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Cohomology for small categories: \(k\)-graphs and groupoids. (English) Zbl 1451.18010
Summary: Given a higher-rank graph \(\Lambda\), we investigate the relationship between the cohomology of \(\Lambda\) and the cohomology of the associated groupoid \(\mathcal{G}_{\Lambda}\). We define an exact functor between the abelian category of right modules over a higher-rank graph \(\Lambda\) and the category of \(\mathcal{G}_{\Lambda}\)-sheaves, where \(\mathcal{G}_{\Lambda}\) is the path groupoid of \(\Lambda\). We use this functor to construct compatible homomorphisms from both the cohomology of \(\Lambda\) with coefficients in a right \(\Lambda\)-module, and the continuous cocycle cohomology of \(\mathcal{G}_{\Lambda}\) with values in the corresponding \(\mathcal{G}_{\Lambda}\)-sheaf, into the sheaf cohomology of \(\mathcal{G}_{\Lambda}\).

MSC:
18B40 Groupoids, semigroupoids, semigroups, groups (viewed as categories)
55N30 Sheaf cohomology in algebraic topology
22E41 Continuous cohomology of Lie groups
46L05 General theory of \(C^*\)-algebras
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References:
[1] A. an Huef, M. Laca, I. Raeburn, and A. Sims, KMS states on \(C^{*}\)-algebras associated to higher-rank graphs, J. Funct. Anal. 266 (2014), no. 1, 265–283. · Zbl 1305.82028
[2] A. an Huef, M. Laca, I. Raeburn, and A. Sims, KMS states on the \({C}^{*}\)-algebra of a higher-rank graph and periodicity in the path space, J. Funct. Anal. 268 (2015), no. 7, 1840–1875. · Zbl 1329.46057
[3] H.-J. Baues and G. Wirsching, Cohomology of small categories, J. Pure Appl. Algebra 38 (1985), no. 2–3, 187–211. · Zbl 0587.18006
[4] J. H. Brown, L. O. Clark, C. Farthing, and A. Sims, Simplicity of algebras associated to étale groupoids, Semigroup Forum 88 (2014), no. 2, 433–452. · Zbl 1304.46046
[5] J. H. Brown, G. Nagy, S. Reznikoff, A. Sims, and D. P. Williams, Cartan subalgebras in \(C^{*}\)-algebras of Hausdorff étale groupoids, Integral Equations Operator Theory 85 (2016), no. 1, 109–126. · Zbl 1360.46046
[6] L. O. Clark, A. an Huef, and A. Sims, AF-embeddability of 2-graph algebras and quasidiagonality of \(k\)-graph algebras, J. Funct. Anal. 271 (2016), no. 4, 958–991. · Zbl 1358.46051
[7] K. R. Davidson and D. Yang, Periodicity in rank 2 graph algebras, Canad. J. Math. 61 (2009), no. 6, 1239–1261. · Zbl 1177.47087
[8] C. Farthing, Removing sources from higher-rank graphs, J. Operator Theory 60 (2008), no. 1, 165–198. · Zbl 1164.46026
[9] A. Grothendieck, Sur quelques points d’algèbre homologique, Tôhoku Math. J. (2) 9 (1957), 119–221. · Zbl 0118.26104
[10] A. Haefliger, “Differential cohomology” in Differential Topology (Varenna, 1976), Liguori, Naples, 1979, 19–70.
[11] F. Hirzebruch, Topological methods in algebraic geometry, Springer, Berlin, 1995. · Zbl 0843.14009
[12] S. Kang and D. Pask, Aperiodicity and primitive ideals of row-finite \(k\)-graphs, Internat. J. Math. 25 (2014), no. 3, art. ID 1450022. · Zbl 1297.46039
[13] A. Kumjian, On equivariant sheaf cohomology and elementary \({C}^{*}\)-bundles, J. Operator Theory 20 (1988), no. 2, 207–240. · Zbl 0692.46066
[14] A. Kumjian and D. Pask, Higher rank graph \({C}^{*}\)-algebras, New York J. Math. 6 (2000), 1–20.
[15] A. Kumjian, D. Pask, and A. Sims, Homology for higher-rank graphs and twisted \({C}^{*}\)-algebras, J. Funct. Anal. 263 (2012), no. 6, 1539–1574. · Zbl 1253.55006
[16] A. Kumjian, D. Pask, and A. Sims, On twisted higher-rank graph \({C}^{*}\)-algebras, Trans. Amer. Math. Soc. 367 (2015), no. 7, 5177–5216. · Zbl 1329.46050
[17] S. Mac Lane, Homology, Springer, Berlin, 1995.
[18] D. Pask, I. Raeburn, M. Rørdam, and A. Sims, Rank-two graphs whose \(C^{*}\)-algebras are direct limits of circle algebras, J. Funct. Anal. 239 (2006), no. 1, 137–178. · Zbl 1112.46042
[19] I. Raeburn, A. Sims, and T. Yeend, The \({C}^{*}\)-algebras of finitely aligned higher-rank graphs, J. Funct. Anal. 213 (2004), no. 1, 206–240. · Zbl 1063.46041
[20] J. Renault, A groupoid approach to \({C}^{*}\)-algebras, Lecture Notes in Math. 793, Springer, Berlin, 1980.
[21] D. I. Robertson and A. Sims, Simplicity of \({C}^{*}\)-algebras associated to higher-rank graphs, Bull. Lond. Math. Soc. 39 (2007), no. 2, 337–344. · Zbl 1125.46045
[22] G. Robertson and T. Steger, Affine buildings, tiling systems and higher rank Cuntz-Krieger algebras, J. Reine Angew. Math. 513 (1999), 115–144. · Zbl 1064.46504
[23] A. Skalski and J. Zacharias, Entropy of shifts on higher-rank graph \(C^{*}\)-algebras, Houston J. Math. 34 (2008), no. 1, 269–282. · Zbl 1160.46046
[24] J. Spielberg, Graph-based models for Kirchberg algebras, J. Operator Theory 57 (2007), no. 2, 347–374. · Zbl 1164.46028
[25] C. E. Watts, “A homology theory for small categories” in Proceedings of the Conference on Categorical Algebra (La Jolla, 1965), Springer, New York, 1966, 331–335.
[26] F. Xu, On the cohomology rings of small categories, J. Pure Appl. Algebra 212 (2008), no. 11, 2555–2569. · Zbl 1153.18013
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