Library UniMath.CategoryTheory.ElementaryTopos

** Following Saunders Mac Lane & Ieke Moerdijk Sheaves in Geometry and Logic - A First Introduction to Topos theory. Chapter IV.1 and IV.2
Contents :

Require Import UniMath.Foundations.PartD.
Require Import UniMath.MoreFoundations.Tactics.
Require Import UniMath.CategoryTheory.Core.Categories.
Require Import UniMath.CategoryTheory.Core.NaturalTransformations.
Require Import UniMath.CategoryTheory.Core.Isos.
Require Import UniMath.CategoryTheory.Core.Functors.
Require Import UniMath.CategoryTheory.limits.terminal.
Require Import UniMath.CategoryTheory.limits.pullbacks.
Require Import UniMath.CategoryTheory.limits.binproducts.
Require Import UniMath.CategoryTheory.SubobjectClassifier.
Require Import UniMath.CategoryTheory.exponentials.
Require Import UniMath.CategoryTheory.PowerObject.
Require Import UniMath.CategoryTheory.Monics.
Require Import UniMath.CategoryTheory.exponentials.
Require Import UniMath.CategoryTheory.Adjunctions.Core.
Require Import UniMath.CategoryTheory.Subobjects.
Require Import UniMath.CategoryTheory.opp_precat.
Require Import UniMath.CategoryTheory.categories.HSET.MonoEpiIso.

Local Open Scope cat.


Definition Topos_Structure (C : category) :=
   (PB : Pullbacks C) (T : Terminal C) (Ω : subobject_classifier T),
    (PowerObject (BinProductsFromPullbacks PB T) Ω).

Definition Topos := (C:category), Topos_Structure C.

Definition make_Topos_Structure {C:category} (PB : Pullbacks C) (T : Terminal C)
  (Ω: subobject_classifier T) (P: PowerObject (BinProductsFromPullbacks PB T) Ω)
  : Topos_Structure C.
Show proof.
  split with PB.
  split with T.
  split with Ω.
  exact P.

Definition make_Topos {C:category} (str: Topos_Structure C)
  : Topos.
Show proof.
  split with C.
  exact str.

Section ToposAccessor.

Context (C : Topos).

Definition Topos_category : category := pr1 C.
Definition Topos_Pullbacks : Pullbacks Topos_category := pr1 (pr2 C).
Definition Topos_Terminal : Terminal Topos_category := pr1 (pr2 (pr2 C)).
Definition Topos_SubobjectClassifier : subobject_classifier (Topos_Terminal) := pr1 (pr2 (pr2 (pr2 C))).
Definition Topos_BinProducts : BinProducts Topos_category := BinProductsFromPullbacks (Topos_Pullbacks) (Topos_Terminal).
Definition Topos_PowerObject : PowerObject (Topos_BinProducts) (Topos_SubobjectClassifier) := pr2 (pr2 (pr2 (pr2 C))).

End ToposAccessor.

Coercion Topos_category : Topos >->category.

Section Topos.

Context {C:Topos}.
Let T := Topos_Terminal C.
Let PB := Topos_Pullbacks C.
Let BinProd := Topos_BinProducts C.
Let P := Topos_PowerObject C.
Let Ω := Topos_SubobjectClassifier C.
Local Notation "c ⨉ d"
  := (BinProductObject C (BinProd c d))(at level 5).
Local Notation "f ⨱ g"
  := (BinProductOfArrows _ (BinProd _ _) (BinProd _ _) f g) (at level 10).

Section KroneckerDelta.

Definition KroneckerDelta (B : C) : C B B , Ω.
Show proof.
  use characteristic_morphism.
  + exact B.
  + use diagonalMap.
Local Notation "'δ' B" := (KroneckerDelta B)(at level 8).

Definition SingletonArrow (B : C) : C B , (PowerObject_on_ob P) B.
Show proof.
Local Notation "'{⋅}' B" := (SingletonArrow B)(at level 8).


Local Definition auxpb {X B: C} (b: C X , B) : Pullback (identity B b) (diagonalMap BinProd B).
Show proof.
  use make_Pullback.
  + exact X.
  + use BinProductArrow.
    - exact b.
    - exact (identity X).
  + exact b.
  + simpl.
    rewrite postcompWithBinProductArrow.
    use pathsinv0.
    use BinProductArrowUnique.
    - rewrite !assoc'.
      unfold diagonalMap'.
      now rewrite BinProductPr1Commutes.
    - rewrite !assoc'.
      unfold diagonalMap'.
      now rewrite BinProductPr2Commutes, id_left, id_right.
  + use make_isPullback.
    intros Y y1 y2 r.
    set (y11 := (y1 · (BinProductPr1 _ (BinProd B X)))).
    set (y12 := (y1 · (BinProductPr2 _ (BinProd B X)))).
    assert (r1 := maponpaths (λ f, compose f (BinProductPr1 C (BinProd B B))) r).
    assert (r2 := maponpaths (λ f, compose f (BinProductPr2 C (BinProd B B))) r).
    simpl in r1, r2.
    unfold diagonalMap' in r1, r2.
    rewrite
      !assoc',
      BinProductOfArrowsPr1,
      BinProductPr1Commutes,
      !id_right
      in r1.
    rewrite
      !assoc',
      BinProductOfArrowsPr2,
      BinProductPr2Commutes,
      !id_right,
      assoc in r2.
    fold y11 in r1.
    fold y12 in r2.
    use make_iscontr.
    - split with y12.
      use (tpair _ _ r2).
      rewrite precompWithBinProductArrow.
      use pathsinv0.
      use BinProductArrowUnique.
      * now rewrite r2, <-r1.
      * now rewrite id_right.
    - intro t.
      induction t as (t,(tri1,tri2)).
      use subtypePath.
      * unfold isPredicate.
        intro.
        use isofhleveldirprod.
        ++ use homset_property. ++ use homset_property.
      * cbn.
        assert (Ts := maponpaths (λ f, compose f (BinProductPr2 C (BinProd B X))) tri1).
        simpl in Ts.
        rewrite assoc', BinProductPr2Commutes, id_right in Ts.
        exact Ts.

Lemma SingletonArrow_isMonic (B: C) : isMonic ({⋅} B).
Show proof.
  use make_isMonic.
  intros X b b' p.
  assert (q : (((identity B) b) · (δ B) = ((identity B) b') · (δ B))). {
    rewrite (PowerObject_transpose_tri P (δ B)).
    fold BinProd.
    rewrite !assoc, !BinProductOfArrows_idxcomp.
    use (maponpaths (( nat_z_iso_inv
    (PowerObject_nat_z_iso P)) (X,,B))).
    exact p.
  }
  
Consider this diagram b ! X --------> B --------> T | | | b x id | diagonalMap| | true v v v B x X --------> B x B --------> Ω id x b δB >> We prove the left-hand square is a pullback (even when we substitue b with b') in pbl and pbl', the right-hand square is the definition of KroneckerDelta and it is a Pullback, so the whole rectangle is a pullback (pb pb')
using q we note b x id and b' x id are pullbackPr1 of the same diagram, thus they differ by an isomorphism h
  set (pbr := subobject_classifier_pullback Ω (diagonalMap BinProd B)).
  set (pbl := auxpb b).
  set (pbl' := auxpb b').
  set (pb := pullback_glue_pullback C pbr pbl).
  set (pb' := pullback_glue_pullback C pbr pbl').
  fold δ B in pb, pb'.
  transparent assert (pb'' : (Pullback ((identity B) b' · δ B) Ω)). {
    use (Pullback_mor_paths q (idpath _)).
    exact pb.
  }
  induction (pullbackiso _ pb' pb'') as (h,(h_tri1,h_tri2)).
  cbn - [BinProd] in h, h_tri1.
  rewrite (precompWithBinProductArrow _ (BinProd B X)) in h_tri1.
  assert (h_tri11 := (maponpaths (λ f, compose f (BinProductPr1 C (BinProd B X))) h_tri1)).
  cbn beta in h_tri11.
  rewrite !BinProductPr1Commutes in h_tri11.
  assert (h_tri12 := (maponpaths (λ f, compose f (BinProductPr2 C (BinProd B X))) h_tri1)).
  cbn beta in h_tri12.
  rewrite !BinProductPr2Commutes, id_right in h_tri12.
  rewrite h_tri12, id_left in h_tri11.
  exact h_tri11.

Definition SingletonArrow_Monic (B: C) := make_Monic C (SingletonArrow B) (SingletonArrow_isMonic B).

Definition SingletonPred (B: C) : C PowerObject_on_ob P B , Ω .
Show proof.
  use characteristic_morphism.
  { exact B. }
  use SingletonArrow_Monic.
End KroneckerDelta.

Section Exponentials.

Local Notation "'δ' B" := (KroneckerDelta B)(at level 8).
Local Notation "'{⋅}' B" := (SingletonArrow B)(at level 8).

Let v (b c : C)
  : C (b (PowerObject_on_ob P (c b))) , (PowerObject_on_ob P c) .
Show proof.

Let u (b c : C)
  : C PowerObject_on_ob P (c b) , (PowerObject_on_ob P b) .
Show proof.
  use (PowerObject_transpose).
  use (compose (b:= (PowerObject_on_ob P c))).
  - use v.
  - use SingletonPred.

Let name_true (b : C) : C T, PowerObject_on_ob P b .
Show proof.


Let G0 (b c : C) : Subobjectscategory
  (PowerObject_on_ob P (c b)).
Show proof.
  use (PullbackSubobject PB).
  { exact (PowerObject_on_ob P b). }
  { use Subobjectscategory_ob.
    - exact T.
    - exact (name_true b).
    - use from_terminal_isMonic.
  }
  - use u.

Local Lemma G0_Sqr (b c : C) : (Subobject_mor (G0 b c) · (u b c) = (TerminalArrow T _ ) · (name_true b)).
Show proof.
  cbn.
  rewrite PullbackSqrCommutes.
  repeat use cancel_postcomposition.
  use TerminalArrowUnique.

Consider this diagram ev
| | | id x G0 v {·} v b x c^b --------> b x P(c x b) ------> Pc <--------- C | | | | id x ! | id x u | SingletonPred| |! v v v v b x T --------> b x Pb -----------> Ω <--------- T | id x name_true inmap true ʌ | |
! >> The left-hand square is b x (def of c^b), the middle square is the definition of v from u, the right-hand square is the definition of SingletonPred and it is a Pullback, the bottom distorted square is the definition of name_true. Every square commutes. We define ev as the PullbackArrow of the right-hand square

Local Definition ev_aux (b c: C) :
  (identity b) (Subobject_mor (G0 b c)) · v b c · characteristic_morphism Ω (SingletonArrow_Monic c) =
  (identity b) (TerminalArrow T (Subobject_dom (G0 b c))) · TerminalArrow T (BinProd b T) · Ω.
Show proof.
  rewrite assoc'.
    assert (p :
      (identity b) (u b c) · PowerObject_inPred P b =
      v b c · characteristic_morphism Ω (SingletonArrow_Monic c)).
    { use pathsinv0.
      use (PowerObject_transpose_tri P). }
    induction p.
    rewrite assoc, BinProductOfArrows_idxcomp.
    rewrite G0_Sqr.
    rewrite !assoc', <-BinProductOfArrows_idxcomp, !assoc'.
    use cancel_precomposition.
    unfold name_true.
    rewrite (PowerObject_charname_nat_z_iso_tri(b:=b) P).
    rewrite !assoc.
    use cancel_postcomposition.
    use TerminalArrowUnique.

Let ev (b c: C) : C b (Subobject_dom (G0 b c)), c .
Show proof.
  assert (p : PullbackObject (subobject_classifier_pullback Ω (SingletonArrow_Monic c)) = c).
  { apply idpath. }
  induction p.
  use PullbackArrow.
  - exact ((identity _) (Subobject_mor (G0 b c)) · (v b c)).
  - exact ((identity b) (TerminalArrow T (Subobject_dom (G0 b c)))·
      (TerminalArrow T _)).
  - apply ev_aux.

Local Lemma ev_tri (b c : C) : ev b c · SingletonArrow_Monic c =
                                 (identity _) ( Subobject_mor (G0 b c)) · (v b c).
Show proof.

Let h {c b a : C} (f: C b a, c )
    := PowerObject_transpose P ((z_iso_inv (BinProduct_assoc BinProd c b a)) ·
                                  (identity c) f · (δ c)).

Local Lemma h_sq {c b a : C} (f: C b a, c ) :
  f · SingletonArrow_Monic c = (identity b) (h f) · v b c.
Show proof.
  use (invmaponpathsweq (hset_z_iso_equiv _ _ (nat_z_iso_pointwise_z_iso (nat_z_iso_inv (PowerObject_nat_z_iso P)) (b a,,c)))).
  simpl. fold BinProd.
  rewrite <-!BinProductOfArrows_idxcomp, !assoc'.
  intermediate_path ((identity c) f · KroneckerDelta c).
  { use cancel_precomposition.
    apply pathsinv0.
    use PowerObject_transpose_tri. }
  use pathsinv0.
  intermediate_path (
    (identity c) ((identity b) (h f)) ·
    ((BinProduct_assoc BinProd c b (PowerObject_on_ob P c b)) ·
    PowerObject_inPred P c b)).
  { use cancel_precomposition.
    use pathsinv0.
    use (PowerObject_transpose_tri P). }
  rewrite assoc.
  rewrite BinProduct_OfArrows_assoc.
  use pathsinv0.
  rewrite !assoc'.
  use z_iso_inv_to_left.
  rewrite assoc.
  rewrite BinProductOfArrows_id.
  use PowerObject_transpose_tri.

Local Lemma g_aux (c b a : C) (f: C constprod_functor1 BinProd b a, c ) : h f · u b c =
  TerminalArrow T a
  · Subobject_mor
      (Subobjectscategory_ob (name_true b)
         (from_terminal_isMonic T (PowerObject_on_ob P b) (name_true b))).
Show proof.
  assert (p : name_true b = Subobject_mor
    (Subobjectscategory_ob (name_true b)
        (from_terminal_isMonic T (PowerObject_on_ob P b) (name_true b)))). { apply idpath. }
    induction p.
    use (invmaponpathsweq (hset_z_iso_equiv _ _ (nat_z_iso_pointwise_z_iso (nat_z_iso_inv (PowerObject_nat_z_iso P)) (a,,b)))).
    simpl.
    fold BinProd.
    rewrite <-!BinProductOfArrows_idxcomp, !assoc'.
    intermediate_path ((identity b) (h f) ·
    (v b c · SingletonPred c)).
    { use cancel_precomposition.
      use pathsinv0.
      use PowerObject_transpose_tri. }
    use pathsinv0.
    intermediate_path ((identity b) (TerminalArrow T a)
    · (BinProductPr1 C (BinProd b T) · ((TerminalArrow T b · Ω)))).
    { use cancel_precomposition.
      use PowerObject_charname_nat_z_iso_tri. }
    intermediate_path (f · SingletonArrow_Monic c · SingletonPred c).
    {
      rewrite !assoc', subobject_classifier_square_commutes, !assoc.
      use cancel_postcomposition.
      use TerminalArrowEq. }
    rewrite !assoc.
    use cancel_postcomposition.
    use h_sq.

Let g (c b a : C) (f: C constprod_functor1 BinProd b a, c ) : C a, Subobject_dom (G0 b c) .
Show proof.
  use PullbackArrow.
  + exact (h f).
  + use TerminalArrow.
  + apply g_aux.

Local Lemma h_tri {c b a : C} (f: C b a, c ): (g c b a f
· Subobject_mor (G0 b c) = h f).
Show proof.

Local Lemma g_tri {c b a : C} (f: C b a, c ) :
  f = (identity b) (g c b a f) · ev b c.
Show proof.
  use (MonicisMonic _ (SingletonArrow_Monic c)).
  now rewrite !assoc',
    ev_tri,
    !assoc,
    BinProductOfArrows_idxcomp,
    h_tri,
    h_sq.


Local Lemma universality (b c : C): is_universal_arrow_from (constprod_functor1 BinProd b) c (Subobject_dom (G0 b c)) (ev b c).
Show proof.
  intros a f.
  use unique_exists.
  + exact (g c b a f).
  + use g_tri.
  + intro.
    use homset_property.
  + intros g' g_tri'.
    simpl in g_tri'.
    unfold BinProduct_of_functors_mor in g_tri'.
    simpl in g_tri'.
    use (MonicisMonic _ (Subobject_Monic (G0 b c))).
    use (invmaponpathsweq (hset_z_iso_equiv _ _ (nat_z_iso_pointwise_z_iso (nat_z_iso_inv (PowerObject_nat_z_iso P)) (a,,c b)))).
    unfold hset_z_iso_equiv.
    cbn - [BinProd G0 Subobject_mor isBinProduct_Pullback].
    fold BinProd.
    rewrite <-BinProductOfArrows_id.
    use (cancel_z_iso' (BinProduct_assoc BinProd _ _ _)).
    rewrite !assoc.
    intermediate_path (
      (identity c) ((identity b) (g' · Subobject_mor (G0 b c)))·
      BinProduct_assoc BinProd _ _ _ · PowerObject_inPred _ _).
    { use cancel_postcomposition.
      use pathsinv0.
      use (BinProduct_OfArrows_assoc BinProd). }
    use pathsinv0.
    intermediate_path (
      (identity c) ((identity b) ((g c b a f) · Subobject_mor (G0 b c)))·
      BinProduct_assoc BinProd _ _ _ · PowerObject_inPred _ _).
    { use cancel_postcomposition.
      use pathsinv0.
      use (BinProduct_OfArrows_assoc BinProd). }
    rewrite !assoc'.
    use (invmaponpathsweq (hset_z_iso_equiv _ _ (nat_z_iso_pointwise_z_iso ( (PowerObject_nat_z_iso P)) (ba,,c)))).
    unfold hset_z_iso_equiv.
    cbn - [BinProd G0 Subobject_mor BinProduct_assoc].
    unfold PowerObject_nt_data.
    rewrite !PowerObject_transpose_precomp.
    fold (v b c).
    rewrite <-!(BinProductOfArrows_idxcomp _ BinProd _ (Subobject_mor _)).
    rewrite !assoc'.
    rewrite <-!ev_tri.
    rewrite !assoc.
    use cancel_postcomposition.
    rewrite <-g_tri.
    use g_tri'.

Definition Exponentials_from_Topos : Exponentials (BinProd).
Show proof.
  intro b.
  use left_adjoint_from_partial.
  + intro c.
    exact (Subobject_dom (G0 b c)).
  + exact (ev b).
  + exact (universality b).

End Exponentials.

End Topos.