Groupoids with an Involution

Introduction

A groupoid is a category in which every morphism is invertible. The assignment of the inverse, $f \mapsto f^{-1}$, is the canonical involution on the morphisms of a groupoid: it is of order two, it fixes every identity, and it reverses the composition, $(g \circ f)^{-1} = f^{-1} \circ g^{-1}$. It is therefore an anti-automorphism of the partial multiplication, and in the language of the preceding article it makes every groupoid a dagger category with the inversion as dagger. This article studies that involution: its orbits, which are the pairs $\{f, f^{-1}\}$ together with the self-inverse morphisms, its fixed elements, which are the morphisms of order at most two, and the non-abelian case, in which inversion is not a homomorphism and the fixed elements are not closed under composition.

The article presupposes Universal Properties and Categories — categories, functors, the opposite category, isomorphisms — and Involutive Categories and the Dagger Functor, which precedes it in this group and treats the general contravariant involution on the morphisms. The group is the one-object case, developed in Groups, later in this Part; the article uses the symmetric group $S_3$ as a one-object example and does not develop the general theory of groups. The relation algebra of a groupoid and the convolution algebra are Groupoid C*-Algebras, in another Part, and are named only.

Three boundaries are observed. No measure and no representation of a groupoid is used. No topology is used; the topological and smooth groupoids belong to later Parts and are named only. No form is used and no C*-algebra is formed.

Groupoids and Inversion

The Groupoid

Definition. A groupoid is a category $\mathcal{G}$ in which every morphism is an isomorphism. The objects of $\mathcal{G}$ are its points, the morphisms are its arrows, and the composition is defined for the composable pairs; for each arrow $f : A \to B$ there is a unique arrow $f^{-1} : B \to A$ with $f \circ f^{-1} = \mathrm{id}_B$ and $f^{-1} \circ f = \mathrm{id}_A$. A groupoid with one object is a group, the arrows being its elements.

Theorem. In a groupoid the inversion is an involution on the arrows, fixes the identities, and reverses the composition:

$$ (f^{-1})^{-1} = f, \qquad \mathrm{id}_A^{-1} = \mathrm{id}_A, \qquad (g \circ f)^{-1} = f^{-1} \circ g^{-1}, $$

and it exchanges the hom-sets, carrying $\mathrm{Hom}(A,B)$ to $\mathrm{Hom}(B,A)$. Hence inversion is an anti-automorphism of the partial multiplication, and it is the dagger of Involutive Categories and the Dagger Functor; the groupoid with inversion is a dagger category in which every morphism is unitary.

Proof. The inverse is unique in a category, so $(f^{-1})^{-1} = f$ because $f$ is an inverse of $f^{-1}$; the identity is its own inverse; and $f^{-1} \circ g^{-1}$ is a two-sided inverse of $g \circ f$ because $(g \circ f) \circ (f^{-1} \circ g^{-1}) = \mathrm{id}$ and the composite in the other order is $\mathrm{id}$. The exchange of hom-sets is the definition of the target and source of $f^{-1}$.

Example. A group $G$ with one object is a groupoid, and the inversion is the group inversion, an anti-automorphism; the abelian groups are exactly those for which it is an automorphism, as the next section proves.

The Involution and the Orbit Decomposition

Theorem (orbits of inversion). The orbits of the involution $f \mapsto f^{-1}$ on the arrows of a groupoid are of size one or two; an orbit is a singleton exactly when $f = f^{-1}$, and then $f$ is an endomorphism with $f \circ f = \mathrm{id}$, that is, $f$ is an involution of its object. The fixed elements of the inversion are exactly the self-inverse arrows, and they include all the identities.

Proof. An orbit $\{f, f^{-1}\}$ is a singleton exactly when $f = f^{-1}$; then $A = B$ because the sources and targets must agree, and $f \circ f = f \circ f^{-1} = \mathrm{id}_A$, so $f^2 = \mathrm{id}$. Conversely $f^2 = \mathrm{id}$ with $f$ an endomorphism gives $f = f^{-1}$ by uniqueness of inverses.

Corollary. For a finite group $G$, the number of orbits of the inversion is $(|G| + #{g : g^2 = e})/2$, where the second term is the number of elements of order at most two.

Proof. The orbits are the fixed elements, one each, and the remaining $|G| - \#\{g : g^2 = e\}$ elements, paired.

Example ($S_3$). In the symmetric group $S_3$ the elements of order at most two are the identity and the three transpositions, so the inversion has $4$ fixed points and $(6 + 4)/2 = 5$ orbits; the two $3$-cycles form one orbit, because the inverse of a $3$-cycle is the other $3$-cycle.

The Non-Abelian Case

Theorem. In a group $G$ the inversion is an automorphism if and only if $G$ is abelian; for a non-abelian group it is an anti-automorphism that is not a homomorphism.

Proof. Inversion is a homomorphism exactly when $(gh)^{-1} = g^{-1}h^{-1}$ for all $g, h$; but $(gh)^{-1} = h^{-1}g^{-1}$, so the two agree for all pairs exactly when the group is abelian. An automorphism is a homomorphism that is bijective, and the inversion is always bijective.

Theorem (the fixed elements are not a subgroup). In a non-abelian group the set of fixed elements of the inversion is not closed under multiplication, and it is not a subgroup. It contains the identity and, when the group is finite, the elements of order two, which in a non-abelian group need not commute.

Proof. In $S_3$ the transpositions $(12)$ and $(13)$ are fixed by the inversion because they are of order two, while their product $(12)(13) = (132)$ is a $3$-cycle and is not fixed; so the fixed set is not closed under multiplication.

Theorem (what the involution does not fix). In a groupoid the inversion fixes exactly the self-inverse arrows and exchanges each remaining arrow with its inverse, so it fixes no arrow of order greater than two, and on a non-abelian group it reverses the order of every product that is not commutative. The involution therefore carries information only about the self-inverse arrows and the pairing of the rest.

Proof. The fixed elements are the self-inverse arrows by the orbit theorem, and on a non-abelian group there are pairs $g, h$ with $gh \neq hg$, for which $(gh)^{-1} = h^{-1}g^{-1} \neq g^{-1}h^{-1}$, so inversion does not preserve the product.

Example (a group with a second involution). Let $G$ be a group and let $\sigma$ be an automorphism of $G$ with $\sigma^2 = \mathrm{id}$; then $\sigma$ is an involution of the group, the fixed subgroup is $G^{\sigma} = \{g : \sigma g = g\}$, and the orbits of $\sigma$ on $G$ are the pairs $\{g, \sigma g\}$. Inversion and $\sigma$ commute when $\sigma(g^{-1}) = (\sigma g)^{-1}$, which holds for every automorphism, so the two involutions generate a group of order at most four acting on $G$; the transformations of the form $g \mapsto (\sigma g)^{-1}$ are the anti-involutions of the group.

Summary

A groupoid is a category with every morphism invertible, and its inversion is the canonical involution on the arrows: it is of order two, fixes the identities, exchanges the hom-sets and reverses the composition, so it is an anti-automorphism and the dagger of the groupoid. Its orbits are the pairs $\{f, f^{-1}\}$ and the self-inverse arrows, and its fixed elements are exactly the arrows of order at most two, which include the identities. For a finite group the number of orbits is $(|G| + #{g : g^2 = e})/2$.

In the non-abelian case the inversion is not an automorphism of the group, and the fixed elements are not closed under multiplication, as $S_3$ shows; the involution fixes the elements of order at most two and pairs the rest, so it carries no information about the order beyond that pairing. A second involution of the group, an automorphism of order two, commutes with the inversion and with it generates a group of involutions on the group.

Summary of Notation

Symbol	Meaning
$\mathcal{G}$	A groupoid
$f^{-1}$	The inverse of $f$; the canonical involution on the arrows
$\mathrm{id}_A$	The identity at the object $A$; fixed by the inversion
self-inverse	$f = f^{-1}$, equivalently $f^2 = \mathrm{id}$; the fixed elements
orbit	$\{f, f^{-1}\}$, of size one or two
$G^{\sigma}$	The fixed subgroup of an automorphism $\sigma$ of order two