## Research interests

Set theory, especially its applications to category theory and algebraic topology.

## Current postgraduate students

Stamatis Dimopoulos (2017)

## Publications

**Brooke-Taylor A, Rosický J** Accessible images revisited *Proceedings of the American Mathematical Society*, **145**, 1327, 2017

DOI:10.1090/proc/13190

View abstract

*© 2016 American Mathematical Society. We extend and improve the result of Makkai and Paré (1989) that the powerful image of any accessible functor F is accessible, assuming there exists a sufficiently large strongly compact cardinal. We reduce the required large cardinal assumption to the existence of Lμ,ω -compact cardinals for sufficiently large μ, and also show that under this assumption the λ-pure powerful image of F is accessible. From the first of these statements, we obtain that the tameness of every Abstract Elementary Class follows from a weaker large cardinal assumption than was previously known. We provide two ways of employing the large cardinal assumption to prove each result - one by a direct ultraproduct construction and one using the machinery of elementary embeddings of the set-theoretic universe.*

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**Brooke-Taylor AD, Fischer V, Friedman SD, Montoya DC** Cardinal characteristics at in a small u (κ) model *Annals of Pure and Applied Logic*, **168**, 37-49, 2017

DOI:10.1016/j.apal.2016.08.004

View abstract

*We provide a model where u(κ)<2κu(κ)<2κ for a supercompact cardinal κ. [10] provides a sketch of how to obtain such a model by modifying the construction in [6]. We provide here a complete proof using a different modification of [6] and further study the values of other natural generalizations of classical cardinal characteristics inour model. For this purpose we generalize some standard facts that hold in the countable case as well as some classical forcing notions and their properties.*

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**Brooke-Taylor AD, Friedman SD** Subcompact cardinals, squares, and stationary reflection *Israel Journal of Mathematics*, **197**, 453-473, 2013

DOI:10.1007/s11856-013-0007-x

View abstract

*We generalise Jensen's result on the incompatibility of subcompactness with□. We show that α + -subcompactness of some cardinal less than or equal to α precludes □ α , but also that square may be forced to hold everywhere where this obstruction is not present. The forcing also preserves other strong large cardinals. Similar results are also given for stationary reflection, with a corresponding strengthening of the large cardinal assumption involved. Finally, we refine the analysis by considering Schimmerling's hierarchy of weak squares, showing which cases are precluded by α + -subcompactness, and again we demonstrate the optimality of our results by forcing the strongest possible squares under these restrictions to hold. © 2013 Hebrew University Magnes Press.*

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**Bagaria J, Brooke-Taylor A** On colimits and elementary embeddings *Journal of Symbolic Logic*, **78**, 562-578, 2013

DOI:10.2178/jsl.7802120

View abstract

*We give a sharper version of a theorem of Rosicky, Trnkova and Adamek [13], and a new proof of a theorem of Rosicky [12] , both about colimits in categories of structures. Unlike the original proofs, which use category-theoretic methods, we use set-theoretic arguments involving elementary embeddings given by large cardinals such as -strongly compact and C(n)-extendible cardinals.© 2013, Association for Symbolic Logic.*

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**Brendle J; Brooke-Taylor A; Ng KM; Nies A ** *An analogy between cardinal characteristics and highness properties of oracles*Proceedings of the 13th Asian Logic Conference, ALC 2013, 1-28 2013

View abstract

*© 2015 by World Scientific Publishing Co. Pte. Ltd. We present an analogy between cardinal characteristics from set theory and highness properties from computability theory, which specify a sense in which a Turing oracle is computationally strong. While this analogy was first studied explicitly byRupprecht (Effective correspondents to cardinal characteristics in Cichoń’s diagram, PhD thesis, University of Michigan, 2010), many prior results can be viewed from this perspective. After a comprehensive survey of the analogy for characteristics from Cichoń’s diagram, we extend it to Kurtz randomness and the analogue of the Specker-Eda number.*

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**Brooke-Taylor AD** Indestructibility of Vopěnka's Principle *Archive for Mathematical Logic*, **50**, 515-529, 2011

DOI:10.1007/s00153-011-0228-9

View abstract

*Vopěnka's Principle is a natural large cardinal axiom that has recently found applications in category theory and algebraic topology. We show that Vopěnka's Principle and Vopěnka cardinals are relatively consistent with a broad range of other principles known to be independent of standard (ZFC) settheory, such as the Generalised Continuum Hypothesis, and the existence of a definable well-order on the universe of all sets. We achieve this by showing that they are indestructible under a broad class of forcing constructions, specifically, reverse Easton iterations of increasingly directed closedpartial orders. © 2011 Springer-Verlag.*

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**Brooke-Taylor AD** Large cardinals and definable well-orders on the universe *Journal of Symbolic Logic*, **74**, 641-654, 2009

DOI:10.2178/jsl/1243948331

View abstract

*We use a reverse Easton forcing iteration to obtain a universe with a definable well-order, while preserving the GCH and proper classes of a variety of very large cardinals. This is achieved by coding using the principle 0+ at a proper class of cardinals k. By choosing the cardinals at which coding occurs sufficiently sparsely, we are able to lift the embeddings witnessing the large cardinal properties without having to meet any non-trivial master conditions.©2009. Association.*

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**Brooke-Taylor AD, Friedman SD** Large cardinals and gap-1 morasses *Annals of Pure and Applied Logic*, **159**, 71-99, 2009

DOI:10.1016/j.apal.2008.10.007

View abstract

*We present a new partial order for directly forcing morasses to exist that enjoys a significant homogeneity property. We then use this forcing in a reverse Easton iteration to obtain an extension universe with morasses at every regular uncountable cardinal, while preserving all n-superstrong (1≤ n ≤ ω), hyperstrong and 1-extendible cardinals. In the latter case, a preliminary forcing to make the GCH hold is required. Our forcing yields morasses that satisfy an extra property related to the homogeneity of the partial order; we refer to them as mangroves and prove that their existenceis equivalent to the existence of morasses. Finally, we exhibit a partial order that forces universal morasses to exist at every regular uncountable cardinal, and use this to show that universal morasses are consistent with n-superstrong, hyperstrong, and 1-extendible cardinals. This all contributesto the second author's outer model programme, the aim of which is to show that L-like principles can hold in outer models which nevertheless contain large cardinals. © 2008 Elsevier B.V. All rights reserved.*

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**Brooke-Taylor AD, Lowe B, Richter B** Inhabitants of interesting subsets of the Bousfield lattice *Journal of Pure and Applied Algebra*

DOI:10.1016/j.jpaa.2017.09.012

View abstract

*In 1979, Bousfield defined an equivalence relation on the stable homotopy category. The set of Bousfield classes has some important subsets such as the distributive lattice, DL, of all classes〈E〉 which are smash idempotent and the complete Boolean algebra, cBA, of closed classes. We provide examples of spectra that are in DL, but not in cBA; in particular, for every prime p , the Bousfield class of the Eilenberg-MacLane spectrum 〈HFp〉 is in DL∖cBA.*

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Brooke-Taylor A; Miller S *Complexity of a knot invariant*PAMM, 899-900

DOI:10.1002/pamm.201610438

View abstract

*The algebraic structures called quandles constitute a complete invariant for tame knots. However, determining when two
quandles are isomorphic is an empirically hard problem, so there is some dissatisfaction with quandles as knot invariants.
We have confirmed this apparent difficulty, showing within the framework of Borel reducibility that the general isomorphism
problem for quandles is as complex as possible.*

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**Brooke-Taylor AD, Brendle J, Friedman SD, Montoya D** Cicho n's Diagram for uncountable cardinals *Israel Journal of Mathematics*