title: Voltage lifts of graphs from a category theory viewpoint
authors: Gejza Jenča
text: 
  
  
  We prove that the notion of a voltage graph lift comes from an adjunction between the
  category of voltage graphs and the category of group labeled graphs.
  
  
  
  gejza.jenca@stuba.sk
  
  Department of Mathematics and Descriptive Geometry
  
  Faculty of Civil Engineering,
  Slovak University of Technology,
  	Slovak Republic
  
  
  This research is supported by grants VEGA 2/0142/20 and 1/0006/19,
  Slovakia and by the Slovak Research and Development Agency under the contracts
  APVV-18-0052 and APVV-20-0069.
  
  
  
  Voltage lifts of graphs from a category theory viewpoint
      Gejza Jenča
      Received December 30, 2023; accepted December 30, 2023
  ==========================================================
  
  
  
  
  
  § INTRODUCTION
  
  
  In this paper, a graph means a structure sometimes called a 
  symmetric multidigraph – that means that it may have multiple darts
  with the same source and target, and the set of all darts of the graph is equipped 
  with an involutive mapping λ that maps every dart to a dart
  with source and target swapped.
  
  A voltage graph is a graph in which every dart is
  labeled with an element of a group in a way that respects the involutive symmetry λ,
  so that the label of a dart d is inverse to the label of λ(d).
  Similarly, a group labeled graph has all vertices labeled with elements of a group. 
  
  In <cit.> Gross introduced the construction of a derived
  graph of a voltage graph.  Nowadays, derived voltage graphs are called (ordinary) voltage graph lifts – this is the terminology we will use in the
  present paper. 
  Let us mention in passing that in
  <cit.>, voltage graphs were generalized to a more general
  notion of permutation voltage graphs, in which the darts are labelled
  with permutations.
  
  After their discovery, voltage graph lifts were extensively investigated in
  many papers. Voltage graph lifts were applied for example in the research
  concerning the degree-diameter problem
  <cit.>, lifting graph automorphisms
  <cit.> and several other areas of graph theory.
  
  In the present paper, we prove that there is an adjunction
  
      @/^1.2pc/[rr]^-L
            @/^1.2pc/[ll]^-R
  
  between the category   of voltage graphs and a
  category  of group labeled graphs. 
  We prove that for every object G of , the underlying graph of the voltage graph
  LR(G) is isomorphic to the voltage graph lift of G.
  
  
  
  § PRELIMINARIES
  
  
  We assume basic knowledge of category theory; for notions not explained here see
  <cit.>.
  
  
  
   §.§ Adjunctions
  
  
  There are several different but equivalent definitions of an adjoint
  pair of functors. For our purposes, the following is the most convenient one.
  <cit.>
  Let , be categories and let F→ and
  G→ 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 
  
      {ϵ_Y FG(Y)→ Y}_Y∈()
   of -morphisms,
  such that for every -object X and a -morphism f F(X)→ Y
  there is a unique -morphism u X→ G(Y) such that 
  
      F(X)
      	[rd]^f@.>[d]_F(u)
      
      FG(Y)
      	[r]_-ϵ_Y   
      Y
  
  commutes.
  
  
  The family {ϵ_Y}_Y∈() then forms a natural transformation of functors 
  ϵ FG→𝕀_, 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 O_1,O_2 of a category ℰ, write ℰ(O_1,O_2) for the
  set of all morphisms from O_1 to O_2 in ℰ.
  Let , be categories, let F→ and
  G→ be functors. Then F G if and only if there is
  a bijection, natural in X and Y,
  
      (F(X),Y)≃(X,G(Y)).
  
  See <cit.>.
  
  
  
   §.§ Pullbacks
  
  
  Let f X→ A, q B→ A be a pair of morphisms in a category  with a common codomain A,
  sometimes called a cospan in 
  
      X
      	[d]^f
      
      
      B
      	[r]_q
         
      A
  
  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 
  cell:pullback in the diagram
  
      V
      	@/^1.4pc/[rrd]^v_X@/_1.4pc/[rdd]_v_B@.>[rd]^urrdcell:toptrianglepullbackrddcell:bottomtrianglepullback
          
      X×_A B
      	[r]^-f^*(q)[d]_q^*(f)@[rd]|cell:pullback   
      X
      	[d]^f
      
          
      B
      	[r]_q
         
      A
  
  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
  (<ref>) commutes,
  there is a unique morphism u V→ X×_A B such that both triangles
  cell:toptrianglepullback and
  cell:bottomtrianglepullback 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 .
  
  Consider a diagram of shape (<ref>) in .
  A pullback X×_A B can be constructed
  as a subset of the direct product of sets
  X×B, given by
  
      X×_A B={(x,b):f(x)=q(b)}
  
  and the maps f^*(q) and q^*(f) are the projections: 
  
      f^*(q)(x,b)=x     q^*(f)(x,b)=b.
  
  Note that, whenever A is a singleton, f(x)=q(b) for all pairs
  (x,b)∈ X×B, so in this case X×_A B=X×B.
  
  
  
  
   §.§ 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=𝕀_D.
  
    * s=t.
  
  The mapping λ is called the dart-reversing involution of G.
  Note that t=s=s∘𝕀_D=s.
  
  All the data in a graph (V,D,s,t,λ) can be expressed graphically by a commutative diagram:
  
      D
      	@/^/[rr]^𝕀_D[rd]^λ@/_/[rdd]_s       
      D
      	@/^/[ldd]^s
          
      D
      	[d]^-t[ru]^λ
          
      V
  
  
  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 ℤ_2 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 . 
  
  
  
   §.§ Graphs are functors
  
  As outlined above, every graph is a diagram in . This can be formulated as follows:
  a graph is a functor from a certain finite category  to the category
  .  This category  has two objects {D,V}, and three non-identity morphisms
  {s,t,λ} that behave as in the diagram (<ref>). The morphisms of graphs
  can then be represented as natural transformations of functors from  to , so
  the category  can be identified with a category of functors [,].
  
  
  
   §.§ Pullbacks of graphs
  
  Since graphs are functors, it follows that limits/colimits in  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 f_1,f_2
  
      G_1
      	[d]^f_1
      
      G_2
      	[r]_f_2   
      H
  
  in , we can compute the pullback simply as
  
      V(G_1HG_2)   =V(G_1)V(H)V(G_2)
      
      D(G_1HG_2)   =D(G_1)D(H)D(G_2)
      
      s(d_1,d_2)   =(s(d_1),s(d_2))
      
      t(d_1,d_2)   =(t(d_1),t(d_2))
      
      λ(d_1,d_2)   =(λ(d_1),λ(d_2)).
  
  The projections f_1^*(f_2), f_2^*(f_1) in 
  
      G_1HG_2[r]^-f_2^*(f_1)[d]_f_1^*(f_2)   
      G_1
      	[d]^f_1
      
      G_2
      	[r]_f_2   
      H
  
  are computed in the obvious way:
  
      (f_2^*(f_1))^V(v_1,v_2)   =v_1   (f_2^*(f_1))^D(d_1,d_2)   =d_1
      
      (f_1^*(f_2))^V(v_1,v_2)   =v_2   (f_1^*(f_2))^D(d_1,d_2)   =d_2.
  
  
  
  
  
   §.§ 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:
  (f_1,h_1)∘(f_2,h_2)=(f_1∘ f_2,h_1∘ h_2).
  Clearly, the class of all group labeled graphs equipped with their morphisms
  forms a category, which we denote by .
  
  Let X be a set. Let (̨X) be the complete graph with semiedges on the vertex set X,
  that means, a graph with V((̨X))=X, D((̨X))=X×X, and structure maps
  s(x_1,x_2)=x_1, t(x_1,x_2)=x_2 and λ(x_1,x_2)=(x_2,x_1). 
  Clearly, $̨ is a functor fromto. 
  
  Let us writeU→for the “forgetful” functor that maps
  a group to its underlying set and denote=∘̨U, so thatis the complete
  graph with semiedges with vertices labelled by the elements of the groupΓ.
  
  The functor  is a right adjoint.
  
  Obviously, for every set X and a graph G,
  
      (V(G),X)≃(G,(̨X)),
  
  hence V$̨.
  It is well known thatUis a right adjoint functor withF U,
  where the left adjointF→maps every setXto the free group generated byX.
  Right adjoint functors are closed with respect to composition, hence=∘̨Uis a right adjoint.
  
  For every pair Γ_1,Γ_2 of groups,
  (Γ_1×Γ_2)≃(Γ_1)×(Γ_2).
  
  Every right adjoint functor preserves limits.
  
  
  AΓ-labelingβon
  a graphGis the same thing as a morphism of graphsG→. Moreover, a morphism(f,h) (G,Γ,β)→(G',Γ',β')incan be identified with a commutative square inG
  	[r]^f[d]_β   
  G'
  	[d]^β'
  [r]_(h)   (Γ')Composition of morphisms incorresponds to horizontal pasting of such commutative squares. This shows that the
  categoryis isomorphic to the  comma category↓, see
  <cit.>.
  
  
  
   §.§ Voltage graphs
  
  
  A  voltage graph is a triple(G,Γ,α),
  whereGis a graph andα D(V)→ Γis a mapping such thatα(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),
  wheref G→ G'is a morphism of graphs andh Γ→ Γ'is a morphism of groups such that,
  for alld∈ D(G),h(α(d))=α'(f^D(d)). The composition is defined similarly
  as in.
  The class of all voltage graphs equipped with morphisms of voltage graphs forms a category,
  which we denote by.
  
  Similarly as for, it is possible to representas a certain
  category of morphisms in. Indeed, consider the digraphwith a single
  vertexvandD()=Γ. Bothsandtare just constant maps with the constantvandλ D(ℓ(Γ))→ D(ℓ(Γ))is given byλ(a)=a^-1;ℓis
  then a functor fromto. Note that the edge{a,λ(a)}ofℓ(Γ)is a semiedge fora=a^-1, otherwise it is a loop.
  ℓ is a right adjoint functor.
  
  The proof is very similar to the proof of <Ref>, however one needs
  to replace the intermediate category  with the category (ℤ_2) of actions of ℤ_2 equipped
  with equivariant maps. 
  The functor F_ℓ→ takes a graph to the group
  with the set of generators D(G) and the set of relations given by d.λ(d)=1, for
  all d∈ D(G) and it is easy to check that F_ℓℓ.
  
  For every pair Γ_1,Γ_2 of groups,
  ℓ(Γ_1×Γ_2)≃ℓ(Γ_1)×ℓ(Γ_2).
  
  Every right adjoint functor preserves limits.
  
  
  A voltageαon
  a graphGis the same thing as a morphism of graphsα G→. Under this identification, a morphism
  in(f,h) (G,Γ,α)→ (G',Γ',α')is the same thing as a commutative square
  inG
  	[r]^f[d]_α   
  G'
  	[d]^α'
  [r]_ℓ(h)   ℓ(Γ')and composition of morphisms corresponds to horizontal pasting of such squares. This shows that
  the categoryis is isomorphic to the comma category↓ℓ.
  
  
  
   §.§ Derived voltage graphs
  <cit.> 
  Let (G,Γ,α) be a voltage graph. There is a  voltage graph lift of
  (G,Γ,α),
  denoted by (G^α,Γ,α')
    * V(G^α)=V(G)×Γ
    * D(G^α)=D(G)×Γ
    * s(d,x)=(s(d),x)
    * t(d,x)=(t(d),x.α(d))
    * λ(d,x)=(λ(d),x.α(d))
    * α'(d,x)=α(d)
  
  Let us remark that in the original definition in <cit.>, 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 morphismϵ_(G,Γ,α)of graphs
  fromG^αtoGgiven by the projectionϵ_(G,Γ,α)^V(v,x)=v,ϵ_(G,Γ,α)^D(d,x)=d. Clearly, this is a morphism in. We will prove in
  the next section that this morphism 
  is a component of the counit of an adjunction betweenand.
  
  
  Consider the ℤ_3-voltage graph at the bottom of
  Figure <ref>; for every edge we draw only one of its
  two darts. The voltage graph lift is pictured at the top of the figure.
  
  
   §.§ Fibrations and covers
  
  
  An  in-neighbourhoodN(v)of a vertexvof a graph is the set
  of darts with targetv.
  A morphism of graphsf G'→ Gis a  fibration
  if for every vertexv∈ V(G'),f^Erestricted toN(v)is a bijection fromN(v)toN(f(v)).
  A fibration is a  covering if and only if it is surjective on vertices.
  The following proposition is well-known.
  
  
  For every voltage graph (G,Γ,α), the canonical projection p G^α→ G
  given by p^V(v,x)=v and p^D(d,x)=d is a covering.
  
  
  A coveringf G'→ Gis  regular if there is a groupΓthat acts freely onG'and an isomorphismi G'/Γ→ Gsuch thati∘ω_Γ=f, whereω_Γ G'→ G'/Γis the quotient map of the action.
  
  For every voltage graph(G,Γ,α),Γacts freely onG^αand the
  canonical projectionp G^α→ Gis a regular covering associated with this
  action. Moreover, it can be proved that every regular coveringf G'→
  Gis isomorphic to the canonical projection for some voltage
  onG(see <cit.> or <cit.>,
  Theorems 2.2.1 and 2.2.2).
  
  The more general notion of permutation voltage graphs can be used to represent
  all coverings of graphs <cit.>.
  
  
  
  § THE ADJUNCTION BETWEEN  AND 
  
  
  
  Consider a group labeled graph(G,Γ,β). If we want to construct a voltage
  graph from(G,Γ,β), it is natural to equip darts ofGwith a voltage given
  by the “quotient” of labels along the edge. Formally, there is a voltage
  graphL(G,Γ,β)=(G,Γ,α), with the voltageαgiven by the ruleα(d)=β(s(d))^-1β(t(d)), see Figure <ref>. For an abelian
  groupΓ, this construction of a voltage graph from aΓ-labeled graph is
  well-known in the theory of flows on graphs <cit.>; in this
  context, the vertex labels are called  potentials. It is clear that everyL(G,Γ,β)satisfies the Kirchhoff laws. Moreover, it is easy to check that
  every voltage graph(G,Γ,α)that satisfies the Kirchhoff laws is in the range
  of the functorL.
  
  
  
  For every groupΓ, there is a morphism of graphsq_Γ→;q_Γ^Vis the
  only possible map andq_Γ^D(u,v)=u^-1v. 
  
  If we
  identify≃↓and≃↓ℓ, 
  thenL(G,Γ,β)is the voltage graph(G,Γ,q_Γ∘ β).G
  	[rd]^q_Γ∘ β[d]_β
  [r]_q_ΓSo in what follows, we sometimes writeL(β)forq_Γ∘ β.
  
  
  The family of morphisms {q_Γ}_Γ∈() is a natural transformation from  to ℓ.
  
  Let h Γ→ Γ' be a morphism of groups. We need to prove that the 
   naturality square at h(Γ)
  	[r]^(h)[d]_q_Γ   (Γ')
  	[d]^q_Γ'
  ℓ(Γ)
  	[r]_ℓ(h)   ℓ(Γ')
  
  commutes. For vertex components of the morphisms, this is trivial
  because ℓ(Γ') has only one vertex. For every d∈ D((Γ)), that means,
  d=(u,v)∈ Γ×Γ we can compute
  
      (ℓ(h))^D(q_Γ^D(u,v))   =(ℓ(h))^D(u^-1v)=h(u^-1v)=(h(u))^-1h(v)
      
      q_Γ'^D((h)^D(u,v))   =q_Γ'^D(h(u),h(v))=(h(u))^-1h(v)
  
  
  Let(f,h)(G,Γ,β)→(G',Γ',β')be a morphism in.
  Consider the diagram
  
  
      G
      	[r]^f
      	[d]_βrdcell:fhmorphinside   
      G'
      	[d]_β'
      (Γ)
      	[r]_K(h)[d]^q_Γrdcell:natinside   (Γ')
      	[d]^q_Γ'
      ℓ(Γ)
      	[r]_ℓ(h)   ℓ(Γ')
  
  
  Since(f,h)is a morphism in, the square cell:fhmorphinside commutes.
  By <Ref>, the square cell:natinside commutes.  The left and right vertical
  compositesq_Γ∘ βandq_Γ'∘ β'areL(β)andL(β'), respectively.  So the
  whole diagram (<ref>) commutes and we see that(f,h)is a morphism fromL(β)toL(β')in.  Thus, we may putL(f,h)=(f,h), and it is then 
  clear thatLis a functor.
  
  L is a left adjoint functor.
  
  
  Let us describe a right adjoint functor R→ 
  associated to the functor L. For every voltage graph (G,Γ,α), we put
  R(G,Γ,α)=(Gℓ(Γ)(Γ),Γ,q_Γ^*(α)):
  
      Gℓ(Γ)(Γ)[r]^-α^*(q_Γ)[d]_q_Γ^*(α)   
      G
      	[d]^α
      [r]_-q_Γ
  
  
  
  To specify R on morphisms we use the fact that pullback is a
  limit. In detail, let 
  
  (f,h)(G,Γ,α)→ (G',Γ',α')
   
  be a morphism of
  voltage graphs. Consider the diagram in 
      G[rd]_α^*(q_Γ)[ddd]_q_Γ^*(α)@.>[rrr]^urdddcell:Gpullbackrrrdcell:topsquare           G'ℓ(Γ')(Γ')[ddd]^q_Γ'^*(α')[ld]^α'^*(q_Γ')
          
      G
      	[r]^f[d]_αrdcell:hmorph   
      G'
      	[d]^α'rdcell:Gprimepullback    
          [r]_ℓ(h)rdcell:naturalityofq   ℓ(Γ')
          
      [ur]^q_Γ[rrr]_(h)           (Γ')
      	[ul]_q_Γ'
  
  The cell cell:Gpullback is a pullback square, the cell
  cell:naturalityofq is a naturality square for q at h, and the middle cell
  cell:hmorph is just the (f,h) morphism in . Therefore, the
  boundary of the diagram consisting of
  cell:Gpullback, cell:naturalityofq and cell:hmorph
  commutes, meaning that 
  
  α'∘ f∘ (α^*(q_Γ))=q_Γ'∘(h)∘ (q_Γ^*(α)).
  
  Since cell:Gprimepullback is a pullback square over the span (α',q_Γ'), there is a unique morphism 
  u such that both cell:topsquare and the outer square of
  (<ref>) commute. Since the outer square of (<ref>) commutes, (u,h) is a
  morphism from R(α) to R(α') in , and we may put 
  R(f,h)=(u,h). We omit the proof of functoriality of R since it it just a
  straightforward exercise in the “universality of the pullback”.
  
  However, it is also possible to observe that R is a functor 
  by describing u explicitly:
  
  u^V(v,x)=(f^V(v),h(x))   u^D(d,(x_1,x_2))=(f^D(d),(h(x_1),h(x_2))).
  
  
  To specify the counit, we first note that for a voltage graph
  (G,Γ,α), LR(G,Γ,α) is the voltage graph
  (G,Γ,q_Γ∘ q_Γ^*(α)).
  For every object (G,Γ,α)
  of , we define ϵ_(G,Γ,α) LR(G,Γ,α)→ (G,Γ,α) 
  to be the morphism (α^*(q_Γ),𝕀_Γ) in :
  
      G[r]^-α^*(q_Γ)[d]_q_Γ^*(α)   
      G
      	[dd]^α
      [d]_q_Γ    
      [r]_ℓ(𝕀_Γ)
  
  
  Note that the commutativity of (<ref>) 
  follows from the commutativity of (<ref>).
  
  To prove that the family of all these ϵ_(G,Γ,α) is a counit of the adjunction
  L R,
  we need to prove that for every group labelled graph (G',Γ',β) and
  every morphism of voltage graphs 
  
  (f,h) L(G',Γ',β)→ (G,Γ,α')
   
  there is
  a unique morphism of group labelled graphs 
  (u,w) (G',Γ',β)→ R(G,Γ,α) such that
  the diagram in 
      L(G',Γ',β)
      	[rd]^(f,h)[d]_L(u,w)
      
      LR(G,Γ,α)
      	[r]_-ϵ_(G,Γ,α)   
      (G,Γ,α)
  
  commutes. If such (u,w) exists, then ϵ_(G,Γ,α)∘ L(u,w)=
  (f,h) in  implies that 𝕀_Γ∘ w=h in , so w=h, and the uniqueness of w is thus
  clear. What remains to prove is the existence and uniqueness of u, under the assumption w=h.
  
  The assumption that (f,h) in (<ref>) is a morphism of voltage graphs
  means that the diagram 
  
      G'
      	[r]^f[d]_β   
      G
      	[dd]^α
      (Γ')
      	[d]^q_Γ'
      ℓ(Γ')
      	[r]_ℓ(h)   ℓ(Γ)
  
  in  commutes.
  Consider the diagram
  
      4em
      G'
      	@/^1.5pc/[rrd]^f@/_1.5pc/[rddd]_β@.>[rd]^-urdddcell:betatopullbackrrdcell:toptriangle
          G[r]^-α^*(q_Γ)[d]_q_Γ^*(α)rdcell:epsilonalpha   
      G
      	[rd]^α
          [r]_q_Γrdcell:nat   [r]_ł(𝕀_G)   
          (Γ')
      	[r]_q_Γ'[u]^(h)   ℓ(Γ')
      	[u]_ℓ(h)
  
  The outer border of (<ref>) is the commutative square (<ref>)–
  the (f,h) morphism we want to express as in (<ref>). 
  In particular, we already know that the outer border
  of (<ref>) commutes.
  The square cell:nat commutes because it
  is the naturality square for q at h. From this, we obtain
  
  q_Γ∘(h)∘ β=α∘ f
  
  and by the universality of the pullback G
  there is a unique u G'→G such that
  
      q_Γ^*(α)∘ u   =(h)∘ β    cell:betatopullback
      
      α^*(q_Γ)∘ u   =f    cell:toptriangle
  
  Note that the commutative square cell:betatopullback is just a -morphism
  (u,h)(G',Γ',β)→ R(G,Γ,α). Applying the functor L on 
  (u,h), that means, pasting of the squares cell:betatopullback and cell:nat
  gives us a morphism L(u,h) L(G',Γ',β)→ LR(G,Γ,α).
  The cell cell:epsilonalpha is just ϵ_(G,Γ,α). Composing in ϵ_(G,Γ,α)∘ L(u,h) gives us the -morphism 
  
  (α^*(q_Γ)∘ u,h) L(G',Γ',β)→ α
  
  and we already know that α^*(q_Γ)∘ u=f, so
  ϵ_(G,Γ,α)∘ L(u,h)=(f,h). It remains to note that we have already proved the uniqueness of u.
  
  
  Let us examine the structure ofR(G,Γ,α)for the case of a single-vertex
  graphG. By our main result <Ref> and <cit.>, we see thatR(G,Γ,α)is a Cayley graph.
  However, it is perhaps interesting to describe the behaviour of our construction
  in this case.
  
  
  Let Γ be a group and let S be a subset of Γ that is closed under 
  taking inverses. The  Cayley graph of Γ induced by S<cit.>) is the graph
  𝒞(Γ,S) with vertices V(𝒞(Γ,S))=Γ and darts
  
  D(𝒞(Γ,S))={(x,y)∈ Γ×Γ:x^-1y∈ S}.
  
  The structural maps of 𝒞(Γ,S) are
  
      s(x,y)   =x
      
      t(x,y)   =y
      
      λ(x,y)   =(y,x)
  
  Naturally, every𝒞(Γ,S)is aΓ-labeled graph, with 
  the labeling given byβ(x)=x.
  
  
  Let Γ be a group and let S be a subset of Γ that is closed with respect
  to taking inverses.
  Let (G,Γ,α) be a voltage graph with a single vertex v, such that D(G)=S and
  α(x)=x. As a group-labeled graph, R(G,Γ,α) is isomorphic to the Cayley graph 
  of Γ induced by S.
  
  Let us compute R(G,Γ,α) as a pullback (<ref>).
  Clearly, since V(ℓ(Γ)) is a singleton,
  
      V(G)=V(G)×V()={v}×Γ
  
  For darts, we can compute a pullback in 
      D(G)=D(G)D())D()=SΓ(Γ×Γ).
  
  The pullback square in  is
  SΓ(Γ×Γ)[r]^-j^*(m)[d]_-m^*(j)   
  S
  	[d]^j
  
  
  Γ×Γ
  	[r]_q_Γ^D   
  Γ
  
  where j is the inclusion of S into Γ. We have 
  SΓ(Γ×Γ)={(u,(x,y)):u∈ S and x^-1y=u}
  The structure maps of the pullback graph are 
  
      s(u,(x,y))   =(s(u),s(x,y))=(v,x)
      
      t(u,(x,y))   =(t(u),t(x,y))=(v,y)
      
      λ(u,(x,y))   =(λ(u),λ(x,y))=(u^-1,(y,x))
  
  and its labeling is given by β(v,x)=x.
  Moreover, note that there are obvious isomorphisms of sets
  {v}×Γ≃ Γ and
  {(u,(x,y)):u∈ S and x^-1y=u}≃{(x,y):x^-1y∈ S}
  and it is easy to see that this pair of isomorphisms of sets give us an isomorphism
  R(G,Γ,α)≃𝒞(Γ,S) in .
  
  For every voltage graph (G,Γ,α),
  the graph LR(G,Γ,α) is isomorphic to the voltage graph lift of (G,Γ,α). 
  
  We adopt the notations of <Ref> here.
  The underlying graph of LR(G,Γ,α) is G. Let us examine
  its structure and prove that there is an isomorphism j LR(G)→ G^α. 
  
  As V() is a singleton, 
  
  V(G)=V(G)×V()=V(G)×Γ,
   
  so V(LR(G,Γ,α))=V(G^α) and we may put j^V(v,x)=(v,x).
  
  For darts, we see that
  
  D(G)={(d,(x_1,x_2))∈ D(G)×D():q_Γ^D(x_1,x_2)=α(d)}.
  
  The structure maps of G are
  
      s(d,(x_1,x_2))=   (s(d),s(x_1,x_2))=(s(d),x_1)
      
      t(d,(x_1,x_2))=   (t(d),t(x_1,x_2))=(t(d),x_2)
  
  Let us note that q_Γ^D(x_1,x_2)=x_1^-1x_2, so
  q_Γ^D(x_1,x_2)=α(d) is equivalent to
  x_2=x_1α(d).
  Therefore, the mapping
  j^D D(G)→ D(G^α) given by
  j^D(d,(x_1,x_2))=(d,x_1) is a bijection. Moreover j=(j^V,j^D) is a morphism
  of graphs, because
  
      j^V(s(d,(x_1,x_2)))=j^V(s(d),x_1)=s(d,x_1)=s(j^D(d,(x_1,x_2))).
  
  and
  
      j^V(t(d,(x_1,x_2)))=j^V(t(d),x_2)=(t(d),x_2)=
      (t(d),x_1α(d))=t(d,x_1)=
      t(j^D(d,(x_1,x_2)))
  
  
  It remains to prove that the graph isomorphism j preserves voltages and is thus a
  morphism in .
  The voltage of a dart (d,(x_1,x_2)) of LR(G,Γ,α) is equal to
  
      q_Γ(q_Γ^*(α)(d,(x_1,x_2)))=q_Γ(x_1,x_2)=α(d)=
      α'(d,x_1)=α'(j^D(d,(x_1,x_2)))
  
  
  Let us collect some consequences of <Ref>.
  
  Using <Ref>, we may construct a product of a pair of voltage
  graphs(G_1,Γ_1,α_1),(G_2,Γ_2,α_2)inas the voltage graph(G_1×G_2,Γ_1×
  Γ_2,α_1×α_2), where(α_1×α_2)(d_1,d_2)=(α_1(d_1),α_2(d_2)). Due to
  <Ref>, the products incan be described similarly.
  
  Let (G_1,Γ_1,α_1), (G_2,Γ_2,α_2) be voltage graphs.
  Then 
  
      G_1^α_1×G_2^α_2≃(G_1×G_2)^α_1×α_2.
  
  Let us compute
  
      R((G_1,Γ_1,α_1)×(G_2,Γ_2,α_2))≃ 
      R(G_1×G_2,α_1×α_2,Γ_1×Γ_2)≃
      
      ((G_1×G_2)^α_1×α_2,Γ_1×Γ_2,q_Γ_1×Γ_2^*(α_1×α_2))
  
  Since R is a right adjoint functor, it preserves limits, therefore
  
      R((G_1,Γ_1,α_1)×(G_2,Γ_2,α_2))≃ 
      R(G_1,Γ_1,α_1)×R(G_2,Γ_2,α_2)≃
      
      (G_1^α_1,Γ_1,q_Γ_1^*(α_1))×(G_2^α_2,Γ_2,q_Γ_2^*(α_2))≃
      (G_1^α_1×G_2^α_2,Γ_1×Γ_2,q_Γ_1^*(α_1)×q_Γ_2^*(α_2))
  
  
  
  A nice characterization of a fibration using pullback can be found in
  <cit.> (the authors attribute this observation to
  Frank Piessens) a morphismf G→ G'is a fibration if and only if the
  square
  
      D(G)
      	[r]^t
      	[d]_f^D   
      V(G)
      	[d]^f^V
      
      D(G')
      	[r]_t
         
      V(G')
  
  is a pullback in. The proof of the following theorem is then an
  easy consequence of the so-called  two-pullbacks lemma.
  <cit.>
  A pullback of a fibration in  along an arbitrary morphism is a fibration.
  
  
  From this, we obtain a new proof of the fact that the canonical projectionp_α G^α→ Gis a covering.
  
  
  By <Ref>, G^α is isomorphic to the pullback G.
  The morphism q_Γ(Γ)→ℓ(Γ) is a fibration
  and it is clear that p is isomorphic to α^*(q_Γ)G→ G. By <Ref>,
  p is then a fibration. Clearly, p^V is surjective, so p is a covering.
  
  
  § CONCLUSION
  
  
  Let us outline a possible direction for future research, concerning the voltage graphs
  and the voltage graph lift construction.
  
  
  
   §.§ Group actions instead of groups
  
  
  For every groupΓ, one can construct the category(Γ)of all
  actions ofΓon sets, equipped with equivariant maps. Moreover, for every
  morphism of groupsh Γ_1→ Γ_2there is an obviously 
  defined functor(h)(Γ_2)→(Γ_1)and this gives us
  a functor^op→.
  
  Let us mention that the categories(Γ)are sometimes called 
  permutational categories<cit.> and figure prominently
  in the theory of cellular embeddings of graphs
  <cit.>. 
  
  From the functor, we may construct the category∫of
  all group actions on sets, via the Grothendieck
  construction.
  
  In this context, thefunctor then naturally generalizes to the  action
  groupoid<cit.> of a given group action⊙ X×
  Γ→ X, considered as a graph.  The morphismq_Γcan be generalized to a
  morphismq_(X,Γ,⊙)(an interested reader can fill in the details here). 
  
  The pullback of a voltageα G→ℓ(Γ)along
  theq_(X,Γ,⊙)morphism is then a generalization of the permutation voltage graph lift construction
  <cit.>, sometimes called a  voltage space<cit.>.
  It seems that this more general construction arises from an adjunction, as
  well.
  
  
  
  
  
   §.§ Bifibrational viewpoint
  
  
  The obvious projection functors→and→arise as a
  pullback (in the category of all categories) of the codomain fibration
  
      cod^→→
   along the functorsandℓ. 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 theqtransformation. One could then attempt to apply the well
  established theory of indexed categories/fibrations
  <cit.> and possibly even categorical logic to better
  understand the voltage graph lifts.
  
  
  The author is indebted to both anonymous referees for their valuable comments 
  that helped to improve the paper.
  
  10boldi2002fibrations
  Paolo Boldi and Sebastiano Vigna.
   Fibrations of graphs.
   Discrete Mathematics, 243(1-3):21–66, 2002.
  
  bollobas1998modern
  Béla Bollobás and Bela Bollobas.
   Modern graph theory, volume 184.
   Springer Science & Business Media, 1998.
  
  brankovic1998large
  Ljiljana Brankovic, Mirka Miller, Ján Plesník, Joe Ryan, and Jozef Širáň.
   Large graphs with small degree and diameter: A voltage assignment
    approach.
   Australasian Journal of Combinatorics, 18:65–76, 1998.
  
  brankovic1998note
  Ljiljana Branković, Mirka Miller, Ján Plesník, Joe Ryan, and Jozef
    Širáň.
   A note on constructing large Cayley graphs of given degree and
    diameter by voltage assignments.
   The Electronic Journal of Combinatorics, page R9, 1998.
  
  brown2006topology
  Ronald Brown.
   Topology and groupoids, 2006.
  
  bryant1985foundations
  Robin P Bryant and David Singerman.
   Foundations of the theory of maps on surfaces with boundary.
   The Quarterly Journal of Mathematics, 36(1):17–41, 1985.
  
  cayley1878desiderata
  Arthur Cayley.
   Desiderata and suggestions: No. 2. The Theory of Groups:
    Graphical Representation.
   American Journal of Mathematics, 1(2):174–176, 1878.
  
  gross1974voltage
  Jonathan L Gross.
   Voltage graphs.
   Discrete mathematics, 9(3):239–246, 1974.
  
  gross1977generating
  Jonathan L Gross and Thomas W Tucker.
   Generating all graph coverings by permutation voltage assignments.
   Discrete Mathematics, 18(3):273–283, 1977.
  
  gross2001topological
  Jonathan L Gross and Thomas W Tucker.
   Topological graph theory.
   Courier Corporation, 2001.
  
  jacobs1999categorical
  Bart Jacobs.
   Categorical logic and type theory.
   Elsevier, 1999.
  
  jones2015combinatorial
  Gareth A Jones.
   Combinatorial categories and permutation groups.
   Ars Mathematica Contemporanea, 11, 2015.
  
  jones1978theory
  Gareth A Jones and David Singerman.
   Theory of maps on orientable surfaces.
   Proceedings of the London Mathematical Society, 3(2):273–307,
    1978.
  
  mac1998categories
  Saunders Mac Lane.
   Categories for the Working Mathematician.
   Number 5 in Graduate Texts in Mathematics. Springer-Verlag, 1971.
  
  malnivc2000lifting
  Aleksander Malnič, Roman Nedela, and Martin Škoviera.
   Lifting graph automorphisms by voltage assignments.
   European Journal of Combinatorics, 21(7):927–947, 2000.
  
  riehl2016category
  Emily Riehl.
   Category theory in context.
   Courier Dover Publications, 2016.