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 Groupsπ
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 : β β β Thin-structure β Group-on Group-structure β .is-hom f G Gβ² = el! (is-group-hom G Gβ² f) Group-structure β .id-is-hom .pres-β x y = refl Group-structure β .β-is-hom f g Ξ± Ξ² .pres-β x y = ap f (Ξ² .pres-β x y) β Ξ± .pres-β _ _ Group-structure β .id-hom-unique {s = s} {t = t} Ξ± Ξ² i = record { _β_ = Ξ» x y β Ξ± .pres-β x y i ; has-is-group = is-propβpathp (Ξ» i β is-group-is-prop {_*_ = Ξ» x y β Ξ± .pres-β x y i}) (s .has-is-group) (t .has-is-group) i } Groups : β β β Precategory (lsuc β) β Groups β = Structured-objects (Group-structure β) Groups-is-category : β {β} β is-category (Groups β) Groups-is-category = Structured-objects-is-category (Group-structure _) instance Groups-equational : β {β} β is-equational (Group-structure β) Groups-equational .is-equational.invert-id-hom x .pres-β a b = sym (x .pres-β a b) module Groups {β} = Cat (Groups β) Group : β β β Type (lsuc β) Group _ = Groups.Ob to-group : β {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 setπ
The category of groups admits a faithful functor into the category of sets, written , 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 _)