Group

definition example

A group is a Monoid in which every element has an inverse — equivalently, a one-object Category in which every morphism is an Isomorphism (a one-object Groupoid).

Sources: 7 Sketches Exercise 3.32, Example 3.18 ($\mathbb{Z}/2\mathbb{Z}$), §3.2.4 ($\mathbf{Grp}$: “reversible action, symmetry”), Example 3.74 (abelianization is a left adjoint); Kittenlab Lecture 7 (natural transformations between group homomorphisms are conjugations); CTfS §3.2 (Definition 3.2.1.1, Proposition 3.2.1.2, Examples 3.2.1.3–3.2.1.10, Application 3.2.1.6, Exercises 3.2.1.7–3.2.1.14), Slogan 4.2.1.5, Application 4.1.2.4

• Examples (CTfS §3.2): $(\mathbb{Z},0,+)$; the clock $\mathbb{Z}/12$ (the inverse of $Q^5$ is $Q^7$); the eight symmetries $\{\mathrm{id},\rho ,{\rho }^2,{\rho }^3,\phi ,\phi \rho ,\phi {\rho }^2,\phi {\rho }^3\}$ of a square with ${\rho }^4={\phi }^2=\mathrm{id}$, ${\rho }^3\phi =\phi \rho$; invertible matrices $G{L}_3$, the orthogonal group $O_3$ (symmetries of the sphere) and the Euclidean group $E(3)$ of isometries of ${\mathbb{R}}^3$; the permutations ${\Sigma }_X$ of a set; the circle group $U(1)$ of angles. In crystallography the space group of an arrangement of atoms $A\subseteq {\mathbb{R}}^3$ consists of the isometries $f$ of ${\mathbb{R}}^3$ that map $A$ into $A$ (CTfS Application 3.2.1.6). Inverses are unique: $m^{\prime }=m^{\prime }(m{m}^{\prime \prime })=(m^{\prime }m)m^{\prime \prime }=m^{\prime \prime }$ (CTfS Proposition 3.2.1.2). The cyclic groups are $\mathbb{Z}$ and $\mathbb{Z}/n$ — exactly the cyclic monoids “without a tail” (Presentation of a Monoid).
• Time-reversibility (CTfS §3.2): “monoids are likely useful in thinking about diffusion, in which time plays a role and things cannot be undone; groups are more likely useful in thinking about mechanics, where actions are time-reversible”. When a symmetry breaks, pass along the forgetful functor $\mathbf{Grp}\to \mathbf{Mon}$ and keep working with the monoid (CTfS Application 4.1.2.4). How groups act on sets — orbits, latitudes on the earth — is in Group Action.
• The monoid $\mathbb{N}$ of Example 3.13 is not a group ($s^n;s=s^{n+1}\ne s^0$); the category presented by one loop $s$ with $s;s=\mathrm{id}$ is the group $\mathbb{Z}/2\mathbb{Z}$ (7S Exercise 3.32).
• Functors between groups-as-categories are group homomorphisms; a Natural Transformation $f\Rightarrow g$ between homomorphisms $G\to H$ is an $h\in H$ with $f(x)=hg(x)h^{-1}$ — for abelian $H$ only $f=g$, but e.g. rotation by $\theta$ and by $-\theta$ in $G{L}_2(\mathbb{R})$ are conjugate by a reflection (Kittenlab Lecture 7).
• $\mathbf{Grp}\to \mathbf{Set}$ (forgetful) has a left adjoint (free group); $\mathbf{Ab}\hookrightarrow \mathbf{Grp}$ has left adjoint the abelianization $G\mapsto G/[G,G]$ (Free-Forgetful Adjunction).
• A dagger structure in which every morphism is unitary makes a category a groupoid; symmetric-monoidal groupoids underlie compact closed structure in physics.

LeanHaskell

#check Group
#check CategoryTheory.SingleObj       -- a group/monoid as a one-object category
#check CategoryTheory.Groupoid        -- every morphism is an iso
#check CategoryTheory.GrpCat          -- the category of groups

class Monoid g => Group g where
  inverse :: g -> g
  -- inverse x <> x = mempty = x <> inverse x
newtype Z2 = Z2 Bool deriving (Eq, Show)
instance Semigroup Z2 where Z2 a <> Z2 b = Z2 (a /= b)
instance Monoid Z2 where mempty = Z2 False
instance Group Z2 where inverse = id