Assaf Hasson

Exercices de style: A homotopy theory for set theory

We construct a model category (in the sense of Quillen) for set theory, starting from two arbitrary, but natural, conventions. It is the simplest category satisfying our conventions and modelling the notions of finiteness, countability and infinite equi-cardinality. We argue that from the homotopy theoretic point of view our construction is essentially automatic following basic existing methods, and so is (almost all) the verification that the construction works.We use the posetal model category to introduce homotopy-theoretic intuitions to set theory. Our main observation is that the homotopy invariant version of cardinality is the covering number of Shelah’s PCF theory, and that other combinatorial objects, such as Shelah’s revised power function—the cardinal function featuring in Shelah’s revised GCH theorem—can be obtained using similar tools. We include a small “dictionary” for set theory in QtNaamen, hoping it will help in finding more meaningful homotopy-theoretic intuitions in set theory.

Some questions concerning Hrushovski's amalgamation constructions

In order to construct a counterexample to Zilbers conjecture—that a strongly minimal set has a degenerate, affine or field-like geometry—Ehud Hrushovski invented an amalgamation technique which has yielded all the exotic geometries so far. We shall present a framework for this construction in the language of standard geometric stability and show how some of the recent constructions fit into this setting. We also ask some fundamental questions concerning this method.

Interpreting structures of finite Morley Rank in strongly minimal sets

Abstract We show that any structure of finite Morley Rank having the definable multiplicity property (DMP) has a rank and multiplicity preserving interpretation in a strongly minimal set. In particular, every totally categorical theory admits such an interpretation. We also show that a slightly weaker version of the DMP is necessary for a structure of finite rank to have a strongly minimal expansion. We conclude by constructing an almost strongly minimal set which does not have the DMP in any rank preserving expansion, and ask whether this structure is interpretable in a strongly minimal set.

ON DEFINABLE SKOLEM FUNCTIONS IN WEAKLY O-MINIMAL NONVALUATIONAL STRUCTURES

We prove that all known examples of weakly o-minimal non-valuational structures have no definable Skolem functions. We show, however, that such structures eliminate imaginaries up to (definable families of) definable cuts. Along the way we give some new examples of weakly o-minimal non-valuational structures.

The Univalence Axiom in posetal model categories.

In this note we interpret Voevodskys Univalence Axiom in the language of (abstract) model categories. We then show that any posetal locally Cartesian closed model category

Embedded o-minimal structures

Qt

Fusion over sublanguages

in which the mapping

Stable types in rosy theories

Hom^{(w)}(Z\times B,C):Qt\longrightarrow Sets

Strongly minimal expansions of (ℂ, +) definable in o-minimal fields

is functorial in

Definable structures in o-minimal theories: One dimensional types

Z