Voltage Lifts of Graphs from a Category Theory Viewpoint

2.1. Adjunctions

There are several different but equivalent definitions of an adjoint pair of functors. For our purposes, the following is the most convenient one.

Definition 2.1

([14: (ii) of Theorem IV.1]). Let C,D be categories and let F:C→D and G:D→C be functors. We say that F is left adjoint to G, or that G is right adjoint to F, in symbols F ⊣ G, if there is a family

{ ∊Y:FG(Y)→Y}Y∈obj(D)

of D-morphisms, such that for every C-object X and a D-morphism f: F(X) → Y there is a unique C-morphism u: X → G(Y) such that

commutes.

The family { ϵY}Y∈obj(D) then forms a natural transformation of functors ϵ:FG→idD, called the counit of the adjunction F ⊣ G.

An important fact concerning the notion of an adjoint pair of functors is that each of the functors F, G determines the other one and the counit, up to isomorphism.

We will need another (perhaps more familiar) characterization of an adjoint pair of functors. For objects O₁, O₂ of a category ℰ, write ℰ(O1, O2) for the set of all morphisms from O to O₂ in ℰ. Let C,D be categories, let F:C→D and G:D→C be functors. Then F ⊣ G if and only if there is a bijection, natural in X and Y,

C(F(X),Y)≃D(X,G(Y)).

See [14: section IV.1].

2.2. Pullbacks

Let f: X → A, q: B → A be a pair of morphisms in a category C with a common codomain A, sometimes called a cospan in C

(1)

Then a pullback is the limit of this diagram. In other words, it is an object (denoted by X ×_A B) equipped with morphisms q*(f): X ×_A B → B and f*(q): X ×_A B → X such that the square (2.3) in the diagram

(2)

commutes and for every object V and a pair of morphisms v_X : V → X and v_B: V → B such the outer square of the diagram (2) commutes, there is a unique morphism u: V → X ×_A B such that both triangles (2.1) and (2.2) commute. We say that q*(f) is a pullback of f along q and that f*(q) is a pullback of q along f.

Let us describe pullbacks in the usual category of sets and mappings, denoted by Set.

2.3. Graphs

A graph is a quintuple G = (V, D, s, t, λ), where

• D is the set of darts of G;

• V is the set of vertices of G;

• s, t: D → V are the source and target maps, respectively;

• λ: D → D is a mapping such that λ ○ λ = id_D;

• s ○ λ = t.

The mapping λ is called the dart-reversing involution of G. Note that t ○ λ = s ○ λ ○λ = s ○id_D = s.

All the data in a graph (V, D, s, t, λ) can be expressed graphically by a commutative diagram:

(3)

We write V(G) for the set of vertices of G and D(G) for the set of darts of G. Usually we will identify G with the pair (V(G), D(G)) and discard s, t, λ from the signature. We say that s, t, λ are the structure maps of G.

Note that λ comes from an action of ℤ2 on D. The orbits of λ are the edges of G. We write E(G) for the set of all edges. There are three types of edges {d, λ(d)}:

• semiedges: λ(d) = d;

• loops: non-semiedges with s(d) = t(d);

• links: all the other edges, that means λ(d) ≠ d and s(d) ≠ t(d).

A morphism of graphs f: G → H is a pair of mappings (f^V, f^D), where f^V : V(G) → V(H) and f^D: D(G) → D(H) are such that for every dart d ∈ D(G), s(f^D(d)) = f^V(s(d)), t(f^D(d)) = f^V(t(d)) and λ(f^D(d)) = f^D(λ(d)). Clearly, graphs equipped with morphisms form a category, denoted by Graph.

2.4. Graphs are functors

As outlined above, every graph is a diagram in Set. This can be formulated as follows: a graph is a functor from a certain finite category gph to the category Set. This category gph has two objects {D, V}, and three non-identity morphisms {s, t, λ} that behave as in the diagram (3). The morphisms of graphs can then be represented as natural transformations of functors from gph to Set, so the category Graph can be identified with a category of functors [gph, Set].

2.5. Pullbacks of graphs

Since graphs are functors, it follows that limits/colimits in Graph can be computed pointwise: we can compute a limit/colimit separately for vertices and darts and then equip the resulting sets with structure maps to obtain a graph.

In particular, given a pair of morphisms f₁, f₂

in Graph, we can compute the pullback simply as

The projections f1*(f2),  f2*(f1) in

are computed in the obvious way:

2.6. Group labeled graphs

A group labeled graph is a triple (G, Γ, β), where G is a graph, Γ is a group and β: V(G) → Γ is a mapping, called a Γ-labeling on G.

A morphism of group labeled graphs (G, Γ, β) → (G′, Γ′, β′) is a pair (f, h), where f: G → G′ is a morphism of graphs and h: Γ → Γ′ is a morphism of groups such that, for all v ∈ V(G), h(β(v)) = β′(f^V(v)). The composition of morphisms is defined in a straightforward way: (f₁, h₁) ○ (f₂, h₂) = (f₁ ○ f₂, h₁ ○ h₂). Clearly, the class of all group labeled graphs equipped with their morphisms forms a category, which we denote by Lab.

Let X be a set. Let k∘(X) be the complete graph with semiedges on the vertex set X, that means, a graph with V(k∘(X))=X,  D(k∘(X))=X×X, and structure maps s(x₁, x₂) = x₁, t(x₁, x₂) = x₂ and λ(x₁, x₂) = (x₂, x₁). Clearly, k∘ is a functor from Set to Graph.

Let us write U: Grp → Set for the “forgetful” functor that maps a group to its underlying set and denote K∘=k∘∘U, so that K∘(Γ) is the complete graph with semiedges with vertices labelled by the elements of the group Γ.

Corollary 2.4.

For every pair Γ₁, Γ₂ of groups, K∘(Γ1×  Γ2)≃K∘(Γ1)  ×K∘(Γ2).

Proof.

Every right adjoint functor preserves limits. □

A Γ-labeling β on a graph G is the same thing as a morphism of graphs G→K∘(Γ). Moreover, a morphism (f, h): (G, Γ, β) → (G′, Γ′, β′) in Lab can be identified with a commutative square in Graph

Composition of morphisms in Lab corresponds to horizontal pasting of such commutative squares. This shows that the category Lab is isomorphic to the comma category Graph↓K∘, see [14: Section II.6].

2.7. Voltage graphs

A voltage graph is a triple (G, Γ, α), where G is a graph and α: D(V) → Γ is a mapping such that α(λ(d)) = (α(d))⁻¹, called a Γ-voltage on G.

A morphism of voltage graphs (G, Γ, α) → (G′, Γ′, α′) is a pair (f, h), where f: G → G′ is a morphism of graphs and h: Γ → Γ′ is a morphism of groups such that, for all d ∈ D(G), h(α(d)) = α′(f^D(d)). The composition is defined similarly as in Lab. The class of all voltage graphs equipped with morphisms of voltage graphs forms a category, which we denote by Volt.

Similarly as for Lab, it is possible to represent Volt as a certain category of morphisms in Graph. Indeed, consider the digraph ℓ(Γ) with a single vertex v and D(ℓ(Γ)) = Γ. Both s and t are just constant maps with the constant v and λ: D(ℓ(Γ)) → D(ℓ(Γ)) is given by λ(a) = a⁻¹; ℓ is then a functor from Grp to Graph. Note that the edge {a, λ(a)} of ℓ(Γ) is a semiedge for a = a⁻¹, otherwise it is a loop.

A voltage α on a graph G is the same thing as a morphism of graphs α: G → ℓ(Γ). Under this identification, a morphism in Volt (f, h): (G, Γ, α) → (G′, Γ′, α′) is the same thing as a commutative square in Graph

and composition of morphisms corresponds to horizontal pasting of such squares. This shows that the category Volt is isomorphic to the comma category Graph ↓ ℓ.

2.8. Derived voltage graphs

Let us remark that in the original definition in [8], the voltage graph lift of a voltage graph is just a graph (not a voltage graph).

For every voltage graph (G, Γ, α), there is a morphism ϵ_(G,Γ,α) of graphs from G^α to G given by the projection ϵ(G,Γ,α)V(v,  x)=v,  ϵ(G,Γ,α)D(d,  x)=d. Clearly, this is a morphism in Volt. We will prove in the next section that this morphism is a component of the counit of an adjunction between Volt and Lab.

Example 2.8.

Consider the ℤ3 graph at the bottom of Figure 1; for every edge we draw only one of its two darts. The voltage graph lift is pictured at the top of the figure.

Figure 1.

A ℤ3-voltage graph and its voltage graph lift

2.9. Fibrations and covers

An in-neighbourhood N(v) of a vertex v of a graph is the set of darts with target v. A morphism of graphs f: G′ → G is a fibration if for every vertex v ϵ V(G′), f^D restricted to N(v) is a bijection from N(v) to N(f(v)). A fibration is a covering if and only if it is surjective on vertices. The following proposition is well known.

4.1. Group actions instead of groups

For every group Γ, one can construct the category Act(Γ) of all actions of Γ on sets, equipped with equivariant maps. Moreover, for every morphism of groups h: Γ₁ → Γ₂ there is an obviously defined functor Act (h): Act (Γ₂) → Act (Γ₁) and this gives us a functor Act : Grp^op → Cat.

Let us mention that the categories Act (Γ) are sometimes called permutational categories [12] and figure prominently in the theory of cellular embeddings of graphs [6,13].

From the functor Act, we may construct the category ∫​​Act of all group actions on sets, via the Grothendieck construction.

In this context, the K∘ functor then naturally generalizes to the action groupoid [5] of a given group action ☉: X × Γ → X, considered as a graph. The morphism q_Γ can be generalized to a morphism q_{(X, Γ, ☉)} (an interested reader can fill in the details here).

The pullback of a voltage α: G → ℓ(Γ) along the q_{(X, Γ, ☉)} morphism is then a generalization of the permutation voltage graph lift construction [9], sometimes called a voltage space [15]. It seems that this more general construction arises from an adjunction, as well.

4.2. Bifibrational viewpoint

The obvious projection functors Volt → Grp and Lab → Grp arise as a pullback (in the category of all categories) of the codomain fibration

along the functors K∘ and ℓ. Therefore, both projection functors are Grothendieck fibrations. It is easy to prove that the functors are cofibrations as well. In this context, it would be interesting to examine the properties of the q transformation. One could then attempt to apply the well established theory of indexed categories/fibrations [11] and possibly even categorical logic to better understand the voltage graph lifts.