# Coverings of k-graphs - COnnecting REpositories In Section 6, we construct universal coverings using

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Journal of Algebra 289 (2005) 161–191

www.elsevier.com/locate/jalgebr

Coverings ofk-graphs✩

David Paska, John Quiggb, Iain Raeburna,∗

a School of Mathematical and Physical Sciences, University of Newcastle, NSW 2308, Australia b Department of Mathematics and Statistics, Arizona State University, Tempe, AZ 85287, USA

Received 30 July 2004

Available online 9 April 2005

Communicated by Michel Broué

Abstract

k-graphs are higher-rank analogues of directed graphs which were first developed to p combinatorial models for operator algebras of Cuntz–Krieger type. Here we develop the the covering spaces fork-graphs, obtaining a satisfactory version of the usual topological class tion in terms of subgroups of a fundamental group. We then use this classification to descr C∗-algebras of coveringk-graphs as crossed products by coactions of homogeneous spaces, alizing recent results on theC∗-algebras of graphs. 2005 Elsevier Inc. All rights reserved.

Keywords:k-Graph; Small category; Covering; Fundamental group;C∗-algebra; Coaction

1. Introduction

k-graphsare combinatorial structures which arek-dimensional analogues of (directe graphs. They were introduced by Kumjian and the first author [18] to help understand of Robertson and Steger on higher-rank analogues of the Cuntz–Krieger algebras [ The theory ofk-graphs and theirC∗-algebras parallels in many respects that of graphs

✩ This research was supported by grants from the Australian Research Council and the University of Ne * Corresponding author.

E-mail addresses:davidp@maths.newcastle.edu.au (D. Pask), quigg@math.asu.edu (J. Quigg),

iain@maths.newcastle.edu.au, iain.raeburn@newcastle.edu.au (I. Raeburn).

0021-8693/$ – see front matter 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.jalgebra.2005.01.051

162 D. Pask et al. / Journal of Algebra 289 (2005) 161–191

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their Cuntz–Krieger algebras [18,23,24]. Here we investigate to what extent there analogue fork-graphs of the theory of coverings of graphs, and the implications of theory for theC∗-algebras ofk-graphs.

A coveringof a graphF is by definition a surjective graph morphismp :E → F which is a local isomorphism. As for coverings of topological spaces, the coverings ofF are clas- sified by the conjugacy classes of subgroups of the fundamental groupπ1(F ), and every connected covering arises as a quotient of a universal covering (see [4,12], for exa This last theorem has interesting ramifications for the Cuntz–Krieger algebrasC∗(E) of covering graphs: ifp :E → F is a covering, then there is a coactionδ of the fundamen- tal groupπ1(F ) on C∗(F ) and a subgroupH of π1(F ) such thatC∗(E) is isomorphic to the crossed productC∗(F ) ×δ (π1(F )/H) [4, Theorem 3.2]. This theorem has in tu been of considerable interest in nonabelian duality forC∗-algebras: it provided a family o crossed products by homogeneous spaces which we could analyse using our und ing of graph algebras, and this analysis inspired substantial improvements in Mans Imprimitivity Theorem [14].

We seek, therefore, an analogue of this theory of covering graphs fork-graphs, and a generalisation of [4, Theorem 3.2] which describes theC∗-algebras of the coveringk- graphs. Any theory of coverings must involve the fundamental group, and a majority authors prefer to use the whole fundamental groupoid. We showed in [19] that the mental groupoids ofk-graphs do not behave as well as one might hope, and in parti that the path category need not embed faithfully in the fundamental groupoid whenk > 1. So it is something of a relief that our final results on coverings mirror in every respe classical topological theory.

Our approach is to exploit an equivalence between the coverings of ak-graph and ac tions of its fundamental groupoid, under which the connected coverings correspo transitive actions. Thus we deduce many of our main theorems from a classification transitive actions of an arbitrary groupoid.

Because every small category is isomorphic to a quotient of a path category, it w clear from the proofs that all our results carry over to arbitrary small categories; how we eschew such a generalization since we have no useful applications.

After we completed this paper, we learned of the existence of [1,3,13,16], which tain results similar to some of ours. In [1, Appendix], Bridson and Haefliger develo elementary theory of the fundamental group and coverings of a small category and results similar to some of ours. Bridson and Haefliger concentrate on the fundam group—indeed, they stop just short of defining the fundamental groupoid. In [3,13], B and Higgins investigate coverings of groupoids, and prove the equivalence with gro actions. Our work was done completely independently of these other sources, and lieve our methods are of interest, especially our use of skew products. In [16], Ku develops, in the specific context ofk-graphs, the fundamental groupoid and the existe of the universal covering, and proves that, under reasonable hypotheses, theC∗-algebra of the universal coveringk-graph is Rieffel–Morita equivalent to a commutative algebra. thank Kumjian for bringing [1] to our attention.

We begin in Section 2 by introducing our notion of covering, and stating our main sification theorems. Analogues of these theorems for coverings of groupoids were

in [3, Chapter 9]. In Section 3, we briefly discuss actions of groupoids on sets, and prove

D. Pask et al. / Journal of Algebra 289 (2005) 161–191 163

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the equivalence between the category of coverings of ak-graphΛ and the category of ac tions of its fundamental groupoidG(Λ) (Theorem 3.5). The main theorem of this sect is a technical result (Theorem 3.7) which implies that the connected coverings ofΛ corre- spond to transitive actions ofG(Λ), and that the fundamental groups of coverings ofΛ can be identified with the stability groups of the corresponding actions ofG(Λ). In Section 4, we state and prove analogues for groupoid actions of most of our main theorems, a in 5 we prove the main theorems themselves. Many of them follow from the general r in the previous section, but when it seemed easier to prove a result about coverings d we did so.

In Section 6, we construct universal coverings using skew products. We also sho every connected covering is arelative skew product(Corollary 6.10), and prove a versio of the Gross–Tucker Theorem which identifies thek-graphs which admit free actions of group as skew products.

It seems to us thatk-graphs are likely to be of interest in their own right, so we h been careful to limit our discussion ofC∗-algebras to a final section on the applicatio of our theory. Our sought-after generalisation of [4, Theorem 3.2] is Corollary 7.2 main idea in the proof of Corollary 7.2 comes from [15]: every group-valued cocycleη on a k-graphΛ induces a normal and maximal coactionδη of the group onC∗(Λ), and every k-graph carries a suitable cocycle with values in the fundamental group. We also p decomposition theorem which generalises [4, Corollary 3.6], prove that theC∗-algebra of everyk-graph is nuclear, and prove that theC∗-algebra of the skew product by the degr map is always AF.

2. Main results

For k-graphs and groupoids we adopt the conventions of [18,19,23], except that not require them to be countable. Briefly, ak-graph is a small categoryΛ equipped with a functord :Λ → Nk satisfying thefactorization property: for all α ∈ Λ andn, l ∈ Nk such thatd(α) = n + l there exist uniqueβ,γ ∈ Λ such thatd(β) = n, d(γ ) = l, andα = βγ . Whend(α) = n we sayα hasdegreen. A groupoid is a small category in which ever morphism has an inverse. All groupoids and groups in this paper are discrete, in the that they carry no topology.

If C is either ak-graph or a groupoid, theverticesare the objects, andC0 denotes the se of vertices. Forα ∈ C, thesources(α) is the domain, and theranger(α) is the codomain For u,v ∈ C0 we writeuC = r−1(u), Cv = s−1(v), anduCv = uC ∩ Cv. C is connectedif the equivalence relation onC0 generated by{(u, v) | uCv �= ∅} is C0 × C0; for a groupoid this just meansuCv �= ∅ for all u,v ∈ C0. If Λ is ak-graph,u,v ∈ Λ0, andn ∈ Nk , we write Λn = d−1(n), uΛn = uΛ ∩ Λn, andΛnv = Λv ∩ Λn. A morphismbetweenk-graphs is a degree-preserving functor.

In general we often write composition of maps as juxtaposition, especially when w chasing around commutative diagrams.

Definition 2.1. A coveringof a k-graphΛ is a surjectivek-graph morphismp :Ω → Λ such that for allv ∈ Ω0, p mapsΩv 1–1 ontoΛp(v) andvΩ 1–1 ontop(v)Λ. If (Ω,p) and(Σ,q) are coverings ofΛ, amorphismfrom (Ω,p) to (Σ,q) is ak-graph morphism

φ :Ω → Σ making the diagram

164 D. Pask et al. / Journal of Algebra 289 (2005) 161–191

o, s de-

uncto-

ry,

Ω φ

p

Σ

q

Λ

commute; we writeφ : (Ω,p) → (Σ,q). A coveringp :Ω → Λ is connectedif Ω (hence k-graphΛ) is connected.

Remark. If Λ is connected then surjectiv

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