reduce).

Operations on BAut

Multiplying by a fixed type

Definition baut_prod_r (X A : Type)
  : BAut X → BAut (X × A)
  := fun Z:BAut X ⇒
       (Z × A ; Trunc_functor (-1) (ap (fun W ⇒ W × A)) (pr2 Z))
       : BAut (X × A).

Definition ap_baut_prod_r `{Univalence} (X A : Type)
           {Z W : BAut X} (e : Z <~> W)
  : ap (baut_prod_r X A) (path_baut Z W e)
    = path_baut (baut_prod_r X A Z) (baut_prod_r X A W) (equiv_functor_prod_r e).
Proof.
  cbn.
  apply moveL_equiv_M; cbn; unfold pr1_path.
  rewrite <- (ap_compose (baut_prod_r X A) pr1 (path_sigma_hprop Z W _)).
  rewrite <- ((ap_compose pr1 (fun Z ⇒ Z × A) (path_sigma_hprop Z W _))^).
  rewrite ap_pr1_path_sigma_hprop.
  apply moveL_equiv_M; cbn.
  apply ap_prod_r_path_universe.
Qed.

Centers

The following lemma says that to define a section of a family P of hsets over BAut X, it is equivalent to define an element of P X which is fixed by all automorphisms of X.

Lemma baut_ind_hset `{Univalence} X

It ought to be possible to allow more generally P : BAut X → Type, but the proof would get more complicated, and this version suffices for present applications.

      (P : Type → Type) `{∀ (Z : BAut X ), IsHSet (P Z )}
: { e : P (point (BAut X)) &
    ∀ g : X <~> X , transport P (path_universe g) e = e }
  <~> (∀ (Z:BAut X ), P Z ).
Proof.
  refine (equiv_sig_ind _ oE _).

We use the fact that maps out of a propositional truncation into an hset are equivalent to weakly constant functions.

  refine ((equiv_functor_forall'
             (P := fun Z ⇒ { f : (Z =X) → P Z & WeaklyConstant f })
             1
             (fun Z ⇒ equiv_merely_rec_hset _ _)) oE _); simpl.
  { intros p. change (IsHSet (P (BAut_pr1 X (Z ; tr p)))). exact _. }
  unfold WeaklyConstant.

Now we peel away a bunch of contractible types.

  refine (equiv_sig_coind _ _ oE _).
  srapply equiv_functor_sigma'.
  1:apply (equiv_paths_ind_r X (fun x _ ⇒ P x)).
  intros p; cbn.
  refine (equiv_paths_ind_r X _ oE _).
  srapply equiv_functor_forall'.
  1:apply equiv_equiv_path.
  intros e; cbn.
  refine (_ oE equiv_moveL_transport_V _ _ _ _).
  apply equiv_concat_r.
  rewrite path_universe_transport_idmap, paths_ind_r_transport.
  reflexivity.
Defined.

This implies that if X is a set, then the center of BAut X is the set of automorphisms of X that commute with every other automorphism (i.e. the center, in the usual sense, of the group of automorphisms of X).

Definition center_baut `{Univalence} X `{IsHSet X}
: { f : X <~> X & ∀ g:X <~>X , g o f == f o g }
  <~> (∀ Z:BAut X , Z = Z ).
Proof.
  refine (equiv_functor_forall_id
            (fun Z ⇒ equiv_path_sigma_hprop Z Z) oE _).
  refine (baut_ind_hset X (fun Z ⇒ Z = Z) oE _).
  simpl.
  refine (equiv_functor_sigma' (equiv_path_universe X X) _); intros f.
  apply equiv_functor_forall_id; intros g; simpl.
  refine (_ oE equiv_path_arrow _ _).
  refine (_ oE equiv_path_equiv (g oE f) (f oE g)).
  revert g. equiv_intro (equiv_path X X) g.
  revert f. equiv_intro (equiv_path X X) f.
  refine (_ oE equiv_concat_l (equiv_path_pp _ _) _).
  refine (_ oE equiv_concat_r (equiv_path_pp _ _)^ _).
  refine (_ oE (equiv_ap (equiv_path X X) _ _)^-1).
  refine (equiv_concat_l (transport_paths_lr _ _) _ oE _).
  refine (equiv_concat_l (concat_pp_p _ _ _) _ oE _).
  refine (equiv_moveR_Vp _ _ _ oE _).
  refine (equiv_concat_l _ _ oE equiv_concat_r _ _).
  - apply concat2; apply eissect.
  - symmetry; apply concat2; apply eissect.
Defined.

We show that this equivalence takes the identity equivalence to the identity in the center. We have to be careful in this proof never to simpl or unfold too many things, or Coq will produce gigantic terms that take it forever to compute with.

Definition id_center_baut `{Univalence} X `{IsHSet X}
: center_baut X (exist
                   (fun (f:X <~>X) ⇒ ∀ (g:X <~>X ), g o f == f o g)
                   (equiv_idmap X)
                   (fun (g:X <~>X) (x:X) ⇒ idpath (g x)))
  = fun Z ⇒ idpath Z.
Proof.
  apply path_forall; intros Z.
  assert (IsHSet (Z.1 = Z.1)) by exact _.
  baut_reduce.
  exact (ap (path_sigma_hprop _ _) path_universe_1
            @ path_sigma_hprop_1 _).
Defined.

Similarly, if X is a 1-type, we can characterize the 2-center of BAut X.

Coq is too eager about unfolding some things appearing in this proof.

Section Center2BAut.
  Local Arguments equiv_path_equiv : simpl never.
  Local Arguments equiv_path2_universe : simpl never.

  Definition center2_baut `{Univalence} X `{IsTrunc 1 X}
  : { f : ∀ x:X , x =x & ∀ (g:X <~>X) (x:X ), ap g (f x) = f (g x) }
      <~> (∀ Z:BAut X , (idpath Z ) = (idpath Z )).
  Proof.
    refine ((equiv_functor_forall_id
               (fun Z ⇒ (equiv_concat_lr _ _)
                           oE (equiv_ap (equiv_path_sigma_hprop Z Z) 1%path 1%path))) oE _).
    { symmetry; apply path_sigma_hprop_1. }
    { apply path_sigma_hprop_1. }
    assert (∀ Z:BAut X, IsHSet (idpath Z .1 = idpath Z .1)) by exact _.
    refine (baut_ind_hset X (fun Z ⇒ idpath Z = idpath Z) oE _).
    simple refine (equiv_functor_sigma' _ _).
    { refine (_ oE equiv_path2_universe 1 1).
      apply equiv_concat_lr.
      - symmetry; apply path_universe_1.
      - apply path_universe_1. }
    intros f.
    apply equiv_functor_forall_id; intros g.
    refine (_ oE equiv_path3_universe _ _).
    refine (dpath_paths2 (path_universe g) _ _ oE _).
    cbn.
    change (equiv_idmap X == equiv_idmap X) in f.
    refine (equiv_concat_lr _ _).
    - refine (_ @ (path2_universe_postcompose_idmap f g)^).
      abstract (rewrite !whiskerR_pp, !concat_pp_p; reflexivity).
    - refine (path2_universe_precompose_idmap f g @ _).
      abstract (rewrite !whiskerL_pp, !concat_pp_p; reflexivity).
  Defined.

Once again we compute it on the identity. In this case it seems to be unavoidable to do some simpling (or at least cbning), making this proof somewhat slower.

  Definition id_center2_baut `{Univalence} X `{IsTrunc 1 X}
  : center2_baut X (exist
                      (fun (f:∀ x:X , x =x) ⇒
                         ∀ (g:X <~>X) (x:X ), ap g (f x) = f (g x))
                      (fun x ⇒ idpath x)
                      (fun (g:X <~>X) (x:X) ⇒ idpath (idpath (g x))))
    = fun Z ⇒ idpath (idpath Z).
  Proof.
    apply path_forall; intros Z.
    assert (IsHSet (idpath Z.1 = idpath Z.1)) by exact _.
    baut_reduce.
    cbn. unfold functor_forall, sig_rect, merely_rec_hset. cbn.
    rewrite equiv_path2_universe_1.
    rewrite !concat_p1, !concat_Vp.
    simpl.
    rewrite !concat_p1, !concat_Vp.
    reflexivity.
  Defined.

End Center2BAut.

Section ClassifyingMaps.

Maps into BAut F classify bundles with fiber F

The property of being merely equivalent to a given type F defines a subuniverse.

  Definition subuniverse_merely_equiv (F : Type) : Subuniverse.
  Proof.
    rapply (Build_Subuniverse (fun E ⇒ merely (E <~> F))).
    intros T U mere_eq f iseq_f.
    strip_truncations.
    pose (feq:=Build_Equiv _ _ f iseq_f).
    exact (tr (mere_eq oE feq^-1)).
  Defined.

The universe of O-local types for subuniverse_merely_equiv F is equivalent to BAut F.

  Proposition equiv_baut_typeO `{Univalence} {F : Type}
    : BAut F <~> Type_ (subuniverse_merely_equiv F).
  Proof.
    srapply equiv_functor_sigma_id; intro X; cbn.
    rapply Trunc_functor_equiv.
    exact (equiv_path_universe _ _)^-1%equiv.
  Defined.

Consequently, maps into BAut F correspond to bundles with fibers merely equivalent to F.

  Corollary equiv_map_baut_fibration `{Univalence} {Y : pType} {F : Type}
    : (Y → BAut F ) <~> { p : Slice Y & ∀ y:Y , merely (hfiber p .2 y <~> F) }.
  Proof.
    refine (_ oE equiv_postcompose' equiv_baut_typeO).
    refine (_ oE equiv_sigma_fibration_O).
    snrapply equiv_functor_sigma_id; intro p.
    rapply equiv_functor_forall_id; intro y.
    by apply Trunc_functor_equiv.
  Defined.

The pointed version of equiv_baut_typeO above.

  Proposition pequiv_pbaut_typeOp@{u v +} `{Univalence} {F : Type@{u}}
    : pBAut@{u v } F <~>* [Type_ (subuniverse_merely_equiv F), (F ; tr equiv_idmap )].
  Proof.
    snrapply Build_pEquiv'; cbn.
    1: exact equiv_baut_typeO.
    by apply path_sigma_hprop.
  Defined.

  Definition equiv_pmap_pbaut_pfibration `{Univalence} {Y F : pType@{u }}
    : (Y ->* pBAut@{u v } F ) <~> { p : { q : pSlice Y & ∀ y:Y , merely (hfiber q .2 y <~> F) } &
                                      pfiber p .1 .2 <~>* F }
    := (equiv_sigma_pfibration_O (subuniverse_merely_equiv F))
         oE pequiv_pequiv_postcompose pequiv_pbaut_typeOp.

When Y is connected, pointed maps into pBAut F correspond to maps into the universe sending the base point to F.

  Proposition equiv_pmap_pbaut_type_p `{Univalence}
              {Y : pType@{u }} {F : Type@{u}} `{IsConnected 0 Y}
    : (Y ->* pBAut F ) <~> (Y ->* [Type@{u}, F ]).
  Proof.
    refine (_ oE pequiv_pequiv_postcompose pequiv_pbaut_typeOp).
    rapply equiv_pmap_typeO_type_connected.
  Defined.

When Y is connected, pBAut F classifies fiber sequences over Y with fiber F.

  Definition equiv_pmap_pbaut_pfibration_connected `{Univalence} {Y F : pType} `{IsConnected 0 Y}
    : (Y ->* pBAut F ) <~> { X : pType & FiberSeq F X Y }
    := classify_fiberseq oE equiv_pmap_pbaut_type_p.

End ClassifyingMaps.