module Algebra.Group.Action where

Group actionsπŸ”—

A useful way to think about groups is to think of their elements as encoding β€œsymmetries” of a particular object. For a concrete example, consider the group of real numbers under addition, and consider the unit circle1 sitting in Given a real number we can consider the β€œaction” on the circle defined by

which β€œvisually” has the effect of rotating the point so it lands radians around the circle, in the counterclockwise direction. Each rotation has an inverse, given by rotation radians in the clockwise direction; but observe that this is the same as rotating degrees counterclockwise. Additionally, observe that rotating by zero radians does nothing to the circle.

We say that acts on the circle by counterclockwise rotation; What this means is that to each element we assign a map in a way compatible with the group structure: Additive inverses β€œtranslate to” inverse maps, addition translates to function composition, and the additive identity is mapped to the identity function. Note that since is a set, this is equivalently a group homomorphism

where the codomain is the group of symmetries of But what if we want a group to act on an object of a more general category, rather than an object of

Automorphism groupsπŸ”—

The answer is that, for an object of some category the collection of all isomorphisms forms a group under composition, generalising the construction of to objects beyond sets! We refer to a β€œself-isomorphism” as an automorphism, and denote their group by

module _ {o β„“} (C : Precategory o β„“) where
  private module C = Cat C

  Aut : C.Ob β†’ Group _
  Aut X = to-group mg where
    mg : make-group (X C.β‰… X)
    mg = hlevel 2
    mg .make-group.unit =
    mg .make-group.mul g f = g C.∘Iso f
    mg .make-group.inv = C._Iso⁻¹
    mg .make-group.assoc x y z = ext (sym (C.assoc _ _ _))
    mg .make-group.invl x = ext (x .C.invl)
    mg .make-group.idl x = ext (C.idr _)

Suppose we have a category an object and a group A on is a group homomorphism

  Action : Group β„“ β†’ C.Ob β†’ Type _
  Action G X = Groups.Hom G (Aut X)

As functorsπŸ”—

Recall that we defined the delooping of a monoid into a category as the category with a single object and If we instead deloop a group into a group then functors correspond precisely to actions of on the object

    Functor→action : (F : Functor BG C) → Action G (F .F₀ tt)
    Functor→action F .hom it = C.make-iso
        (F .F₁ it) (F .F₁ (it ⁻¹))
        (F.annihilate inversel) (F.annihilate inverser)
        open Group-on (G .snd)
        module F = Functor-kit F
    Functorβ†’action F .preserves .is-group-hom.pres-⋆ x y = ext (F .F-∘ _ _)

    Action→functor : {X : C.Ob} (A : Action G X) → Functor BG C
    Action→functor {X = X} A .F₀ _ = X
    Actionβ†’functor A .F₁ e = (A # e)
    Action→functor A .F-id = ap (is-group-hom.pres-id (A .preserves))
    Actionβ†’functor A .F-∘ f g = ap (is-group-hom.pres-⋆ (A .preserves) _ _)

After constructing these functions in either direction, it’s easy enough to show that they are inverse equivalences, but we must be careful about the codomain: rather than taking β€œ with a ” for any particular we take the total space of equipped with

After this small correction, the proof is almost definitional: after applying the right helpers for pushing paths inwards, we’re left with refl at all the leaves.

    Functor≃action : is-equiv {B = Ξ£ _ (Action G)} Ξ» i β†’ _ , Functorβ†’action i
    Functor≃action = is-isoβ†’is-equiv Ξ» where
      .is-iso.inv (x , act) → Action→functor act
      .is-iso.rinv x β†’ Ξ£-pathp refl $
        total-hom-pathp _ _ _ (funext (Ξ» i β†’ C.β‰…-pathp _ _ refl))
          (is-prop→pathp (λ i → is-group-hom-is-prop) _ _)
      .is-iso.linv x β†’ Functor-path (Ξ» _ β†’ refl) Ξ» _ β†’ refl

  1. this is not the higher inductive type , but rather the usual definition of the circle as a subset of β†©οΈŽ