Library UniMath.CategoryTheory.categories.commrings

Category of commrings

Contents

  • Precategory of commrings
  • Category of commrings

Category of commrings

Section def_commring_precategory.

  Definition commring_fun_space (A B : commring) : hSet := make_hSet (ringfun A B) (isasetrigfun A B).

  Definition commring_precategory_ob_mor : precategory_ob_mor :=
    tpair (λ ob : UU, ob -> ob -> UU) commring (λ A B : commring, commring_fun_space A B).

  Definition commring_precategory_data : precategory_data :=
    make_precategory_data
      commring_precategory_ob_mor (λ (X : commring), (rigisotorigfun (idrigiso X)))
      (fun (X Y Z : commring) (f : ringfun X Y) (g : ringfun Y Z) => rigfuncomp f g).

  Local Lemma commring_id_left (X Y : commring) (f : ringfun X Y) :
    rigfuncomp (rigisotorigfun (idrigiso X)) f = f.
  Show proof.
    use rigfun_paths. use idpath.
  Opaque commring_id_left.

  Local Lemma commring_id_right (X Y : commring) (f : ringfun X Y) :
    rigfuncomp f (rigisotorigfun (idrigiso Y)) = f.
  Show proof.
    use rigfun_paths. use idpath.
  Opaque commring_id_right.

  Local Lemma commring_assoc (X Y Z W : commring) (f : ringfun X Y) (g : ringfun Y Z)
        (h : ringfun Z W) : rigfuncomp f (rigfuncomp g h) = rigfuncomp (rigfuncomp f g) h.
  Show proof.
    use rigfun_paths. use idpath.
  Opaque commring_assoc.

  Lemma is_precategory_commring_precategory_data : is_precategory commring_precategory_data.
  Show proof.
    use make_is_precategory_one_assoc.
    - intros a b f. use commring_id_left.
    - intros a b f. use commring_id_right.
    - intros a b c d f g h. use commring_assoc.

  Definition commring_precategory : precategory :=
    make_precategory commring_precategory_data is_precategory_commring_precategory_data.

  Lemma has_homsets_commring_precategory : has_homsets commring_precategory.
  Show proof.
    intros X Y. use isasetrigfun.

End def_commring_precategory.

Category of commrings

(ringiso X Y) ≃ (z_iso X Y)


  Lemma commring_iso_is_equiv (A B : ob commring_category) (f : z_iso A B) : isweq (pr1 (pr1 f)).
  Show proof.
    use isweq_iso.
    - exact (pr1rigfun _ _ (inv_from_z_iso f)).
    - intros x.
      use (toforallpaths _ _ _ (subtypeInjectivity _ _ _ _ (z_iso_inv_after_z_iso f)) x).
      intros x0. use isapropisrigfun.
    - intros x.
      use (toforallpaths _ _ _ (subtypeInjectivity _ _ _ _ (z_iso_after_z_iso_inv f)) x).
      intros x0. use isapropisrigfun.
  Opaque commring_iso_is_equiv.

  Lemma commring_iso_equiv (X Y : ob commring_category) :
    z_iso X Y -> ringiso (X : commring) (Y : commring).
  Show proof.
    intro f.
    use make_ringiso.
    - exact (make_weq (pr1 (pr1 f)) (commring_iso_is_equiv X Y f)).
    - exact (pr2 (pr1 f)).

  Lemma commring_equiv_is_z_iso (X Y : ob commring_category)
        (f : ringiso (X : commring) (Y : commring)) :
    @is_z_isomorphism commring_category X Y (ringfunconstr (pr2 f)).
  Show proof.
    exists (ringfunconstr (pr2 (invrigiso f))).
    use make_is_inverse_in_precat.
    - use rigfun_paths. use funextfun. intros x. use homotinvweqweq.
    - use rigfun_paths. use funextfun. intros y. use homotweqinvweq.
  Opaque commring_equiv_is_z_iso.

  Lemma commring_equiv_iso (X Y : ob commring_category) :
    ringiso (X : commring) (Y : commring) -> z_iso X Y.
  Show proof.
    intros f. exact (_,,commring_equiv_is_z_iso X Y f).

  Lemma commring_iso_equiv_is_equiv (X Y : commring_category) : isweq (commring_iso_equiv X Y).
  Show proof.
    use isweq_iso.
    - exact (commring_equiv_iso X Y).
    - intros x. use z_iso_eq. use rigfun_paths. apply idpath.
    - intros y. use rigiso_paths. use subtypePath.
      + intros x0. use isapropisweq.
      + apply idpath.
  Opaque commring_iso_equiv_is_equiv.

  Definition commring_iso_equiv_weq (X Y : ob commring_category) :
    weq (z_iso X Y) (ringiso (X : commring) (Y : commring)).
  Show proof.
    use make_weq.
    - exact (commring_iso_equiv X Y).
    - exact (commring_iso_equiv_is_equiv X Y).

  Lemma commring_equiv_iso_is_equiv (X Y : ob commring_category) : isweq (commring_equiv_iso X Y).
  Show proof.
    use isweq_iso.
    - exact (commring_iso_equiv X Y).
    - intros y. use rigiso_paths. use subtypePath.
      + intros x0. use isapropisweq.
      + apply idpath.
    - intros x. use z_iso_eq. use rigfun_paths. apply idpath.
  Opaque commring_equiv_iso_is_equiv.

  Definition commring_equiv_weq_iso (X Y : ob commring_category) :
    (ringiso (X : commring) (Y : commring)) (z_iso X Y).
  Show proof.
    use make_weq.
    - exact (commring_equiv_iso X Y).
    - exact (commring_equiv_iso_is_equiv X Y).

Category of commrings


  Definition commring_category_isweq (X Y : ob commring_category) :
    isweq (λ p : X = Y, idtoiso p).
  Show proof.
    use (@isweqhomot
           (X = Y) (z_iso X Y)
           (pr1weq (weqcomp (commring_univalence X Y) (commring_equiv_weq_iso X Y)))
           _ _ (weqproperty (weqcomp (commring_univalence X Y) (commring_equiv_weq_iso X Y)))).
    intros e. induction e.
    use (pathscomp0 weqcomp_to_funcomp_app).
    use total2_paths_f.
    - apply idpath.
    - use proofirrelevance. use isaprop_is_z_isomorphism.
  Opaque commring_category_isweq.

  Definition commring_category_is_univalent : is_univalent commring_category.
  Show proof.
    intros X Y. exact (commring_category_isweq X Y).

  Definition commring_univalent_category : univalent_category :=
    make_univalent_category commring_category commring_category_is_univalent.

End def_commring_category.