Library UniMath.CategoryTheory.AbelianToAdditive

AbelianPreCat is CategoryWithAdditiveStructure

Contents

  • AbelianPreCat is CategoryWithAdditiveStructure
    • Preliminaries
    • AbelianPreCat is CategoryWithAdditiveStructure

AbelianPreCat is CategoryWithAdditiveStructure.

Section abelian_is_additive.

  Variable A : AbelianPreCat.
  Let hs : has_homsets A := homset_property A.

Preliminaries

Some maps we are going to use.
  Definition DiagonalMap {X : A} (BinProd : BinProduct A X X) :
    AX, (BinProductObject A BinProd)⟧ := BinProductArrow A BinProd (identity X) (identity X).

  Definition IdZeroMap {X : A} (BinProd : BinProduct A X X) :
    AX, (BinProductObject A BinProd)⟧ := BinProductArrow A BinProd (identity X)
                                                          (ZeroArrow (to_Zero A) X X).

  Definition ZeroIdMap {X : A} (BinProd : BinProduct A X X) :
    AX, (BinProductObject A BinProd)⟧ := BinProductArrow A BinProd (ZeroArrow (to_Zero A) X X)
                                                          (identity X).

Proofs that these maps are monics.
  Lemma DiagonalMap_isMonic {X : A} (BinProd : BinProduct A X X) :
    isMonic (DiagonalMap BinProd).
  Show proof.
    intros x u v H.
    apply (maponpaths (λ f : _, f · (BinProductPr1 A BinProd))) in H.
    repeat rewrite <- assoc in H. unfold DiagonalMap in H.
    repeat rewrite (BinProductPr1Commutes A _ _ BinProd _ (identity X) _) in H.
    repeat rewrite id_right in H.
    exact H.

  Lemma IdZeroMap_isMonic {X : A} (BinProd : BinProduct A X X) :
    isMonic (IdZeroMap BinProd).
  Show proof.
    intros x u v H.
    apply (maponpaths (λ f : _, f · (BinProductPr1 A BinProd))) in H.
    repeat rewrite <- assoc in H. unfold IdZeroMap in H.
    repeat rewrite (BinProductPr1Commutes A _ _ BinProd _ (identity X) _) in H.
    repeat rewrite id_right in H.
    exact H.

  Lemma ZeroIdMap_isMonic {X : A} (BinProd : BinProduct A X X) :
    isMonic (ZeroIdMap BinProd).
  Show proof.
    intros x u v H.
    apply (maponpaths (λ f : _, f · (BinProductPr2 A BinProd))) in H.
    repeat rewrite <- assoc in H. unfold ZeroIdMap in H.
    repeat rewrite (BinProductPr2Commutes A _ _ BinProd _ _ (identity X)) in H.
    repeat rewrite id_right in H.
    exact H.

We show that Pr1 and Pr2 of BinProduct are epimorphisms.
  Lemma BinProductPr1_isEpi {X : A} (BinProd : BinProduct A X X) :
    isEpi (BinProductPr1 A BinProd).
  Show proof.
    use isEpi_precomp.
    - exact X.
    - exact (IdZeroMap BinProd).
    - unfold IdZeroMap. rewrite (BinProductPr1Commutes A _ _ BinProd _ (identity X) _).
      apply identity_isEpi.

  Lemma BinProductPr2_isEpi {X : A} (BinProd : BinProduct A X X) :
    isEpi (BinProductPr2 A BinProd).
  Show proof.
    use isEpi_precomp.
    - exact X.
    - exact (ZeroIdMap BinProd).
    - unfold ZeroIdMap. rewrite (BinProductPr2Commutes A _ _ BinProd _ _ (identity X)).
      apply identity_isEpi.

We construct kernels of BinProduct Pr1 and Pr2.
  Lemma KernelOfPr1_Eq {X : A} (BinProd : BinProduct A X X) :
    ZeroIdMap BinProd · BinProductPr1 A BinProd = ZeroArrow (to_Zero A) X X.
  Show proof.
    unfold ZeroIdMap.
    exact (BinProductPr1Commutes A _ _ BinProd _ _ (identity X)).

  Local Lemma KernelOfPr1_isKernel_comm (X : A) (BinProd : BinProduct A X X) (w : A)
        (h : A w, BinProductObject A BinProd )
        (H' : h · BinProductPr1 A BinProd = ZeroArrow (to_Zero A) _ _) :
    h · (BinProductPr2 A BinProd)
      · BinProductArrow A BinProd (ZeroArrow (to_Zero A) X X) (identity X) = h.
  Show proof.
    apply BinProductArrowsEq.
    - rewrite <- assoc. rewrite (BinProductPr1Commutes A _ _ BinProd _ _ (identity X)).
      rewrite H'. rewrite ZeroArrow_comp_right. apply idpath.
    - rewrite <- assoc. rewrite (BinProductPr2Commutes A _ _ BinProd _ _ (identity X)).
      apply id_right.

  Local Lemma KernelOfPr1_isKernel_unique (X : A) (BinProd : BinProduct A X X) (w : A)
        (h : A w, BinProductObject A BinProd )
        (H' : h · BinProductPr1 A BinProd = ZeroArrow (to_Zero A) _ _)
        (y : A w, X )
        (H : y · BinProductArrow A BinProd (ZeroArrow (to_Zero A) X X) (identity X) = h) :
    y = h · BinProductPr2 A BinProd.
  Show proof.
    rewrite <- H. rewrite <- assoc.
    rewrite (BinProductPr2Commutes A _ _ BinProd _ _ (identity X)).
    rewrite id_right. apply idpath.

  Lemma KernelOfPr1_isKernel {X : A} (BinProd : BinProduct A X X) :
    isKernel (to_Zero A) (ZeroIdMap BinProd) (BinProductPr1 A BinProd) (KernelOfPr1_Eq BinProd).
  Show proof.
    use (make_isKernel).
    intros w h H'.
    unfold ZeroIdMap.
    use unique_exists.
    - exact (h · (BinProductPr2 A BinProd)).
    - exact (KernelOfPr1_isKernel_comm X BinProd w h H').
    - intros y. apply hs.
    - exact (KernelOfPr1_isKernel_unique X BinProd w h H').

  Definition KernelOfPr1 {X : A} (BinProd : BinProduct A X X) :
    kernels.Kernel (to_Zero A) (BinProductPr1 A BinProd).
  Show proof.
    exact (make_Kernel (to_Zero A) (ZeroIdMap BinProd) _ (KernelOfPr1_Eq BinProd)
                     (KernelOfPr1_isKernel BinProd)).

  Lemma KernelOfPr2_Eq {X : A} (BinProd : BinProduct A X X) :
    IdZeroMap BinProd · BinProductPr2 A BinProd = ZeroArrow (to_Zero A) X X.
  Show proof.
    unfold IdZeroMap. rewrite (BinProductPr2Commutes A _ _ BinProd _ (identity X) _).
    apply idpath.

  Local Lemma KernelOfPr2_isKernel_comm (X : A) (BinProd : BinProduct A X X) (w : A)
        (h : A w, BinProductObject A BinProd)
        (H' : h · BinProductPr2 A BinProd = ZeroArrow (to_Zero A) _ _) :
    h · (BinProductPr1 A BinProd)
      · BinProductArrow A BinProd (identity X) (ZeroArrow (to_Zero A) X X) = h.
  Show proof.
    apply BinProductArrowsEq.
    - rewrite <- assoc. rewrite (BinProductPr1Commutes A _ _ BinProd _ (identity X) _).
      apply id_right.
    - rewrite <- assoc. rewrite (BinProductPr2Commutes A _ _ BinProd _ (identity X) _).
      rewrite H'. rewrite ZeroArrow_comp_right. apply idpath.

  Local Lemma KernelOfPr2_isKernel_unique (X : A) (BinProd : BinProduct A X X) (w : A)
        (h : A w, BinProductObject A BinProd )
        (H' : h · BinProductPr2 A BinProd = ZeroArrow (to_Zero A) _ _)
        (y : A w, X )
        (H : y · BinProductArrow A BinProd (identity X) (ZeroArrow (to_Zero A) X X) = h) :
    y = h · BinProductPr1 A BinProd.
  Show proof.
    rewrite <- H. rewrite <- assoc.
    rewrite (BinProductPr1Commutes A _ _ BinProd _ (identity X) _).
    rewrite id_right. apply idpath.

  Definition KernelOfPr2_isKernel {X : A} (BinProd : BinProduct A X X) :
    isKernel (to_Zero A) (IdZeroMap BinProd) (BinProductPr2 A BinProd) (KernelOfPr2_Eq BinProd).
  Show proof.
    use (make_isKernel).
    intros w h H'.
    unfold IdZeroMap.
    use unique_exists.
    - exact (h · (BinProductPr1 A BinProd)).
    - exact (KernelOfPr2_isKernel_comm X BinProd w h H').
    - intros y. apply hs.
    - intros y H. exact (KernelOfPr2_isKernel_unique X BinProd w h H' y H).

  Definition KernelOfPr2 {X : A} (BinProd : BinProduct A X X) :
    kernels.Kernel (to_Zero A) (BinProductPr2 A BinProd).
  Show proof.
    exact (make_Kernel (to_Zero A) (IdZeroMap BinProd) _ (KernelOfPr2_Eq BinProd)
                     (KernelOfPr2_isKernel BinProd)).

From properties of abelian categories, it follows that Pr1 and Pr2 are cokernels of the above kernels since they are epimorphisms.
  Definition CokernelOfKernelOfPr1 {X : A} (BinProd : BinProduct A X X) :
    cokernels.Cokernel (to_Zero A) (KernelArrow (KernelOfPr1 BinProd)).
  Show proof.
    exact (EpiToCokernel' A (make_Epi A _ (BinProductPr1_isEpi BinProd))
                          (KernelOfPr1 BinProd)).

  Definition CokernelOfKernelOfPr2 {X : A} (BinProd : BinProduct A X X) :
    cokernels.Cokernel (to_Zero A) (KernelArrow (KernelOfPr2 BinProd)).
  Show proof.
    exact (EpiToCokernel' A (make_Epi A _ (BinProductPr2_isEpi BinProd))
                          (KernelOfPr2 BinProd)).

We construct a cokernel of the DiagonalMap. The CokernelOb is the object X.
  Lemma CokernelOfDiagonal_is_iso {X : A} (BinProd : BinProduct A X X) :
    is_z_isomorphism ((BinProductArrow A BinProd (identity X) (ZeroArrow (to_Zero A) X X))
                        · (CokernelArrow (Cokernel (DiagonalMap BinProd)))).
  Show proof.
    set (coker := Cokernel (DiagonalMap BinProd)).
    set (r := (BinProductArrow A BinProd (identity X) (ZeroArrow (to_Zero A) X X))
                · (CokernelArrow coker)).
    set (M := make_Monic A _ (DiagonalMap_isMonic BinProd)).
    set (ker := MonicToKernel M).
    use monic_epi_is_iso.
    - use (@KernelZeroisMonic A (to_Zero A) _ _ _ (ZeroArrow_comp_left _ _ _ _ _ _)).
      use (make_isKernel).
      intros w h H'.
      use unique_exists.
      + exact (ZeroArrow (to_Zero A) w (to_Zero A)).
      + unfold r in H'. rewrite assoc in H'.
        set (y := KernelIn _ ker _ _ H'). cbn in y.
        set (com1 := KernelCommutes _ ker _ _ H'). cbn in com1. fold y in com1.
        unfold DiagonalMap in com1.
        assert (H : y = ZeroArrow (to_Zero A) w ker).
        {
          rewrite <- (id_right y).
          set (tmp := BinProductPr2Commutes A _ _ BinProd _ (identity X) (identity X)).
          rewrite <- tmp. rewrite assoc. rewrite com1. rewrite <- assoc.
          rewrite (BinProductPr2Commutes A _ _ BinProd _ (identity X) _).
          apply ZeroArrow_comp_right.
        }
        cbn. rewrite ZeroArrow_comp_left. cbn in H. apply pathsinv0 in H.
        use (pathscomp0 H). rewrite <- id_right.
        rewrite <- (BinProductPr1Commutes A _ _ BinProd _ (identity X) (ZeroArrow (to_Zero A) _ _)).
        rewrite assoc. rewrite <- com1. rewrite <- assoc.
        rewrite (BinProductPr1Commutes A _ _ BinProd _ (identity X) (identity X)).
        rewrite id_right. apply idpath.
      + intros y. apply hs.
      + intros y H. apply ArrowsToZero.
    - use (@CokernelZeroisEpi A _ _ (to_Zero A) _ (ZeroArrow_comp_right _ _ _ _ _ _)).
      use make_isCokernel.
      intros w h H'.
      use unique_exists.
      + exact (ZeroArrow (to_Zero A) (to_Zero A) w).
      + set(coker2 := CokernelOfKernelOfPr2 BinProd).
        set(coker2ar := CokernelArrow coker2). cbn in coker2ar.
        unfold r in H'. rewrite <- assoc in H'.
        set (y := CokernelOut _ coker2 _ _ H'). cbn in y.
        set (com1 := CokernelCommutes _ coker2 _ _ H'). cbn in com1. fold y in com1.
        assert (H : y = ZeroArrow (to_Zero A) X w).
        {
          rewrite <- (id_left y).
          set (tmp := BinProductPr2Commutes A _ _ BinProd _ (identity X) (identity X)).
          rewrite <- tmp. rewrite <- assoc. rewrite com1. rewrite assoc.
          rewrite CokernelCompZero.
          apply ZeroArrow_comp_left.
        }
        rewrite H in com1. rewrite ZeroArrow_comp_right in com1.
        rewrite <- (ZeroArrow_comp_right A (to_Zero A) _ _ _ (CokernelArrow coker)) in com1.
        apply CokernelArrowisEpi in com1. rewrite <- com1.
        apply ZeroArrow_comp_left.
      + intros y. apply hs.
      + intros y H. apply ArrowsFromZero.

  Definition CokernelOfDiagonal {X : A} (BinProd : BinProduct A X X) :
    cokernels.Cokernel (to_Zero A) (DiagonalMap BinProd).
  Show proof.
    set (X0 := z_iso_inv (make_z_iso _ _ (CokernelOfDiagonal_is_iso BinProd))).
    exact (Cokernel_up_to_iso A (to_Zero A) _
                              (CokernelArrow (Cokernel (DiagonalMap BinProd)) · X0)
                              (Cokernel (DiagonalMap BinProd)) X0 (idpath _)).

We define the op which makes the homsets of an abelian category to abelian groups.
  Definition Abelian_minus_op {X Y : A} (f g : X --> Y) : AX, Y :=
    (BinProductArrow A (to_BinProducts A Y Y) f g)
      · CokernelArrow (CokernelOfDiagonal (to_BinProducts A Y Y)).

  Definition Abelian_op (X Y : A) : binop (AX, Y) :=
    (λ f : _, λ g : _, Abelian_minus_op f (Abelian_minus_op (ZeroArrow (to_Zero A) _ _) g)).

Construction of a precategory with binops from Abelian category.
  Definition AbelianToprecategoryWithBinops : precategoryWithBinOps.
  Show proof.
    use (make_precategoryWithBinOps A).
    unfold precategoryWithBinOpsData.
    intros x y. exact (Abelian_op x y).

We need the following lemmas to prove that the homsets of an abelian precategory are abelian groups.
  Lemma Abelian_DiagonalMap_eq {X Y : A} (f : X --> Y) :
    f · (DiagonalMap (to_BinProducts A Y Y)) =
    (DiagonalMap (to_BinProducts A X X))
      · (BinProductArrow A (to_BinProducts A Y Y)
                          (BinProductPr1 _ (to_BinProducts A X X) · f)
                          (BinProductPr2 _ (to_BinProducts A X X) · f)).
  Show proof.
    unfold DiagonalMap.
    use BinProductArrowsEq.
    - repeat rewrite <- assoc.
      rewrite BinProductPr1Commutes.
      rewrite BinProductPr1Commutes.
      rewrite assoc. rewrite BinProductPr1Commutes.
      rewrite id_left. apply id_right.
    - repeat rewrite <- assoc.
      rewrite BinProductPr2Commutes.
      rewrite BinProductPr2Commutes.
      rewrite assoc. rewrite BinProductPr2Commutes.
      rewrite id_left. apply id_right.

  Lemma Abelian_op_eq_IdZero (X : A) :
    IdZeroMap (to_BinProducts A X X) · CokernelArrow (CokernelOfDiagonal (to_BinProducts A X X)) =
    identity _.
  Show proof.
    set (ar := (BinProductArrow A (to_BinProducts A X X) (identity X) (ZeroArrow (to_Zero A) X X))
                 · (CokernelArrow (Cokernel (DiagonalMap (to_BinProducts A X X))))).
    set (r := make_z_iso ar _ (CokernelOfDiagonal_is_iso (to_BinProducts A X X))).
    cbn. fold ar. fold r. rewrite assoc. exact (is_inverse_in_precat1 r).

  Lemma Abelian_op_eq {X Y : A} (f : X --> Y) :
    let bp1 := to_BinProducts A X X in
    let bp2 := to_BinProducts A Y Y in
    CokernelArrow (CokernelOfDiagonal bp1) · f =
    (BinProductArrow A bp2 (BinProductPr1 _ bp1 · f) (BinProductPr2 _ bp1 · f))
      · (CokernelArrow (CokernelOfDiagonal bp2)).
  Show proof.
    set (bpX := to_BinProducts A X X).
    set (bpY := to_BinProducts A Y Y).
    cbn beta. cbn zeta.
    set (ar := (BinProductArrow A bpY (BinProductPr1 _ bpX · f) (BinProductPr2 _ bpX · f))
                 · (CokernelArrow (CokernelOfDiagonal bpY))).
    assert (H: DiagonalMap bpX · ar = ZeroArrow (to_Zero A) _ _).
    {
      unfold ar. rewrite assoc.
      unfold bpX, bpY. rewrite <- (Abelian_DiagonalMap_eq f). fold bpY.
      rewrite <- assoc. rewrite CokernelCompZero.
      apply ZeroArrow_comp_right.
    }
    set (g := CokernelOut (to_Zero A) (CokernelOfDiagonal bpX) (CokernelOfDiagonal bpY) ar H).
    set (com := CokernelCommutes
                  (to_Zero A) (CokernelOfDiagonal bpX) (CokernelOfDiagonal bpY) ar H).
    rewrite <- com. apply cancel_precomposition.
    set (e1 := Abelian_op_eq_IdZero X).
    set (e2 := Abelian_op_eq_IdZero Y).
    set (ar' := BinProductArrow A bpY (BinProductPr1 A bpX · f) (BinProductPr2 A bpX · f)).
    assert (e3 : IdZeroMap (to_BinProducts A X X) · ar' = f · IdZeroMap (to_BinProducts A Y Y)).
    {
      unfold ar', IdZeroMap.
      apply BinProductArrowsEq.
      - rewrite <- assoc.
        rewrite BinProductPr1Commutes. rewrite assoc.
        rewrite BinProductPr1Commutes. rewrite id_left.
        rewrite <- assoc. rewrite BinProductPr1Commutes.
        rewrite id_right. apply idpath.
      - rewrite <- assoc.
        rewrite BinProductPr2Commutes. rewrite assoc.
        rewrite BinProductPr2Commutes.
        rewrite <- assoc. rewrite BinProductPr2Commutes.
        rewrite ZeroArrow_comp_right.
        apply ZeroArrow_comp_left.
    }
    rewrite <- (id_right f). rewrite <- e2. rewrite assoc.
    rewrite <- e3. unfold ar'. rewrite <- assoc. fold ar.
    rewrite <- id_left. cbn. rewrite <- e1. rewrite <- assoc.
    apply cancel_precomposition. apply pathsinv0.
    apply com.

Construction of morphisms to BinProducts.
  Definition Abelian_mor_to_BinProd {X Y : A} (f g : X --> Y) :
    AX, (BinProductObject A (to_BinProducts A Y Y))⟧ :=
    BinProductArrow _ (to_BinProducts A Y Y) f g.

  Definition Abelian_mor_from_to_BinProd {X Y : A} (f g : X --> Y) :
    ABinProductObject A (to_BinProducts A X X), BinProductObject A (to_BinProducts A Y Y) :=
    BinProductArrow _ (to_BinProducts A Y Y)
                    (BinProductPr1 _ (to_BinProducts A X X) · f)
                    (BinProductPr2 _ (to_BinProducts A X X) · g).

A few equations Abelian_minus_op and Abelian_op.
  Lemma AbelianPreCat_op_eq1 {X Y : A} (a b c d : X --> Y) :
    Abelian_minus_op (Abelian_mor_to_BinProd a b) (Abelian_mor_to_BinProd c d) =
    Abelian_mor_to_BinProd (Abelian_minus_op a c) (Abelian_minus_op b d).
  Show proof.
    set (com1 := Abelian_op_eq (BinProductPr1 A (to_BinProducts A Y Y))).
    set (com2 := Abelian_op_eq (BinProductPr2 A (to_BinProducts A Y Y))).
    unfold Abelian_minus_op. unfold Abelian_mor_to_BinProd.
    use BinProductArrowsEq.
    - rewrite <- assoc. rewrite com1.
      rewrite BinProductPr1Commutes. rewrite assoc.
      apply cancel_postcomposition.
      use BinProductArrowsEq.
      + rewrite BinProductPr1Commutes. rewrite <- assoc.
        rewrite BinProductPr1Commutes. rewrite assoc.
        rewrite BinProductPr1Commutes.
        rewrite BinProductPr1Commutes.
        apply idpath.
      + rewrite BinProductPr2Commutes. rewrite <- assoc.
        rewrite BinProductPr2Commutes. rewrite assoc.
        rewrite BinProductPr2Commutes.
        rewrite BinProductPr1Commutes.
        apply idpath.
    - rewrite <- assoc. rewrite com2.
      rewrite BinProductPr2Commutes. rewrite assoc.
      apply cancel_postcomposition.
      use BinProductArrowsEq.
      + rewrite BinProductPr1Commutes. rewrite <- assoc.
        rewrite BinProductPr1Commutes. rewrite assoc.
        rewrite BinProductPr1Commutes.
        rewrite BinProductPr2Commutes.
        apply idpath.
      + rewrite BinProductPr2Commutes. rewrite <- assoc.
        rewrite BinProductPr2Commutes. rewrite assoc.
        rewrite BinProductPr2Commutes.
        rewrite BinProductPr2Commutes.
        apply idpath.

  Lemma AbelianPreCat_op_eq2 {X Y : A} (a b c d : X --> Y) :
    (Abelian_mor_to_BinProd (Abelian_mor_to_BinProd a b) (Abelian_mor_to_BinProd c d))
      · (Abelian_mor_from_to_BinProd
            (CokernelArrow (CokernelOfDiagonal (to_BinProducts A Y Y)))
            (CokernelArrow (CokernelOfDiagonal (to_BinProducts A Y Y)))) =
    Abelian_mor_to_BinProd
      ((Abelian_mor_to_BinProd a b) · CokernelArrow (CokernelOfDiagonal (to_BinProducts A Y Y)))
      ((Abelian_mor_to_BinProd c d) · CokernelArrow (CokernelOfDiagonal (to_BinProducts A Y Y))).
  Show proof.
    unfold Abelian_mor_to_BinProd.
    unfold Abelian_mor_from_to_BinProd.
    use BinProductArrowsEq.
    - rewrite BinProductPr1Commutes. rewrite <- assoc.
      rewrite BinProductPr1Commutes. rewrite assoc.
      rewrite BinProductPr1Commutes. apply idpath.
    - rewrite BinProductPr2Commutes. rewrite <- assoc.
      rewrite BinProductPr2Commutes. rewrite assoc.
      rewrite BinProductPr2Commutes. apply idpath.

  Lemma AbelianPreCat_op_eq3 {X Y : A} (a b c d : X --> Y) :
    Abelian_minus_op (Abelian_minus_op a b) (Abelian_minus_op c d) =
    Abelian_minus_op (Abelian_minus_op a c) (Abelian_minus_op b d).
  Show proof.
    unfold Abelian_minus_op at 1.
    set (tmp := AbelianPreCat_op_eq1 a c b d).
    unfold Abelian_mor_to_BinProd in tmp.
    rewrite <- tmp.
    set (com := Abelian_op_eq (CokernelArrow (CokernelOfDiagonal (to_BinProducts A Y Y)))).
    unfold Abelian_minus_op at 1. rewrite <- assoc. rewrite com.
    rewrite assoc.
    set (tmp1 := AbelianPreCat_op_eq2 a c b d).
    unfold Abelian_mor_to_BinProd, Abelian_mor_from_to_BinProd in tmp1.
    rewrite tmp1. apply cancel_postcomposition.
    unfold Abelian_minus_op. apply idpath.

AbelianPreCat is CategoryWithAdditiveStructure

The zero element in a homset of A is given by the ZeroArrow.
  Definition AbelianPreCat_homset_zero (X Y : A) : AX, Y := ZeroArrow (to_Zero A) X Y.

Some equations involving Abelian_minus_op and Abelian_op.
  Lemma AbelianPreCat_homset_zero_right' {X Y : A} (f : X --> Y) :
    Abelian_minus_op f (AbelianPreCat_homset_zero X Y) = f.
  Show proof.
    unfold AbelianPreCat_homset_zero. unfold Abelian_minus_op.
    set (tmp := Abelian_op_eq_IdZero Y). unfold IdZeroMap in tmp.
    assert (e : BinProductArrow A (to_BinProducts A Y Y) f (ZeroArrow (to_Zero A) X Y) =
                f · BinProductArrow A (to_BinProducts A Y Y) (identity _)
                  (ZeroArrow (to_Zero A) Y Y)).
    {
      apply BinProductArrowsEq.
      - rewrite BinProductPr1Commutes. rewrite <- assoc.
        rewrite BinProductPr1Commutes.
        rewrite id_right. apply idpath.
      - rewrite BinProductPr2Commutes. rewrite <- assoc.
        rewrite BinProductPr2Commutes.
        rewrite ZeroArrow_comp_right. apply idpath.
    }
    rewrite e. clear e. rewrite <- assoc. rewrite tmp. apply id_right.

  Lemma AbelianPreCat_homset_zero_right {X Y : A} (f : X --> Y) :
    Abelian_op _ _ f (AbelianPreCat_homset_zero X Y) = f.
  Show proof.
    unfold AbelianPreCat_homset_zero.
    unfold Abelian_op. unfold Abelian_minus_op at 2.
    use (pathscomp0 _ (AbelianPreCat_homset_zero_right' f)).
    set (bpY := to_BinProducts A Y Y).
    assert (e : (BinProductArrow A bpY (ZeroArrow (to_Zero A) X Y) (ZeroArrow (to_Zero A) X Y))
                  · CokernelArrow (CokernelOfDiagonal bpY) = ZeroArrow (to_Zero A) _ _ ).
    {
      use (pathscomp0 _ (ZeroArrow_comp_left A (to_Zero A) _ _ _
                                             (CokernelArrow (CokernelOfDiagonal bpY)))).
      apply cancel_postcomposition.
      use BinProductArrowsEq.
      - rewrite BinProductPr1Commutes.
        rewrite ZeroArrow_comp_left. apply idpath.
      - rewrite BinProductPr2Commutes.
        rewrite ZeroArrow_comp_left. apply idpath.
    }
    rewrite e. apply idpath.

  Definition AbelianPreCat_homset_inv {X Y : A} (f : X --> Y) :
    AX, Y := Abelian_minus_op (ZeroArrow (to_Zero A) _ _) f.

  Lemma AbelianPreCat_homset_inv_left' {X Y : A} (f : X --> Y) :
    Abelian_minus_op f f = (AbelianPreCat_homset_zero X Y).
  Show proof.
    unfold AbelianPreCat_homset_zero. unfold Abelian_minus_op.
    set (bpY := to_BinProducts A Y Y).
    assert (e : BinProductArrow A bpY f f =
                f · BinProductArrow A bpY (identity _ ) (identity _ )).
    {
      use BinProductArrowsEq.
      - rewrite BinProductPr1Commutes. rewrite <- assoc.
        rewrite BinProductPr1Commutes. rewrite id_right.
        apply idpath.
      - rewrite BinProductPr2Commutes. rewrite <- assoc.
        rewrite BinProductPr2Commutes. rewrite id_right.
        apply idpath.
    }
    rewrite e. rewrite <- assoc. rewrite CokernelCompZero. apply ZeroArrow_comp_right.

  Lemma AbelianPreCat_homset_inv_left {X Y : A} (f : X --> Y) :
    Abelian_op _ _ (AbelianPreCat_homset_inv f) f = (AbelianPreCat_homset_zero X Y).
  Show proof.
    unfold AbelianPreCat_homset_inv.
    unfold AbelianPreCat_homset_zero.
    unfold Abelian_op.
    rewrite AbelianPreCat_op_eq3.
    rewrite AbelianPreCat_homset_inv_left'.
    rewrite AbelianPreCat_homset_inv_left'.
    apply AbelianPreCat_homset_zero_right'.

  Lemma AbelianPreCat_homset_zero_left {X Y : A} (f : X --> Y) :
    Abelian_op _ _ (AbelianPreCat_homset_zero X Y) f = f.
  Show proof.
    rewrite <- (AbelianPreCat_homset_inv_left' f).
    set (tmp := AbelianPreCat_homset_zero_right' f).
    apply (maponpaths (λ g : _, Abelian_op X Y (Abelian_minus_op f f) g)) in tmp.
    apply pathsinv0 in tmp.
    use (pathscomp0 tmp).
    unfold Abelian_op.
    rewrite AbelianPreCat_op_eq3.
    rewrite AbelianPreCat_homset_zero_right'.
    rewrite AbelianPreCat_homset_zero_right'.
    rewrite AbelianPreCat_homset_inv_left'.
    rewrite AbelianPreCat_homset_zero_right'.
    apply idpath.

  Lemma Abeliain_precategory_homset_comm' {X Y : A} (f g : X --> Y) :
    Abelian_minus_op (Abelian_minus_op (AbelianPreCat_homset_zero X Y) f) g =
    Abelian_minus_op (Abelian_minus_op (AbelianPreCat_homset_zero X Y) g) f.
  Show proof.
    rewrite <- (AbelianPreCat_homset_zero_right' g).
    rewrite AbelianPreCat_op_eq3.
    rewrite (AbelianPreCat_homset_zero_right' f).
    rewrite (AbelianPreCat_homset_zero_right' g).
    apply idpath.

  Lemma AbelianPreCat_homset_comm {X Y : A} (f g : X --> Y) :
    Abelian_op _ _ f g = Abelian_op _ _ g f.
  Show proof.
    set (tmp1 := AbelianPreCat_homset_zero_left f).
    apply (maponpaths (λ h : _, Abelian_op X Y h g)) in tmp1.
    apply pathsinv0 in tmp1. use (pathscomp0 tmp1). clear tmp1.
    set (tmp2 := AbelianPreCat_homset_zero_left g).
    apply (maponpaths (λ h : _, Abelian_op X Y h f)) in tmp2.
    use (pathscomp0 _ tmp2). clear tmp2.
    unfold Abelian_op.
    rewrite (AbelianPreCat_op_eq3 _ _ _ g).
    rewrite (AbelianPreCat_op_eq3 _ _ _ f).
    rewrite (Abeliain_precategory_homset_comm' _ g).
    apply idpath.

  Lemma AbelianPreCat_homset_inv_minus {X Y : A} (f g : X --> Y) :
    Abelian_op _ _ f (AbelianPreCat_homset_inv g) = Abelian_minus_op f g.
  Show proof.
    unfold Abelian_op.
    unfold AbelianPreCat_homset_inv.
    set (tmp := AbelianPreCat_homset_zero_left g).
    unfold Abelian_op in tmp.
    unfold AbelianPreCat_homset_zero in tmp.
    rewrite tmp. apply idpath.

  Lemma AbelianPreCat_homset_inv_right {X Y : A} (f : X --> Y) :
    Abelian_op _ _ f (AbelianPreCat_homset_inv f) = AbelianPreCat_homset_zero X Y.
  Show proof.

  Lemma AbelianPreCat_homset_assoc_eq1 {X Y : A} (f g : X --> Y) :
    Abelian_minus_op (AbelianPreCat_homset_zero X Y) (Abelian_minus_op f g) =
    Abelian_op _ _ (Abelian_minus_op (AbelianPreCat_homset_zero X Y) f) g.
  Show proof.
    rewrite <- (AbelianPreCat_homset_inv_left' (AbelianPreCat_homset_zero X Y)).
    unfold Abelian_op.
    rewrite AbelianPreCat_op_eq3.
    rewrite (AbelianPreCat_homset_inv_left' (AbelianPreCat_homset_zero X Y)).
    apply idpath.

  Lemma AbelianPreCat_homset_assoc_eq2 {X Y : A} (f g : X --> Y) :
    Abelian_minus_op (AbelianPreCat_homset_zero X Y) (Abelian_op _ _ f g) =
    Abelian_minus_op (Abelian_minus_op (AbelianPreCat_homset_zero X Y) f) g.
  Show proof.
    set (tmp := AbelianPreCat_homset_inv_left' (AbelianPreCat_homset_zero X Y)).
    apply (maponpaths (λ h : _, Abelian_minus_op h (Abelian_op X Y f g))) in tmp.
    apply pathsinv0 in tmp. use (pathscomp0 tmp).
    unfold Abelian_op.
    rewrite AbelianPreCat_op_eq3.
    set (tmp2 := AbelianPreCat_homset_zero_left g).
    unfold Abelian_op in tmp2. rewrite tmp2.
    apply idpath.

  Lemma AbelianPreCat_homset_assoc_eq3 {X Y : A} (f g h : X --> Y) :
     Abelian_op _ _ (Abelian_minus_op f g) h = Abelian_minus_op f (Abelian_minus_op g h).
  Show proof.
    unfold Abelian_op.
    rewrite AbelianPreCat_op_eq3.
    rewrite AbelianPreCat_homset_zero_right'.
    apply idpath.

  Lemma AbelianPreCat_homset_assoc {X Y : A} (f g h : X --> Y) :
     Abelian_op _ _ (Abelian_op _ _ f g) h = Abelian_op _ _ f (Abelian_op _ _ g h).
  Show proof.
    set (tmp := AbelianPreCat_homset_zero_left h).
    apply (maponpaths (λ k : _, Abelian_op X Y (Abelian_op X Y f g) k)) in tmp.
    apply pathsinv0 in tmp. use (pathscomp0 tmp).
    unfold Abelian_op.
    rewrite AbelianPreCat_op_eq3.
    rewrite AbelianPreCat_homset_zero_right'.
    rewrite AbelianPreCat_op_eq3.
    rewrite AbelianPreCat_homset_zero_right'.
    apply idpath.

  Definition AbelianTocategoryWithAbgropsData :
    categoryWithAbgropsData AbelianToprecategoryWithBinops.
  Show proof.
    unfold categoryWithAbgropsData.
    intros x y.
    split.
    - use make_isgrop.
      + split.
        * intros f g h. apply AbelianPreCat_homset_assoc.
        * use make_isunital.
          -- apply AbelianPreCat_homset_zero.
          -- split.
             ++ intros f. apply AbelianPreCat_homset_zero_left.
             ++ intros f. apply AbelianPreCat_homset_zero_right.
      + use tpair.
        * intros f. apply (AbelianPreCat_homset_inv f).
        * split.
          -- intros f. apply AbelianPreCat_homset_inv_left.
          -- intros f. apply AbelianPreCat_homset_inv_right.
    - intros f g. apply AbelianPreCat_homset_comm.

We prove that Abelian_precategories are PrecategoriesWithAbgrops.
  Definition AbelianTocategoryWithAbgrops :
    categoryWithAbgrops := make_categoryWithAbgrops
                             AbelianToprecategoryWithBinops
                             AbelianTocategoryWithAbgropsData.

Hide isPreAdditive behind Qed.
  Lemma AbelianToisPreAdditive :
    isPreAdditive AbelianTocategoryWithAbgrops.
  Show proof.
    use make_isPreAdditive.
    - intros x y z f.
      split.
      + intros x' x'0. unfold to_premor.
        cbn. unfold Abelian_op. unfold Abelian_minus_op.
        cbn in x', x'0. rewrite assoc. apply cancel_postcomposition.
        set (bpz := to_BinProducts A z z).
        assert (e : BinProductArrow
                      A bpz (f · x')
                      ((BinProductArrow A bpz (ZeroArrow (to_Zero A) x z) (f · x'0))
                         · CokernelArrow (CokernelOfDiagonal bpz)) =
                    BinProductArrow
                      A bpz (f · x')
                      ((BinProductArrow A bpz (f · ZeroArrow (to_Zero A) y z) (f · x'0))
                         · CokernelArrow (CokernelOfDiagonal bpz))).
        {
          set (tmp := ZeroArrow_comp_right A (to_Zero A) x y z f).
          apply (maponpaths
                   (λ h : _, BinProductArrow A bpz (f · x')
                                              ((BinProductArrow A bpz h (f · x'0))
                                                 · CokernelArrow (CokernelOfDiagonal bpz))))
            in tmp.
          apply pathsinv0 in tmp. apply tmp.
        }
        use (pathscomp0 _ (!e)). clear e.
        rewrite <- precompWithBinProductArrow. rewrite <- assoc.
        rewrite <- precompWithBinProductArrow. apply idpath.
      + unfold to_premor. cbn.
        unfold AbelianPreCat_homset_zero.
        apply ZeroArrow_comp_right.
    - intros x y z f.
      split.
      + intros x' x'0. unfold to_postmor.
        cbn. unfold Abelian_op. unfold Abelian_minus_op.
        cbn in x', x'0.
        set (bpz := to_BinProducts A z z).
        assert (e : BinProductArrow A bpz (ZeroArrow (to_Zero A) x z) (x'0 · f) =
                    (BinProductArrow A (to_BinProducts A y y) (ZeroArrow (to_Zero A) x y) x'0)
                      · Abelian_mor_from_to_BinProd f f).
        {
          use BinProductArrowsEq.
          - unfold Abelian_mor_from_to_BinProd.
            rewrite <- assoc.
            rewrite BinProductPr1Commutes.
            rewrite BinProductPr1Commutes.
            rewrite assoc.
            rewrite BinProductPr1Commutes.
            rewrite ZeroArrow_comp_left.
            apply idpath.
          - unfold Abelian_mor_from_to_BinProd.
            rewrite <- assoc.
            rewrite BinProductPr2Commutes.
            rewrite BinProductPr2Commutes.
            rewrite assoc.
            rewrite BinProductPr2Commutes.
            apply idpath.
        }
        rewrite e. clear e. unfold Abelian_mor_from_to_BinProd. repeat rewrite <- assoc.
        set (tmp := Abelian_op_eq f). cbn zeta in tmp. unfold bpz. rewrite <- tmp.
        repeat rewrite assoc.
        rewrite <- (postcompWithBinProductArrow A _ (to_BinProducts A y y)).
        repeat rewrite <- assoc.
        apply cancel_precomposition.
        unfold BinProductOfArrows.
        apply tmp.
      + unfold to_postmor. cbn.
        unfold AbelianPreCat_homset_zero.
        apply ZeroArrow_comp_left.

We prove that Abelian_precategories are PreAddtitive.
Finally, we show that Abelian_precategories are CategoryWithAdditiveStructure.
  Definition AbelianToAdditive : CategoryWithAdditiveStructure.
  Show proof.
    use make_Additive.
    - exact AbelianToPreAdditive.
    - use make_AdditiveStructure.
      + exact (to_Zero A).
      + exact (BinDirectSums_from_BinProducts
                 AbelianToPreAdditive (to_Zero A) (to_BinProducts A)).

End abelian_is_additive.