Alexander Campbell

email: alexander.campbell@anu.edu.au

I am an Australian category theorist.
My research interests are in category theory, especially higher and enriched category theory, and categorical aspects of homotopy theory.
I am currently a Postdoctoral Fellow in the Mathematical Sciences Institute at the Australian National University.
I am also a member of the Centre of Australian Category Theory at Macquarie University, where I was a PhD student (2013--2016), a Postdoctoral Research Fellow (2017--2019), and a Research Officer (2020--2021).
I was the Viterbi Endowed Postdoctoral Fellow in the Higher Categories and Categorification program at MSRI in (Northern Hemisphere) Spring 2020.

Publications

6. Joyal's cylinder conjecture, arXiv:1911.02631.
7. A homotopy coherent cellular nerve for bicategories, arXiv:1907.01999, Advances in Mathematics 368 (2020), 107138, 67 pp.
8. A counterexample in quasi-category theory, arXiv:1904.04965, Proceedings of the American Mathematical Society 148 (2020), no. 1, 37--40.
9. On truncated quasi-categories (with Edoardo Lanari), arXiv:1810.11188, Cahiers Topol. Géom. Différ. Catég. 61 (2020), no. 2, 154--207.
10. How strict is strictification?, arXiv:1802.07538, Journal of Pure and Applied Algebra 223 (2019), no. 7, 2948--2976.
11. Skew-enriched categories, arXiv:1709.01222, Applied Categorical Structures 26 (2018), no. 3, 597--615.
12. A higher categorical approach to Giraud's non-abelian cohomology, PhD thesis, Macquarie University, 2016. Supervisor: Ross Street.
13. Linear algebra via complex analysis (with Daniel Daners), The American Mathematical Monthly 120 (2013), no. 10, 877--892.

Papers in preparation:

• The model category of algebraically cofibrant 2-categories. (abstract, slides)
• A modular proof of the straightening theorem. (abstract, slides)
• The gregarious model structure for double categories. (abstract, slides)
• (with Yuki Maehara) A model-independent construction of the Gray monoidal structure for (∞,2)-categories. (slides)
• The new normal lax functors.
• A Chevalley criterion for 2-fibrations.
• A convenient model category for tricategories.

Talks

See here for a list of my talks at the Australian Category Seminar.

Slides:

• A model-independent construction of the Gray monoidal structure for (∞,2)-categories, talk at the Melbourne Topology Seminar, 20 October 2020.
• A modular proof of the straightening theorem, talk at the Australian Category Seminar, 29 July 2020.
• The gregarious model structure for double categories, talk at the Masaryk University Algebra Seminar, 18 June 2020.
• Joyal's cylinder conjecture, talk at the MSRI Online Seminar, 8 April 2020.
• The model category of algebraically cofibrant 2-categories, talk at the Category Theory 2019 conference.
• Enriched algebraic weak factorisation systems, talk at the Category Theory 2017 conference.

Notes:

• Equivalences of complicial sets, handwritten notes, Australian Category Seminar, 28 August & 4 September 2019.
• The fundamental theorem of n-quasi-category theory, handwritten notes, Australian Category Seminar, 26 September 2018.
• Change of base for enriched model categories, an expository talk given at the Australian Category Seminar, 14 February 2018.

Blog post: An exegesis of Yoneda structures, written for the Kan Extension Seminar.

Teaching

I was a co-organiser with Emily Riehl and Brendan Fong of the Kan Extension Seminar II, an online graduate reading course in category theory.

Other material

• AustMS Lift-Off Fellowship Report.