## Rocky Mountain Journal of Mathematics

### Discrete Conduché fibrations and $C^*$-algebras

#### Abstract

The $k$-graphs in the sense of Kumjian and Pask~\cite {KP} are discrete Conduch\'{e} fibrations over the monoid~$\mathbb {N}^k$, satisfying a finiteness condition. We examine the generalization of this construction to discrete Conduch\'{e} fibrations with the same finiteness condition and a lifting property for completions of cospans to commutative squares, over any category satisfying a strong version of the right Ore condition, including all categories with pullbacks and right Ore categories in which all morphisms are monic.

#### Article information

Source
Rocky Mountain J. Math., Volume 47, Number 3 (2017), 711-756.

Dates
First available in Project Euclid: 24 June 2017

https://projecteuclid.org/euclid.rmjm/1498269809

Digital Object Identifier
doi:10.1216/RMJ-2017-47-3-711

Mathematical Reviews number (MathSciNet)
MR3682147

Zentralblatt MATH identifier
1369.18003

#### Citation

Brown, Jonathan H.; Yetter, David N. Discrete Conduché fibrations and $C^*$-algebras. Rocky Mountain J. Math. 47 (2017), no. 3, 711--756. doi:10.1216/RMJ-2017-47-3-711. https://projecteuclid.org/euclid.rmjm/1498269809

#### References

• C. Anantharaman-Delaroche and J. Renault, Amenable groupoids, Mono. Enseign. Math. 36, Geneva, 2000.
• J. Brown, L. Clark, C. Farthing and A. Sims, Simplicity of algebras associated to étale groupoids, Semigroup Forum 88 (2014), 433–452.
• T. Carlsen, S. Kang, J. Shotwell and A. Sims, The primitive ideals of the Cuntz-Krieger algebra of a row-finite higher-rank graph with no sources, J. Funct. Anal. 266 (2014), 2570–2589.
• J. Cuntz and W. Krieger, A class of $C^*$-algebras and topological Markov chains, Invent. Math. 56 (1980), 251–268.
• A. Huef and I. Raeburn, The ideal structure of Cuntz-Krieger algebras, Ergodic Th. Dynam. Syst. 17 (1997), 611–624.
• P. Johnstone, Sketches of an elephant: A topos theory compendium, Volume 1, Oxford Logic Guides 43, Oxford University Press, Oxford, 2002.
• A. Kumjian and D. Pask, Higher rank graph $C^*$-algebras, New York J. Math. 6 (2000), 1–20.
• A. Kumjian, D. Pask and I. Raeburn, Cuntz-Krieger algebras of directed graphs, Pacific J. Math. 184 (1998), 161–174.
• A. Kumjian, D. Pask, I. Raeburn and J. Renault, Graphs, groupoids and Cuntz-Krieger algebras, J. Funct. Anal. 144 (1997), 505–541.
• S. Mac Lane, Categories for the working mathematician, 2nd edition, Grad. Texts Math. 5, Springer-Verlag, New York, 1998.
• D. Pask, M. Rørdam, I. Raeburn and A. Sims, Rank-two graphs whose $C^*$-algebras are direct limits of circle algebras, J. Funct. Anal. 239 (2006), 137–178.
• J. Renault, A groupoid approach to $C^*$-algebras, Lect. Notes Math. 793, Springer, Berlin 1980.
• D. Robertson and A. Sims, Simplicity of $C^*$-algebras associated to higher-rank graphs, Bull. Lond. Math. Soc. 39 (2007), 337–344.
• G. Robertson and T. Steger, $C^*$-algebras arising from group actions on the boundary of a triangle building, Proc. Lond. Math. Soc. 72 (1996), 613–637.
• J. Spielberg, Groupoids and $C^*$-algebras for categories of paths, Trans. Amer. Math. Soc. 366, American Mathematical Society, Providence, 2014.
• D. Yang, The structure of higher rank graph $C^*$-algebras revisited, J. Austral. Math. Soc. 99 (2015), 267–286.