Library UniMath.CategoryTheory.categories.rigs

Rigs category

Contents

  • Precategory of rigs
  • Category of rigs

Precategory of rigs

Section def_rig_precategory.

  Definition rig_fun_space (A B : rig) : hSet := make_hSet (rigfun A B) (isasetrigfun A B).

  Definition rig_precategory_ob_mor : precategory_ob_mor :=
    tpair (λ ob : UU, ob -> ob -> UU) rig (λ A B : rig, rig_fun_space A B).

  Definition rig_precategory_data : precategory_data :=
    make_precategory_data
      rig_precategory_ob_mor (λ (X : rig), (rigisotorigfun (idrigiso X)))
      (fun (X Y Z : rig) (f : rigfun X Y) (g : rigfun Y Z) => rigfuncomp f g).

  Local Definition rig_id_left (X Y : rig) (f : rigfun X Y) :
    rigfuncomp (rigisotorigfun (idrigiso X)) f = f.
  Show proof.
    use rigfun_paths. use idpath.
  Opaque rig_id_left.

  Local Definition rig_id_right (X Y : rig) (f : rigfun X Y) :
    rigfuncomp f (rigisotorigfun (idrigiso Y)) = f.
  Show proof.
    use rigfun_paths. use idpath.
  Opaque rig_id_right.

  Local Definition rig_assoc (X Y Z W : rig) (f : rigfun X Y) (g : rigfun Y Z) (h : rigfun Z W) :
    rigfuncomp f (rigfuncomp g h) = rigfuncomp (rigfuncomp f g) h.
  Show proof.
    use rigfun_paths. use idpath.
  Opaque rig_assoc.

  Lemma is_precategory_rig_precategory_data : is_precategory rig_precategory_data.
  Show proof.
    use make_is_precategory_one_assoc.
    - intros a b f. use rig_id_left.
    - intros a b f. use rig_id_right.
    - intros a b c d f g h. use rig_assoc.

  Definition rig_precategory : precategory :=
    make_precategory rig_precategory_data is_precategory_rig_precategory_data.

  Lemma has_homsets_rig_precategory : has_homsets rig_precategory.
  Show proof.
    intros X Y. use isasetrigfun.

End def_rig_precategory.

Category of rigs

(rigiso X Y) ≃ (iso X Y)


  Lemma rig_iso_is_equiv (A B : ob rig_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 rig_iso_is_equiv.

  Lemma rig_iso_equiv (X Y : ob rig_category) : z_iso X Y -> rigiso (X : rig) (Y : rig).
  Show proof.
    intro f.
    use make_rigiso.
    - exact (make_weq (pr1 (pr1 f)) (rig_iso_is_equiv X Y f)).
    - exact (pr2 (pr1 f)).

  Lemma rig_equiv_is_z_iso (X Y : ob rig_category) (f : rigiso (X : rig) (Y : rig)) :
    @is_z_isomorphism rig_precategory X Y (rigfunconstr (pr2 f)).
  Show proof.
    exists (rigfunconstr (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 rig_equiv_is_z_iso.

  Lemma rig_equiv_z_iso (X Y : ob rig_category) : rigiso (X : rig) (Y : rig) -> z_iso X Y.
  Show proof.
    intros f.
    exists (rigfunconstr (pr2 f)).
    exact (rig_equiv_is_z_iso X Y f).

  Lemma rig_iso_equiv_is_equiv (X Y : rig_category) : isweq (rig_iso_equiv X Y).
  Show proof.
    use isweq_iso.
    - exact (rig_equiv_z_iso X Y).
    - intros x. use z_iso_eq. use rigfun_paths. use idpath.
    - intros y. use rigiso_paths. use subtypePath.
      + intros x0. use isapropisweq.
      + use idpath.
  Opaque rig_iso_equiv_is_equiv.

  Definition rig_iso_equiv_weq (X Y : ob rig_category) :
    weq (z_iso X Y) (rigiso (X : rig) (Y : rig)).
  Show proof.
    use make_weq.
    - exact (rig_iso_equiv X Y).
    - exact (rig_iso_equiv_is_equiv X Y).

  Lemma rig_equiv_iso_is_equiv (X Y : ob rig_category) : isweq (rig_equiv_z_iso X Y).
  Show proof.
    use isweq_iso.
    - exact (rig_iso_equiv X Y).
    - intros y. use rigiso_paths. use subtypePath.
      + intros x0. use isapropisweq.
      + use idpath.
    - intros x. use z_iso_eq. use rigfun_paths. use idpath.
  Opaque rig_equiv_iso_is_equiv.

  Definition rig_equiv_weq_iso (X Y : ob rig_category) :
    (rigiso (X : rig) (Y : rig)) (z_iso X Y).
  Show proof.
    use make_weq.
    - exact (rig_equiv_z_iso X Y).
    - exact (rig_equiv_iso_is_equiv X Y).

Category of rigs


  Definition rig_category_isweq (X Y : ob rig_category) : isweq (λ p : X = Y, idtoiso p).
  Show proof.
    use (@isweqhomot
           (X = Y) (z_iso X Y)
           (pr1weq (weqcomp (rig_univalence X Y) (rig_equiv_weq_iso X Y)))
           _ _ (weqproperty (weqcomp (rig_univalence X Y)
                                     (rig_equiv_weq_iso X Y)))).
    intros e. induction e.
    use (pathscomp0 weqcomp_to_funcomp_app).
    use total2_paths_f.
    - use idpath.
    - use proofirrelevance. use isaprop_is_z_isomorphism.
  Opaque rig_category_isweq.

  Definition rig_category_is_univalent : is_univalent rig_category.
  Show proof.
    intros X Y. exact (rig_category_isweq X Y).

  Definition rig_univalent_category : univalent_category :=
    make_univalent_category rig_category rig_category_is_univalent.

End def_rig_category.