Groupoids

Content created by Fredrik Bakke, Egbert Rijke, Jonathan Prieto-Cubides, Julian KG, fernabnor and louismntnu.

Created on 2022-09-21.
Last modified on 2024-01-31.

module category-theory.groupoids where
Imports
open import category-theory.categories
open import category-theory.functors-categories
open import category-theory.isomorphisms-in-categories
open import category-theory.isomorphisms-in-precategories
open import category-theory.precategories
open import category-theory.pregroupoids

open import foundation.1-types
open import foundation.contractible-types
open import foundation.dependent-pair-types
open import foundation.equivalences
open import foundation.function-types
open import foundation.functoriality-dependent-pair-types
open import foundation.fundamental-theorem-of-identity-types
open import foundation.identity-types
open import foundation.iterated-dependent-pair-types
open import foundation.propositions
open import foundation.sets
open import foundation.type-arithmetic-dependent-pair-types
open import foundation.universe-levels

Idea

A groupoid is a category in which every morphism is an isomorphism.

Definition

is-groupoid-prop-Category :
  {l1 l2 : Level} (C : Category l1 l2)  Prop (l1  l2)
is-groupoid-prop-Category C =
  is-pregroupoid-prop-Precategory (precategory-Category C)

is-groupoid-Category :
  {l1 l2 : Level} (C : Category l1 l2)  UU (l1  l2)
is-groupoid-Category C =
  is-pregroupoid-Precategory (precategory-Category C)

Groupoid : (l1 l2 : Level)  UU (lsuc l1  lsuc l2)
Groupoid l1 l2 = Σ (Category l1 l2) is-groupoid-Category

module _
  {l1 l2 : Level} (G : Groupoid l1 l2)
  where

  category-Groupoid : Category l1 l2
  category-Groupoid = pr1 G

  precategory-Groupoid : Precategory l1 l2
  precategory-Groupoid = precategory-Category category-Groupoid

  obj-Groupoid : UU l1
  obj-Groupoid = obj-Category category-Groupoid

  hom-set-Groupoid : obj-Groupoid  obj-Groupoid  Set l2
  hom-set-Groupoid = hom-set-Category category-Groupoid

  hom-Groupoid : obj-Groupoid  obj-Groupoid  UU l2
  hom-Groupoid = hom-Category category-Groupoid

  id-hom-Groupoid :
    {x : obj-Groupoid}  hom-Groupoid x x
  id-hom-Groupoid = id-hom-Category category-Groupoid

  comp-hom-Groupoid :
    {x y z : obj-Groupoid} 
    hom-Groupoid y z  hom-Groupoid x y  hom-Groupoid x z
  comp-hom-Groupoid = comp-hom-Category category-Groupoid

  associative-comp-hom-Groupoid :
    {x y z w : obj-Groupoid}
    (h : hom-Groupoid z w) (g : hom-Groupoid y z) (f : hom-Groupoid x y) 
    comp-hom-Groupoid (comp-hom-Groupoid h g) f 
    comp-hom-Groupoid h (comp-hom-Groupoid g f)
  associative-comp-hom-Groupoid =
    associative-comp-hom-Category category-Groupoid

  inv-associative-comp-hom-Groupoid :
    {x y z w : obj-Groupoid}
    (h : hom-Groupoid z w) (g : hom-Groupoid y z) (f : hom-Groupoid x y) 
    comp-hom-Groupoid h (comp-hom-Groupoid g f) 
    comp-hom-Groupoid (comp-hom-Groupoid h g) f
  inv-associative-comp-hom-Groupoid =
    inv-associative-comp-hom-Category category-Groupoid

  left-unit-law-comp-hom-Groupoid :
    {x y : obj-Groupoid} (f : hom-Groupoid x y) 
    ( comp-hom-Groupoid id-hom-Groupoid f)  f
  left-unit-law-comp-hom-Groupoid =
    left-unit-law-comp-hom-Category category-Groupoid

  right-unit-law-comp-hom-Groupoid :
    {x y : obj-Groupoid} (f : hom-Groupoid x y) 
    ( comp-hom-Groupoid f id-hom-Groupoid)  f
  right-unit-law-comp-hom-Groupoid =
    right-unit-law-comp-hom-Category category-Groupoid

  iso-Groupoid : (x y : obj-Groupoid)  UU l2
  iso-Groupoid = iso-Category category-Groupoid

  is-groupoid-Groupoid : is-groupoid-Category category-Groupoid
  is-groupoid-Groupoid = pr2 G

Property

The type of groupoids with respect to universe levels l1 and l2 is equivalent to the type of 1-types in l1

The groupoid associated to a 1-type

module _
  {l : Level} (X : 1-Type l)
  where

  obj-groupoid-1-Type : UU l
  obj-groupoid-1-Type = type-1-Type X

  precategory-Groupoid-1-Type : Precategory l l
  pr1 precategory-Groupoid-1-Type = obj-groupoid-1-Type
  pr1 (pr2 precategory-Groupoid-1-Type) = Id-Set X
  pr1 (pr1 (pr2 (pr2 precategory-Groupoid-1-Type))) q p = p  q
  pr1 (pr2 (pr1 (pr2 (pr2 precategory-Groupoid-1-Type))) r q p) =
    inv (assoc p q r)
  pr2 (pr2 (pr1 (pr2 (pr2 precategory-Groupoid-1-Type))) r q p) =
    assoc p q r
  pr1 (pr2 (pr2 (pr2 precategory-Groupoid-1-Type))) x = refl
  pr1 (pr2 (pr2 (pr2 (pr2 precategory-Groupoid-1-Type)))) p = right-unit
  pr2 (pr2 (pr2 (pr2 (pr2 precategory-Groupoid-1-Type)))) p = left-unit

  is-category-groupoid-1-Type :
    is-category-Precategory precategory-Groupoid-1-Type
  is-category-groupoid-1-Type x =
    fundamental-theorem-id
      ( is-contr-equiv'
        ( Σ ( Σ (type-1-Type X)  y  x  y))
            ( λ (y , p) 
              Σ ( Σ (y  x)  q  q  p  refl))
                ( λ (q , l)  p  q  refl)))
        ( ( equiv-tot
            ( λ y 
              equiv-tot
                ( λ p 
                  associative-Σ
                    ( y  x)
                    ( λ q  q  p  refl)
                    ( λ (q , r)  p  q  refl)))) ∘e
          ( associative-Σ
            ( type-1-Type X)
            ( λ y  x  y)
            ( λ (y , p) 
              Σ ( Σ (y  x)  q  q  p  refl))
                ( λ (q , l)  p  q  refl))))
        ( is-contr-iterated-Σ 2
          ( is-torsorial-Id x ,
            ( x , refl) ,
            ( is-contr-equiv
              ( Σ (x  x)  q  q  refl))
              ( equiv-tot  q  equiv-concat (inv right-unit) refl))
              ( is-torsorial-Id' refl)) ,
            ( refl , refl) ,
            ( is-proof-irrelevant-is-prop
              ( is-1-type-type-1-Type X x x refl refl)
              ( refl)))))
      ( iso-eq-Precategory precategory-Groupoid-1-Type x)

  is-groupoid-groupoid-1-Type :
    is-pregroupoid-Precategory precategory-Groupoid-1-Type
  pr1 (is-groupoid-groupoid-1-Type x y p) = inv p
  pr1 (pr2 (is-groupoid-groupoid-1-Type x y p)) = left-inv p
  pr2 (pr2 (is-groupoid-groupoid-1-Type x y p)) = right-inv p

  groupoid-1-Type : Groupoid l l
  pr1 (pr1 groupoid-1-Type) = precategory-Groupoid-1-Type
  pr2 (pr1 groupoid-1-Type) = is-category-groupoid-1-Type
  pr2 groupoid-1-Type = is-groupoid-groupoid-1-Type

The 1-type associated to a groupoid

module _
  {l1 l2 : Level} (G : Groupoid l1 l2)
  where

  1-type-Groupoid : 1-Type l1
  1-type-Groupoid = obj-1-type-Category (category-Groupoid G)

The groupoid obtained from the 1-type induced by a groupoid G is G itself

module _
  {l1 l2 : Level} (G : Groupoid l1 l2)
  where

  functor-equiv-groupoid-1-type-Groupoid :
    functor-Category
      ( category-Groupoid (groupoid-1-Type (1-type-Groupoid G)))
      ( category-Groupoid G)
  pr1 functor-equiv-groupoid-1-type-Groupoid = id
  pr1 (pr2 functor-equiv-groupoid-1-type-Groupoid) {x} {.x} refl =
    id-hom-Groupoid G
  pr1 (pr2 (pr2 functor-equiv-groupoid-1-type-Groupoid)) {x} refl refl =
    inv (right-unit-law-comp-hom-Groupoid G (id-hom-Groupoid G))
  pr2 (pr2 (pr2 functor-equiv-groupoid-1-type-Groupoid)) x = refl

The 1-type obtained from the groupoid induced by a 1-type X is X itself

module _
  {l : Level} (X : 1-Type l)
  where

  equiv-1-type-groupoid-1-Type :
    type-equiv-1-Type (1-type-Groupoid (groupoid-1-Type X)) X
  equiv-1-type-groupoid-1-Type = id-equiv

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