Group Action

definition example theorem

An action of a Group $G$ on a set $X$ is a Monoid Action of the underlying monoid: $\circlearrowleft :G\times X\to X$ with $e\circlearrowleft x=x$ and $g\circlearrowleft (h\circlearrowleft x)=(gh)\circlearrowleft x$. Because every $g$ has an inverse, each map $g\circlearrowleft -:X\to X$ is a bijection — “when a group acts on a set, it has the character of symmetry”. The orbit of $x$ is $Gx:=\{g\circlearrowleft x\mid g\in G\}$; “being in the same orbit” is an Equivalence Relation (reflexive by $e$, symmetric by inverses, transitive by composition), so the orbits partition $X$ (CTfS Exercise 3.2.1.15). Categorically, an action is a Functor $G\to \mathbf{Set}$ from the one-object Groupoid $G$, and the set of orbits is its Colimit.

Sources: CTfS §3.2 (Definitions 3.2.1.9, 3.2.1.12, Examples 3.2.1.4, 3.2.1.10, Applications 3.2.1.6, 3.2.1.13, Exercises 3.2.1.11, 3.2.1.14–3.2.1.15), Slogan 4.2.1.5; §5.2.2.1 (representations: actions on vector spaces).

Examples from Category Theory for Scientists

• Rotations of the earth (CTfS Example 3.2.1.10). The circle group $U(1)$ — angles with $360°=0°$, or unit complex numbers under multiplication — acts on ${\mathbb{R}}^3$ by rotation about the $z$-axis, $\theta \circlearrowleft (x,y,z)=(x\cos \theta +y\sin \theta ,-x\sin \theta +y\cos \theta ,z),$ and preserves the unit sphere since $(x\cos \theta +y\sin \theta {)}^2+(-x\sin \theta +y\cos \theta {)}^2+z^2=x^2+y^2+z^2$. An action table needs infinitely many columns (one per angle); with this formula $(1,0,0)\mapsto (0.71,-0.71,0)$ at $45°$ and $(3,4,2)\mapsto (4,-3,2)$ at $90°$ (CTfS’s printed table rotates the other way).
• Latitudes and climate (CTfS Application 3.2.1.13). The orbits of $U(1)$ on the earth’s surface are the circles of latitude; on a thin atmospheric shell they are latitude-lines-at-altitude. “A simplifying assumption in climatology may be given by assuming that $U(1)$ acts on all currents in the atmosphere”: only motion that looks the same along each orbit is allowed. The orbits on all of ${\mathbb{R}}^3$ are the horizontal circles around the $z$-axis together with the single points on it (CTfS Exercise 3.2.1.14).
• Permutations (CTfS Exercises 3.2.1.7, 3.2.1.11): the permutations ${\Sigma }_X$ of a set form a group acting on $X$ by evaluation, $\sigma \circlearrowleft x=\sigma (x)$; ${\Sigma }_{\{1,2,3\}}$ has a single orbit. Every action of $G$ on $X$ is a homomorphism $G\to {\Sigma }_X$ (Cayley’s view).
• Symmetries of a square and of crystals (CTfS Example 3.2.1.4, Application 3.2.1.6): the dihedral group of 8 symmetries acts on the square; the space group of an arrangement of atoms $A\subseteq {\mathbb{R}}^3$ is the group of isometries $f$ of ${\mathbb{R}}^3$ with $f(A)\subseteq A$ — the dashed arrow in $A\to A$ over $f:{\mathbb{R}}^3\to {\mathbb{R}}^3$.
• Linear symmetries: $G{L}_3$ acts on ${\mathbb{R}}^3$, $O_3$ on the sphere, the Euclidean group $E(3)$ on space. Actions on vector spaces by linear maps are representations, functors $G\to \mathbf{Vect}$ (CTfS §5.2.2.1).

Structure

• Monoids vs. groups (CTfS §3.2): “monoids are likely useful in thinking about diffusion, in which time plays a role and things cannot be undone; groups in mechanics, where actions are time-reversible.” When a symmetry breaks, forget along $\mathbf{Grp}\to \mathbf{Mon}$ and keep the monoid action (CTfS Application 4.1.2.4).
• The orbit set is the Coequalizer of the two maps $G\times X\rightrightarrows X$ (action and projection), i.e. the colimit of the functor $G\to \mathbf{Set}$; the set of fixed points is its Limit.
• Equivariant maps are natural transformations; $G$-sets form a Topos ${\mathbf{Set}}^G$, and the automorphism group $\mathrm{Aut}(X)$ of any object of any category acts on its hom-sets.

JuliaLeanHaskell

Docs: plain Julia — Catlab has no dedicated API for this; related: Catlab v0.16 docs · GATlab standard library

using LinearAlgebra
# U(1) acting on ℝ³ by rotation about the z-axis (CTfS Example 3.2.1.10)
R(θ) = [cosd(θ) sind(θ) 0; -sind(θ) cosd(θ) 0; 0 0 1]
act(θ, p) = R(θ) * p
round.(act(45, [1, 0, 0]); digits = 2)                 # [0.71, -0.71, 0.0] (sign convention of R)
act(190, act(278, [3, 4, 2])) ≈ act(468, [3, 4, 2])     # the action law: θ₁ ⋅ (θ₂ ⋅ p) = (θ₁+θ₂) ⋅ p
norm(act(100, [0.6, 0.0, 0.8])) ≈ 1                      # the sphere is preserved
# the orbit of a point on the sphere is its circle of latitude
orbit(p; n = 8) = [act(360k / n, p) for k in 0:n-1]
all(q -> q[3] ≈ 0.8, orbit([0.6, 0.0, 0.8]))            # same height z: true

import Mathlib
#check @MulAction.orbit           -- the orbit G • x
#check @MulAction.orbitRel        -- "same orbit" as a Setoid (equivalence relation)
#check @MulAction.orbitRel.Quotient
#check @MulAction.stabilizer
#check @MulAction.toPermHom       -- an action is a homomorphism G →* Equiv.Perm X
example (X : Type) : MulAction (Equiv.Perm X) X := inferInstance   -- Σ_X acts on X

import Data.List (nub, sort)
 
-- the dihedral group of the square acting on its corners 0..3
data D4 = Rot Int | Flip Int deriving (Eq, Show)   -- ρ^k and φ ρ^k
actD4 :: D4 -> Int -> Int
actD4 (Rot k)  c = (c + k) `mod` 4
actD4 (Flip k) c = (negate (c + k)) `mod` 4
 
orbit :: [g] -> (g -> x -> x) -> x -> [x]
orbit gs act x = map (`act` x) gs
 
-- orbit [Rot k | k <- [0..3]] actD4 0 == [0,1,2,3]: one orbit, the action is transitive