open import Algebra.Group.Cat.Base open import Algebra.Prelude open import Algebra.Group open import Cat.Instances.Delooping import Cat.Functor.Reasoning as Functor-kit module Algebra.Group.Action where

# Group actionsπ

A useful way to think about groups
is to think of their elements as encoding βsymmetriesβ of a particular
object. For a concrete example, consider the group
$(\mathbb{R}, +, 0)$
of real numbers under addition, and consider the unit circle^{1} sitting in
$\mathbb{C}$.
Given a real number
$x : \mathbb{R}$,
we can consider the βactionβ on the circle defined by

$a \mapsto e^{ix}a\text{,}$

which βvisuallyβ has the effect of rotating the point $a$ so it lands $x$ radians around the circle, in the counterclockwise direction. Each $x$-radian rotation has an inverse, given by rotation $x$ radians in the clockwise direction; but observe that this is the same as rotating $-x$ degrees counterclockwise. Addiitonally, observe that rotating by zero radians does nothing to the circle.

We say that
$\mathbb{R}$
*acts* on the circle by counterclockwise rotation; What this
means is that to each element
$x : \mathbb{R}$,
we assign a map
$\mathbb{C} \to \mathbb{C}$
in a way *compatible* with the group structure: Additive inverses
βtranslate toβ inverse maps, addition translates to function
composition, and the additive identity is mapped to the identity
function. Note that since
$\mathbb{C}$
is a set, this is equivalently a group homomorphism

$\mathbb{R} \to \mathrm{Sym}(\mathbb{C})\text{,}$

where the codomain is the group of symmetries of $\mathbb{C}$. But what if we want a group $G$ to act on an object $X$ of a more general category, rather than an object of $\mathbf{Sets}$?

## Automorphism groupsπ

The answer is that, for an object
$X$
of some category
$\mathcal{C}$,
the collection of all isomorphisms
$X \cong X$
forms a group under composition, generalising the construction of
$\mathrm{Sym}(X)$
to objects beyond sets! We give refer to a βself-isomorphismβ as an
**automorphism**, and denote their group by
$\mathrm{Aut}(X)$.

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 .make-group.group-is-set = C.β -is-set mg .make-group.unit = C.id-iso mg .make-group.mul g f = g C.βIso f mg .make-group.inv = C._Isoβ»ΒΉ mg .make-group.assoc x y z = C.β -pathp refl refl (C.assoc _ _ _) mg .make-group.invl x = C.β -pathp refl refl (x .C.invl) mg .make-group.idl x = C.β -pathp refl refl (C.idr _)

Suppose we have a category
$\mathcal{C}$,
an object
$X : \mathcal{C}$,
and a group
$G$;
A
**$G$-action
on
$X$**
is a group homomorphism
$G \to \mathrm{Aut}(X)$.

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

# As functorsπ

Recall that we defined the delooping of a monoid $M$ into a category as the category $\mathbf{B}M$ with a single object $\bull$, and $\mathbf{Hom}(\bull, \bull) = M$. If we instead deloop a group $G$ into a group $\mathbf{B}G$, then functors $F : \mathbf{B}G \to \mathcal{C}$ correspond precisely to actions of $G$ on the object $F(\bull)$!

Functorβaction : (F : Functor BG C) β Action G (F .Fβ tt) Functorβaction F .hom it = C.make-iso (F .Fβ it) (F .Fβ (it β»ΒΉ)) (F.annihilate inversel) (F.annihilate inverser) where open Group-on (G .snd) module F = Functor-kit F Functorβaction F .preserves .is-group-hom.pres-β x y = C.β -pathp refl refl (F .F-β _ _) Actionβfunctor : {X : C.Ob} (A : Action G X) β Functor BG C Actionβfunctor {X = X} A .Fβ _ = X Actionβfunctor A .Fβ e = (A # e) .C.to Actionβfunctor A .F-id = ap C.to (is-group-hom.pres-id (A .preserves)) Actionβfunctor A .F-β f g = ap C.to (is-group-hom.pres-β (A .preserves) _ _)

After constructing these functions in either direction, itβs easy
enough to show that they are inverse equivalences, but we must be
careful about the codomain: rather than taking
β$X$
with a
$G$-actionβ
for any particular
$X$,
we take the *total space* of
$\mathcal{C}$-objects
equipped with
$G$-actions.

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

Functorβaction : is-equiv {B = Ξ£ _ (Action G)} Ξ» i β _ , Functorβaction i Functorβaction = is-isoβis-equiv Ξ» where .is-iso.inv (x , act) β Actionβfunctor act .is-iso.rinv x β Ξ£-pathp refl $ total-hom-pathp _ _ _ (funext (Ξ» i β C.β -pathp _ _ refl)) (is-propβpathp (Ξ» i β is-group-hom-is-prop) _ _) .is-iso.linv x β Functor-path (Ξ» _ β refl) Ξ» _ β refl