Andrew Brooke-Taylor's Research Page

My main area of interest is in set theory, specifically, large cardinal axioms and forcing. I am also very interested in connections between set theory and other fields of mathematics, especially category theory and algebraic topology, which I studied before switching to set theory.

Large cardinals and forcing

Large cardinal axioms are axioms that extend the usual (ZFC) axioms for set theory, strengthening the theory. Forcing is the standard technique for proving consistency results for other set-theoretic principles; put them together and you can address interesting questions about which strengthenings of ZFC are compatible with which other set-theoretic principles.

Papers

Unprepared Indestructibility (conference paper surveying "Indestructibility of Vopenka's Principle", below). To appear in Proceedings of the RIMS Symposium on Aspects of Descriptive Set Theory, 2011, RIMS Ko-kyu-roku series. (PDF)
With Sy-David Friedman. Subcompact cardinals, squares, and stationary reflection. Submitted. (PDF, arXiv page).
Indestructibility of Vopenka's Principle, Archive for Mathematical Logic 50, no. 5 (2011) , pp 515-529. (PDF, journal page, arXiv)
With Sy-David Friedman. Large cardinals and gap-1 morasses, Annals of Pure and Applied Logic 159, no. 1-2 (2009), pp 71-99. (PDF, journal, arXiv)
Large cardinals and definable well-orders on the universe, Journal of Symbolic Logic 74, no. 2 (June 2009) pp 641-654. (PDF, journal, arXiv)
My Doctoral Thesis: Large Cardinals and L-like Combinatorics (PDF, PS format). For a bit of light relief, I included a couple of figures (PS format).

Talk slides

The limits of large cardinal compatibility for square, at Logic Colloquium 2011

Along similar lines, Subcompact cardinals, squares and stationary reflection, Czech Winter School in Abstract Analysis Section Set Theory, February 2011.

Unprepared Indestructibility, at the Third European Set Theory Conference, Edinburgh, July 2011.

Similarly: Unprepared Indestructibility, RIMS Set Theory Workshop 2011, Kyoto, October 2011.
Indestructibility of Vopenka Cardinals, Czech Winter School in Abstract Analysis Section Topology, February 2010.
Indestructibility of Vopenka's Principle, Indian School in Logic and its Applications, Hyderabad, January 2010.

Definable well-ordering, the GCH, and large cardinals, Logic Colloquium 2007.
Forcing while preserving 1-extendibles, Logic Colloquium 2006.

Applications of large cardinals in category theory

There is a significant literature using large cardinal axioms, especially Vopenka's Principle, in the study of locally presentable and accessible categories. Joan Bagaria and I have been working to advance the area making use of our set-theoretic perspective. We have already been able to extend a colimit preservation theorem of Rosick, Trnkova and Adamek, and simplify the proof of another colimit preservation theorem of Rosicky.

Papers

With Joan Bagaria. On colimits and elementary embeddings. Submitted. (PDF, arXiv page)

Talk slides

Large cardinals and colimits, at the Large cardinal methods in homotopy meeting, Barcelona, September 2011.

Fraisse limits

With Damiano Testa I have done some work on Fraisse limits of Fraisse classes for infinite languages which behave in a "locally finite" manner. In particular we are able to show that there is still a 0-1 law in this case, for example giving a new 0-1 law for simplicial complexes.

Talk slides

Fraisse limits for infinite languages, ICM satellite conference in logic and set theory, Chennai, August 2010.

Other research interests

I have worked with Benedikt Loewe and Birgit Richter on the Bousfield lattice in algebraic topology, and with Sheila Miller on rank-to-rank embeddings and their implications for left self-distributive systems, but don't yet have anything to link to for those!

Other Miscellanea

My Honours Thesis on derived categories (PDF, or PS).
At the Young Logicians Gathering in Japan 2011, Sakae Fuchino asked how many ways there could be to assign 3 distinct cardinals to the cardinal characteristics in Cichon's diagram (that is, respecting the inequalities in Cichon's diagram, but assuming that no other set-theoretic intricacies get in the way). According to this program that I wrote in Sage, the answer is 179.

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Last updated: 15/12/11 ([d]d/[m]m/yy)