Library UniMath.CategoryTheory.Groupoids

Groupoids

Author: Langston Barrett (@siddharthist), March 2018

Contents

  • Definitions
    • Pregroupoid
    • Groupoid
    • Univalent groupoid
  • An alternative characterization of univalence for groupoids
  • Lemmas
  • Subgroupoids
  • Discrete categories

Definitions

A precategory is a pregroupoid when all of its arrows are isos.
Definition is_pregroupoid (C : category) :=
   (x y : C) (f : x --> y), is_z_isomorphism f.

Lemma isaprop_is_pregroupoid (C : category) : isaprop (is_pregroupoid C).
Show proof.
  do 3 (apply impred; intro).
  apply isaprop_is_z_isomorphism.

Definition pregroupoid : UU := C : category, is_pregroupoid C.

Constructors, accessors, and coersions
A category is a groupoid when all of its arrows are isos.
Definition groupoid : UU := C : category, is_pregroupoid C.

Constructors, accessors, and coersions
Constructors, accessors, and coersions
An alternative characterization of univalence for groupoids
The morphism part of an isomorphism is an inclusion.
Lemma morphism_from_iso_is_incl (C : category) (a b : ob C) :
  isincl (@morphism_from_z_iso C a b).
Show proof.
  intro g.
  apply (isofhlevelweqf _ (ezweqpr1 _ _)).
  apply isaprop_is_z_isomorphism.

The alternative characterization implies the normal one. Note that the other implication is missing, it should be completed if possible.
Lemma is_univalent_pregroupoid_is_univalent {pgpd : groupoid} :
  is_univalent_pregroupoid pgpd -> is_univalent pgpd.
Show proof.
  intros ig.
  intros a b.
  use (isofhlevelff 0 idtoiso (morphism_from_z_iso _ _)).
  + use (isweqhomot (idtomor _ _)).
    * intro p; destruct p; reflexivity.
    * apply ig.
  + apply (morphism_from_iso_is_incl pgpd).

Lemmas

In a pregroupoid, the hom-types are equivalent to the type of isomorphisms.
Lemma pregroupoid_hom_weq_iso {pgpd : pregroupoid} (a b : pgpd) : (a --> b) z_iso a b.
Show proof.
  use weq_iso.
  - intros f; refine (f,, _); apply pregroupoid_is_pregroupoid.
  - apply pr1.
  - reflexivity.
  - intro; apply z_iso_eq; reflexivity.

Lemma pregroupoid_hom_weq_iso_idtoiso {pgpd : pregroupoid} (a : pgpd) :
  pregroupoid_hom_weq_iso a a (identity a) = idtoiso (idpath a).
Show proof.
  apply z_iso_eq; reflexivity.

Lemma pregroupoid_hom_weq_iso_comp {pgpd : pregroupoid} {a b c : ob pgpd}
      (f : a --> b) (g : b --> c) :
  z_iso_comp (pregroupoid_hom_weq_iso _ _ f) (pregroupoid_hom_weq_iso _ _ g) =
  (pregroupoid_hom_weq_iso _ _ (f · g)).
Show proof.
  apply z_iso_eq; reflexivity.

If D is a groupoid, then a functor category into it is as well.
Lemma is_pregroupoid_functor_cat {C : precategory} {D : category}
  (gr_D : is_pregroupoid D)
  : is_pregroupoid (functor_category C D).
Show proof.
  intros F G α; apply nat_trafo_z_iso_if_pointwise_z_iso.
  intros c; apply gr_D.

In a univalent groupoid, arrows are equivalent to paths
Lemma univalent_groupoid_arrow_weq_path {ugpd : univalent_groupoid} {a b : ob ugpd} :
  (a --> b) a = b.
Show proof.
  intermediate_weq (z_iso a b).
  - apply (@pregroupoid_hom_weq_iso ugpd).
  - apply invweq; use make_weq.
    + exact idtoiso.
    + apply univalent_category_is_univalent.

Subgroupoids

Every category has a subgroupoid of all the objects and only the isos.
Definition maximal_subgroupoid {C : category} : pregroupoid.
Show proof.
  use make_pregroupoid.
  - use make_category.
    + use make_precategory; use tpair.
      * use tpair.
        -- exact (ob C).
        -- exact (λ a b, f : a --> b, is_z_isomorphism f).
      * unfold precategory_id_comp; cbn.
        use make_dirprod.
        -- exact (λ a, identity a,, identity_is_z_iso _).
        -- intros ? ? ? f g; exact (z_iso_comp f g).
      * use make_dirprod;
        intros;
        apply z_iso_eq.
        -- apply id_left.
        -- apply id_right.
      * use make_dirprod; intros; apply z_iso_eq.
        -- apply assoc.
        -- apply assoc'.
    + cbn. intros a b. cbn. apply isaset_z_iso.
  - intros ? ? f. exists (z_iso_inv_from_z_iso f).
    use make_dirprod; apply z_iso_eq.
    + apply z_iso_inv_after_z_iso.
    + apply z_iso_after_z_iso_inv.

Goal C:category, pregroupoid_to_precategory (@maximal_subgroupoid (C^op))
                      = (@maximal_subgroupoid C)^op.
Show proof.
  Fail reflexivity.
Abort.
Goal (C:category) (a b:C) (f : C b, a ⟧), @is_z_isomorphism C^op a b f = @is_z_isomorphism C b a f.
Proof.
  Fail reflexivity.
Abort.

Discrete categories

See Categories.Categories.StandardCategories for a proof that any discrete category is equivalent to the path groupoid of its objects.
A discrete category is a univalent pregroupoid with an underlying setcategory. Why? In this case, all arrows must be identities: every arrow has an inverse, and isos induce equality (by univalence).

In a discrete category, hom-types are propositions.
A functor between discrete categories is given by a function on their objects.
Lemma discrete_functor {C D : discrete_category} (f : ob C ob D) :
  functor C D.
Show proof.
  use make_functor.
  - use make_functor_data.
    + apply f.
    + intros a b atob.
      pose (aeqb := @univalent_groupoid_arrow_weq_path C _ _ atob).
      exact (transportf (λ z, _ f a, z ) (maponpaths f aeqb) (identity _)).
  - split.
    + intro; apply discrete_category_hom_prop.
    + intros ? ? ? ? ?; apply discrete_category_hom_prop.

Definition discrete_cat_nat_trans {C : precategory} {D : discrete_category}
  {F G : functor C D} (t : x : ob C, F x --> G x) : is_nat_trans F G t.
Show proof.
  intros ? ? ?; apply discrete_category_hom_prop.