arXiv subjects

Andrew Brooke-Taylor

Publications and source records attributed to Andrew Brooke-Taylor.

7 recordsLinked to original sources

arXiv2017

Inhabitants of interesting subsets of the Bousfield lattice

The set of Bousfield classes has some important subsets such as the distributive lattice $\mathbf{DL}$ of all classes $\langle E\rangle$ which are smash idempotent and the complete Boolean algebra $\mathbf{cBA}$ of closed classes. We provide examples of spectra that are in $\mathbf{DL}$, but not in $\mathbf{cBA}$; in particular, for every prime $p$, the Bousfield class of the Eilenberg-MacLane spectrum $\langle H\mathbb{F}_p\rangle\in\mathbf{DL}{\setminus}\mathbf{cBA}$.

math.AT↗

arXiv2016

Cichon's Diagram for uncountable cardinals

We develop a version of Cichon's diagram for cardinal invariants on the generalized Cantor space 2^kappa or the generalized Baire space kappa^kappa where kappa is an uncountable regular cardinal. For strongly inaccessible kappa, many of the ZFC-results about the order relationship of the cardinal invariants which hold for omega generalize; for example we obtain a natural generalization of the Bartoszynski-Raisonnier-Stern Theorem. We also prove a number of independence results, both with <kappa-support iterations and kappa-support iterations and products, showing that we consistently have strict inequality between some of the cardinal invariants.

math.LO↗

arXiv2016

Accessible images revisited

We extend and improve the result of Makkai and Paré 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 $L_{μ,ω}$-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.

math.CT↗

arXiv2014

An analogy between cardinal characteristics and highness properties of oracles

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 by Rupprecht in his PhD thesis, many prior results can be viewed from this perspective. After a comprehensive survey of the analogy for characteristics from Cichon's diagram, we extend it to Kurtz randomness and the analogue of the Specker-Eda number.

math.LO↗

arXiv2013

The infinite random simplicial complex

We study the Fraisse limit of the class of all finite simplicial complexes. Whilst the natural model-theoretic setting for this class uses an infinite language, a range of results associated with Fraisse limits of structures for finite languages carry across to this important example. We introduce the notion of a local class, with the class of finite simplicial complexes as an archetypal example, and in this general context prove the existence of a 0-1 law and other basic model-theoretic results. Constraining to the case where all relations are symmetric, we show that every direct limit of finite groups, and every metrizable profinite group, appears as a subgroup of the automorphism group of the Fraisse limit. Finally, for the specific case of simplicial complexes, we show that the geometric realisation is topologically surprisingly simple: despite the combinatorial complexity of the Fraisse limit, its geometric realisation is homeomorphic to the infinite simplex.

math.LO↗

arXiv2012

Unprepared Indestructibility

This article is based on the talk of the same name which I gave at the "Aspects of Descriptive Set Theory" RIMS Symposium in Kyoto in October 2011; it is essentially just a survey of my article "Indestructibility of Vopenka's Principle". In particular, I present (with a sketch of the proof) a forcing indestructibility theorem for the large cardinal axiom Vopenka's Principle. It is notable in that there is no preparatory forcing required to make the axiom indestructible, unlike other indestructibility results.

math.LO↗

arXiv2012

On colimits and elementary embeddings

We give a sharper version of a theorem of Rosicky, Trnkova and Adamek, and a new proof of a theorem of Rosicky, both about colimit preservation between 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 alpha-strongly compact and C^(n)-extendible cardinals.

math.LO↗