Library UniMath.CategoryTheory.limits.graphs.cokernels

Cokernels defined in terms of colimits.

Contents

  • Definition coincides with direct definition

Definition of cokernels in terms of colimits

Section def_cokernels.

  Variable C : precategory.
  Variable hs: has_homsets C.
  Variable Z : Zero C.

  Definition Cokernel {a b : C} (f : Ca, b) := Coequalizer C f (ZeroArrow Z a b).

Coincides with the direct definiton

  Lemma equiv_Cokernel1_eq {a b : C} (f : Ca, b)
        (CK : limits.cokernels.Cokernel (equiv_Zero2 Z) f) :
    f · (CokernelArrow CK) = ZeroArrow Z a b · (CokernelArrow CK).
  Proof.
    rewrite CokernelCompZero. rewrite postcomp_with_ZeroArrow. apply equiv_ZeroArrow.
  Qed.

  Lemma equiv_Cokernel1_isCoequalizer {a b : C} (f : Ca, b)
        (CK : limits.cokernels.Cokernel (equiv_Zero2 Z) f) :
    isCoequalizer C f (ZeroArrow Z a b) CK (CokernelArrow CK) (equiv_Cokernel1_eq f CK).
  Proof.
    use (mk_isCoequalizer _ hs).
    intros w h H.
    use unique_exists.
    - use CokernelOut.
      + exact h.
      + rewrite postcomp_with_ZeroArrow in H.
        use (pathscomp0 _ (!(equiv_ZeroArrow a w Z))).
        exact H.
    - use CokernelCommutes.
    - intros y. apply hs.
    - intros y T. cbn in T.
      use CokernelOutsEq. unfold CokernelArrow.
      use (pathscomp0 T). apply pathsinv0.
      use CokernelCommutes.
  Qed.

  Definition equiv_Cokernel1 {a b : C} (f : Ca, b)
             (CK : limits.cokernels.Cokernel (equiv_Zero2 Z) f) : Cokernel f.
  Proof.
    use mk_Coequalizer.
    - exact CK.
    - exact (CokernelArrow CK).
    - exact (equiv_Cokernel1_eq f CK).
    - exact (equiv_Cokernel1_isCoequalizer f CK).
  Defined.


  Lemma equiv_Cokernel2_eq {a b : C} (f : Ca, b) (CK : cokernels.Cokernel (equiv_Zero2 Z) f) :
    f · CokernelArrow CK = ZeroArrow Z a b · CokernelArrow CK.
  Proof.
    rewrite CokernelCompZero. rewrite postcomp_with_ZeroArrow. apply equiv_ZeroArrow.
  Qed.

  Lemma equiv_Cokernel2_isCoequalizer {a b : C} (f : Ca, b)
        (CK : cokernels.Cokernel (equiv_Zero2 Z) f) :
    isCoequalizer C f (ZeroArrow Z a b) CK (CokernelArrow CK) (equiv_Cokernel2_eq f CK).
  Proof.
    use (mk_isCoequalizer _ hs).
    intros w h H.
    use unique_exists.
    - use CokernelOut.
      + exact h.
      + use (pathscomp0 H). apply pathsinv0. rewrite postcomp_with_ZeroArrow. apply equiv_ZeroArrow.
    - use CokernelCommutes.
    - intros y. apply hs.
    - intros y T. cbn in T. use CokernelOutsEq. rewrite T. rewrite CokernelCommutes.
      apply idpath.
  Qed.

  Definition equiv_Cokernel2 {a b : C} (f : Ca, b)
             (CK : limits.cokernels.Cokernel (equiv_Zero2 Z) f) : Cokernel f.
  Proof.
    use mk_Coequalizer.
    - exact CK.
    - exact (CokernelArrow CK).
    - exact (equiv_Cokernel2_eq f CK).
    - exact (equiv_Cokernel2_isCoequalizer f CK).
  Defined.

End def_cokernels.