Library UniMath.CategoryTheory.categories.rings

Rings category

Contents

  • Precategory of rings
  • Category of rings

Precategory of rings

Section def_ring_precategory.

  Definition ring_fun_space (A B : ring) : hSet := make_hSet (ringfun A B) (isasetrigfun A B).

  Definition ring_precategory_ob_mor : precategory_ob_mor :=
    tpair (λ ob : UU, ob -> ob -> UU) ring (λ A B : ring, ring_fun_space A B).

  Definition ring_precategory_data : precategory_data :=
    make_precategory_data
      ring_precategory_ob_mor (λ (X : ring), (rigisotorigfun (idrigiso X)))
      (fun (X Y Z : ring) (f : ringfun X Y) (g : ringfun Y Z) => rigfuncomp f g).

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

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

  Local Lemma ring_assoc (X Y Z W : ring) (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. apply idpath.
  Opaque ring_assoc.

  Lemma is_precategory_ring_precategory_data : is_precategory ring_precategory_data.
  Show proof.
    use make_is_precategory_one_assoc.
    - intros a b f. use ring_id_left.
    - intros a b f. use ring_id_right.
    - intros a b c d f g h. use ring_assoc.

  Definition ring_precategory : precategory :=
    make_precategory ring_precategory_data is_precategory_ring_precategory_data.

  Lemma has_homsets_ring_precategory : has_homsets ring_precategory.
  Show proof.
    intros X Y. use isasetrigfun.

End def_ring_precategory.

Category of rings

(ringiso X Y) ≃ (z_iso X Y)


  Lemma ring_iso_is_equiv (A B : ob ring_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 ring_iso_is_equiv.

  Lemma ring_iso_equiv (X Y : ob ring_category) : z_iso X Y -> ringiso (X : ring) (Y : ring).
  Show proof.
    intro f.
    use make_ringiso.
    - exact (make_weq (pr1 (pr1 f)) (ring_iso_is_equiv X Y f)).
    - exact (pr2 (pr1 f)).

  Lemma ring_equiv_is_z_iso (X Y : ob ring_category) (f : ringiso (X : ring) (Y : ring)) :
    @is_z_isomorphism ring_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 ring_equiv_is_z_iso.

  Lemma ring_equiv_iso (X Y : ob ring_category) : ringiso (X : ring) (Y : ring) -> z_iso X Y.
  Show proof.
    intros f. exact (_,,ring_equiv_is_z_iso X Y f).

  Lemma ring_iso_equiv_is_equiv (X Y : ring_category) : isweq (ring_iso_equiv X Y).
  Show proof.
    use isweq_iso.
    - exact (ring_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 ring_iso_equiv_is_equiv.

  Definition ring_iso_equiv_weq (X Y : ob ring_category) :
    weq (z_iso X Y) (ringiso (X : ring) (Y : ring)).
  Show proof.
    use make_weq.
    - exact (ring_iso_equiv X Y).
    - exact (ring_iso_equiv_is_equiv X Y).

  Lemma ring_equiv_iso_is_equiv (X Y : ob ring_category) : isweq (ring_equiv_iso X Y).
  Show proof.
    use isweq_iso.
    - exact (ring_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 ring_equiv_iso_is_equiv.

  Definition ring_equiv_weq_iso (X Y : ob ring_category) :
    (ringiso (X : ring) (Y : ring)) (z_iso X Y).
  Show proof.
    use make_weq.
    - exact (ring_equiv_iso X Y).
    - exact (ring_equiv_iso_is_equiv X Y).

Category of rings


  Definition ring_category_isweq (X Y : ob ring_category) : isweq (λ p : X = Y, idtoiso p).
  Show proof.
    use (@isweqhomot
           (X = Y) (z_iso X Y)
           (pr1weq (weqcomp (ring_univalence X Y) (ring_equiv_weq_iso X Y)))
           _ _ (weqproperty (weqcomp (ring_univalence X Y) (ring_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 ring_category_isweq.

  Definition ring_category_is_univalent : is_univalent ring_category.
  Show proof.
    intros X Y. exact (ring_category_isweq X Y).

  Definition ring_univalent_category : univalent_category :=
    make_univalent_category ring_category ring_category_is_univalent.

End def_ring_category.