Require Import Basics Types ReflectiveSubuniverse Pointed.Core.
[Loading ML file number_string_notation_plugin.cmxs (using legacy method) ... done]

Local Open Scope pointed_scope.

(** * Modalities, reflective subuniverses and pointed types *)

(** So far, everything is about general reflective subuniverses, but in the future results about modalities can be placed here as well. *)

#[export] Instance ispointed_O `{O : ReflectiveSubuniverse} (X : Type)
  `{IsPointed X} : IsPointed (O X) := to O _ (point X).

Definition pto (O : ReflectiveSubuniverse@{u}) (X : pType@{u})
  : X ->* [O X, _]
  := Build_pMap (to O X) idpath.

(** If [A] is already [O]-local, then Coq knows that [pto] is an equivalence, so we can simply define: *)
Definition pequiv_pto `{O : ReflectiveSubuniverse} {X : pType} `{In O X}
  : X <~>* [O X, _] := Build_pEquiv (pto O X) _.

(** Applying [O_rec] to a pointed map yields a pointed map. *)
Definition pO_rec `{O : ReflectiveSubuniverse} {X Y : pType}
  `{In O Y} (f : X ->* Y) : [O X, _] ->* Y
  := Build_pMap (O_rec f) (O_rec_beta _ _ @ point_eq f).

Definition pO_rec_beta `{O : ReflectiveSubuniverse} {X Y : pType}
  `{In O Y} (f : X ->* Y)
  : pO_rec f o* pto O X ==* f.
O: ReflectiveSubuniverse
X, Y: pType
H: In O Y
f: X ->* Y
pO_rec f o* pto O X ==* f
Proof.
O: ReflectiveSubuniverse
X, Y: pType
H: In O Y
f: X ->* Y
pO_rec f o* pto O X ==* f
  srapply Build_pHomotopy.
O: ReflectiveSubuniverse
X, Y: pType
H: In O Y
f: X ->* Y
pO_rec f o* pto O X == f
O: ReflectiveSubuniverse
X, Y: pType
H: In O Y
f: X ->* Y
?p pt =
dpoint_eq (pO_rec f o* pto O X) @ (dpoint_eq f)^
  1: napply O_rec_beta.
O: ReflectiveSubuniverse
X, Y: pType
H: In O Y
f: X ->* Y
O_rec_beta f pt =
dpoint_eq (pO_rec f o* pto O X) @ (dpoint_eq f)^
  cbn.
O: ReflectiveSubuniverse
X, Y: pType
H: In O Y
f: X ->* Y
O_rec_beta f pt =
(1 @ (O_rec_beta f pt @ point_eq f)) @ (dpoint_eq f)^
  apply moveL_pV.
O: ReflectiveSubuniverse
X, Y: pType
H: In O Y
f: X ->* Y
O_rec_beta f pt @ dpoint_eq f =
1 @ (O_rec_beta f pt @ point_eq f)
  exact (concat_1p _)^.
Defined.

(** A pointed version of the universal property. *)
Definition pequiv_o_pto_O `{Funext}
  (O : ReflectiveSubuniverse) (P Q : pType) `{In O Q}
  : ([O P, _] ->** Q) <~>* (P ->** Q).
H: Funext
O: ReflectiveSubuniverse
P, Q: pType
H0: In O Q
([O P, ispointed_O P] ->** Q) <~>* (P ->** Q)
Proof.
H: Funext
O: ReflectiveSubuniverse
P, Q: pType
H0: In O Q
([O P, ispointed_O P] ->** Q) <~>* (P ->** Q)
  snapply Build_pEquiv.
H: Funext
O: ReflectiveSubuniverse
P, Q: pType
H0: In O Q
([O P, ispointed_O P] ->** Q) ->* (P ->** Q)
H: Funext
O: ReflectiveSubuniverse
P, Q: pType
H0: In O Q
IsEquiv ?pointed_equiv_fun
  (* We could just use the map [e] defined in the next bullet, but we want Coq to immediately unfold the underlying map to this. *)
  -
H: Funext
O: ReflectiveSubuniverse
P, Q: pType
H0: In O Q
([O P, ispointed_O P] ->** Q) ->* (P ->** Q) exact (Build_pMap (fun f => f o* pto O P) 1).
  (* We'll give an equivalence that definitionally has the same underlying map. *)
  -
H: Funext
O: ReflectiveSubuniverse
P, Q: pType
H0: In O Q
IsEquiv
  (Build_pMap
     (fun f : [O P, ispointed_O P] ->** Q =>
      f o* pto O P) 1) transparent assert (e : (([O P, _] ->* Q) <~> (P ->* Q))).
H: Funext
O: ReflectiveSubuniverse
P, Q: pType
H0: In O Q
([O P, ispointed_O P] ->* Q) <~> (P ->* Q)
H: Funext
O: ReflectiveSubuniverse
P, Q: pType
H0: In O Q
e:= ?Goal: ([O P, ispointed_O P] ->* Q) <~> (P ->* Q)
IsEquiv
  (Build_pMap
     (fun f : [O P, ispointed_O P] ->** Q =>
      f o* pto O P) 1)
    +
H: Funext
O: ReflectiveSubuniverse
P, Q: pType
H0: In O Q
([O P, ispointed_O P] ->* Q) <~> (P ->* Q) refine (issig_pmap P Q oE _ oE (issig_pmap [O P, _] Q)^-1%equiv).
H: Funext
O: ReflectiveSubuniverse
P, Q: pType
H0: In O Q
{f : [O P, ispointed_O P] -> Q & f pt = pt} <~>
{f : P -> Q & f pt = pt}
      snapply equiv_functor_sigma'.
H: Funext
O: ReflectiveSubuniverse
P, Q: pType
H0: In O Q
([O P, ispointed_O P] -> Q) <~> (P -> Q)
H: Funext
O: ReflectiveSubuniverse
P, Q: pType
H0: In O Q
forall a : [O P, ispointed_O P] -> Q,
(fun f : [O P, ispointed_O P] -> Q => f pt = pt) a <~>
(fun f : P -> Q => f pt = pt) (?f a)
      *
H: Funext
O: ReflectiveSubuniverse
P, Q: pType
H0: In O Q
([O P, ispointed_O P] -> Q) <~> (P -> Q) rapply equiv_o_to_O.
      *
H: Funext
O: ReflectiveSubuniverse
P, Q: pType
H0: In O Q
forall a : [O P, ispointed_O P] -> Q,
(fun f : [O P, ispointed_O P] -> Q => f pt = pt) a <~>
(fun f : P -> Q => f pt = pt) (equiv_o_to_O O P Q a) intro f; cbn.
H: Funext
O: ReflectiveSubuniverse
P, Q: pType
H0: In O Q
f: [O P, ispointed_O P] -> Q
f (ispointed_O P) = pt <~> f (to O P pt) = pt
      (* [reflexivity] works here, but then the underlying map won't agree definitionally with precomposition by [pto P], since pointed composition inserts a reflexivity path here. *)
      apply (equiv_concat_l 1).
    +
H: Funext
O: ReflectiveSubuniverse
P, Q: pType
H0: In O Q
e:= issig_pmap P Q
oE equiv_functor_sigma' (equiv_o_to_O O P Q)
     (fun f : [O P, ispointed_O P] -> Q =>
      equiv_concat_l 1 pt
      :
      (fun f0 : [O P, ispointed_O P] -> Q =>
       f0 pt = pt) f <~>
      (fun f0 : P -> Q => f0 pt = pt)
        (equiv_o_to_O O P Q f))
oE (issig_pmap [O P, ispointed_O P] Q)^-1: ([O P, ispointed_O P] ->* Q) <~> (P ->* Q)
IsEquiv
  (Build_pMap
     (fun f : [O P, ispointed_O P] ->** Q =>
      f o* pto O P) 1) exact (equiv_isequiv e).
Defined.

(** ** Pointed functoriality *)

Definition O_pfunctor `(O : ReflectiveSubuniverse) {X Y : pType}
  (f : X ->* Y) : [O X, _] ->* [O Y, _]
  := pO_rec (pto O Y o* f).

(** Coq knows that [O_pfunctor O f] is an equivalence whenever [f] is. *)
Definition equiv_O_pfunctor `(O : ReflectiveSubuniverse) {X Y : pType}
  (f : X ->* Y) `{IsEquiv _ _ f} : [O X, _] <~>* [O Y, _]
  := Build_pEquiv (O_pfunctor O f) _.

(** Pointed naturality of [O_pfunctor]. *)
Definition pto_O_natural `(O : ReflectiveSubuniverse) {X Y : pType}
  (f : X ->* Y) : O_pfunctor O f o* pto O X ==* pto O Y o* f.
O: ReflectiveSubuniverse
X, Y: pType
f: X ->* Y
O_pfunctor O f o* pto O X ==* pto O Y o* f
Proof.
O: ReflectiveSubuniverse
X, Y: pType
f: X ->* Y
O_pfunctor O f o* pto O X ==* pto O Y o* f
  napply pO_rec_beta.
Defined.

Definition pequiv_O_inverts `(O : ReflectiveSubuniverse) {X Y : pType}
  (f : X ->* Y) `{O_inverts O f}
  : [O X, _] <~>* [O Y, _]
  := Build_pEquiv (O_pfunctor O f) _.