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Action groupoids

Action groupoids

Introduction

In this section, we'll explore the connection between Transitive quivers and groups. We'll see that a transitive quiver that is the Cayley quiver of a group has an important property: its path groupoid is the so-called action groupoid of that group. This has two important consequences: firstly, it gives us a dictionary to translate between the language of quiver geometry and group theory when dealing with Cayley quivers, and secondly, it explains how and in what exact sense quiver geometry can take us beyond group theory, and the spaces that group theory can describe.

Groups and groupoids

We briefly recall the definitions of groups and groupoids.

A group is a structure \(\tuple{\group{G},\Gmult }\) consisting of a set of elements \(\group{G}\) and a multiplication \(\functionSignature{\function{\Gmult }}{\tuple{\group{G},\group{G}}}{\group{G}}\). The multiplication is associative, so \(\groupElement{f}\Gmult \paren{\groupElement{g}\Gmult \groupElement{h}} = \paren{\groupElement{f}\Gmult \groupElement{g}}\Gmult \groupElement{h}\) for all \(\elemOf{\groupElement{f},\groupElement{g},\groupElement{h}}{\group{G}}\). Each element \(\groupElement{g}\) also has an inverse \(\groupInverse{\groupElement{g}}\) satisfying \(\groupInverse{\groupElement{g}}\Gmult \groupElement{g} = \groupElement{g}\Gmult \groupInverse{\groupElement{g}} = \groupElement{e}\), where \(\groupElement{e}\Gmult \groupElement{g} = \groupElement{g}\Gmult \groupElement{e} = \groupElement{g}\) defines the identity or unit element \(\elemOf{\groupElement{e}}{\group{G}}\).

A groupoid is, roughly speaking, a group in which the multiplication \(\Gmult\) becomes a partial function, and so need not be defined for all pairs \(\elemOf{\groupoidElement{g},\groupoidElement{h}}{\groupoid{G}}\). Importantly, a groupoid does not need to have a unique identity: in general, there can be multiple identity elements that satisfy the required properties for different subsets of \(G\). We'll call these the units of the groupoid.

Group actions

A group action \(\function{A}\) is the binding of a group \(\group{G}\) to an object \(\sym{X}\) on which it acts, expressed as a two-argument map \(\functionSignature{\function{A}}{\tuple{\group{G},\sym{X}}}{\sym{X}}\) that encodes how elements of \(\group{G}\) effect elements of \(\sym{X}\). By fixing the first argument of the map \(\function{A}\) we obtain a family of maps \(\functionSignature{\function{\function{A}(\group{G})}}{\sym{X}}{\sym{X}}\), it is these maps that encode the behavior of the group \(\group{G}\), with function composing playing the role of group multiplication.

Self action

Perhaps the most natural group action is given by Cayley's theorem, in which we allow a group to act on itself, so that \(\sym{X} = \group{G}\). The action is defined by \(\mto{\tuple{\groupElement{g},\groupoidElement{h}}}{\groupElement{g}\Gmult \groupElement{h}}\). The representative of a particular group element \(\groupElement{g}\) is the function that left-multiplies by \(\groupElement{g}\):

\[ \function{A_{\groupElement{g}}}\defEqualSymbol \mto{\groupElement{h}}{\groupElement{g}\Gmult \groupElement{h}} \]

This action is faithful, meaning that it fully captures the behavior of the original group. We can state this formally in terms of the curried form of \(\function{A}\), which is then an injective group homorphism \(\functionSignature{\function{A}}{\group{G}}{\symmetricGroup{\group{G}}}\). Cayley's theorem states that \(\group{G}\) is isomorphic to its image under \(\function{A}\): we have "embedded" \(\group{G}\) into the group of permutations of its elements.

Action groupoids

With these terms defined, we can now define the action groupoid of a particular group action \(\function{A}\).

Triad

The relationship between groups, transitive quivers, and groupoids is summarized in the diagram below:

In the section "Paths", we defined the path groupoid of a quiver \(\quiver{Q}\). When the quiver is a transitive quiver, this path groupoid co-incides with action groupoid of a groupoid \(\group{G}\), specifically the self-action groupoid of \(\group{G}\). This particular group \(\group{G}\) is the group for which \(\quiver{Q}\) is the Cayley quiver, or equivalently, the translation group of \(\quiver{Q}\).

The benefit of keeping this triad in mind is that it will allow us to interpret various constructions on quivers in terms of groups. When the quivers are transitive quivers, we will end up describing familiar mathematical constructions on groups. But crucially, for non-transitive quivers, we will find ourselves outside the domain of group theory, but with intuitions to guide us from the transitive case.