open import Algebra.Prelude
open import Algebra.Group

open import Cat.Displayed.Univalence.Thin
open import Cat.Functor.Properties
open import Cat.Prelude

module Algebra.Group.Cat.Base where

private variable
  β„“ : Level
open Cat.Displayed.Univalence.Thin public
open Functor
import Cat.Reasoning as CR

The category of GroupsπŸ”—

The category of groups, as the name implies, has its objects the Groups, with the morphisms between them the group homomorphisms.

open Group-on
open is-group-hom

Group-structure : βˆ€ β„“ β†’ Thin-structure β„“ Group-on
Group-structure β„“ .is-hom f G Gβ€² = el! (is-group-hom G Gβ€² f)

Group-structure β„“ .id-is-hom        .pres-⋆ x y = refl
Group-structure β„“ .∘-is-hom f g Ξ± Ξ² .pres-⋆ x y =
  ap f (Ξ² .pres-⋆ x y) βˆ™ Ξ± .pres-⋆ _ _

Group-structure β„“ .id-hom-unique {s = s} {t = t} Ξ± Ξ² i =
  record
    { _⋆_          = Ξ» x y β†’ Ξ± .pres-⋆ x y i
    ; has-is-group =
      is-propβ†’pathp (Ξ» i β†’ is-group-is-prop {_*_ = Ξ» x y β†’ Ξ± .pres-⋆ x y i})
        (s .has-is-group)
        (t .has-is-group)
        i
    }

Groups : βˆ€ β„“ β†’ Precategory (lsuc β„“) β„“
Groups β„“ = Structured-objects (Group-structure β„“)

Groups-is-category : βˆ€ {β„“} β†’ is-category (Groups β„“)
Groups-is-category = Structured-objects-is-category (Group-structure _)

instance
  Groups-equational : βˆ€ {β„“} β†’ is-equational (Group-structure β„“)
  Groups-equational .is-equational.invert-id-hom x .pres-⋆ a b = sym (x .pres-⋆ a b)

module Groups {β„“} = Cat (Groups β„“)

Group : βˆ€ β„“ β†’ Type (lsuc β„“)
Group _ = Groups.Ob

to-group : βˆ€ {β„“} {A : Type β„“} β†’ make-group A β†’ Group β„“
to-group {A = A} mg = el A (mg .make-group.group-is-set) , (to-group-on mg)

The underlying setπŸ”—

The category of groups admits a faithful functor into the category of sets, written U:Groups→SetsU : \mathrm{Groups} \to \mathbf{Sets}, which projects out the underlying set of the group. Faithfulness of this functor says, in more specific words, that equality of group homomorphisms can be tested by comparing the underlying morphisms of sets.

Forget : Functor (Groups β„“) (Sets β„“)
Forget = Forget-structure (Group-structure _)

Forget-is-faithful : is-faithful (Forget {β„“})
Forget-is-faithful = Structured-hom-path (Group-structure _)