Tibor Beke

Sheafifiable homotopy model categories

If a Quillen model category can be specified using a certain logical syntax (intuitively, ‘is algebraic/combinatorial enough’), so that it can be defined in any category of sheaves, then the satisfaction of Quillens axioms over any site is a purely formal consequence of their being satisfied over the category of sets. Such data give rise to a functor from the category of topoi and geometric morphisms to Quillen model categories and Quillen adjunctions.

Abstract elementary classes and accessible categories

Abstract We investigate properties of accessible categories with directed colimits and their relationship with categories arising from Shelahʼs Abstract Elementary Classes. We also investigate ranks of objects in accessible categories, and the effect of accessible functors on ranks.

When is flatness coherent

ABSTRACT We characterize those small categories with the property that flat (contravariant) functors on them are coherently axiomatized in the language of presheaves on them. They are exactly the categories with the property that every finite diagram into them has a finite set of (weakly) initial cocones.

Isoperimetric inequalities and the Friedlander-Milnor conjecture

Abstract We prove that Friedlander’s generalized isomorphism conjecture on the cohomology of algebraic groups, and hence the Isomorphism Conjecture for the cohomology of the complex algebraic Lie group G(ℂ) made discrete, are equivalent to the existence of an isoperimetric inequality in the homological bar complex of G(F ), where F  is the algebraic closure of a finite field.

Fibrations of Simplicial Sets

There are infinitely many variants of the notion of Kan fibration that, together with suitable choices of cofibrations and the usual notion of weak equivalence of simplicial sets, satisfy Quillen’s axioms for a homotopy model category. The combinatorics underlying these fibrations is purely finitary and seems interesting both for its own sake and for its interaction with homotopy types. To show that these notions of fibration are indeed distinct, one needs to understand how iterates of Kan’s Ex functor act on graphs and on nerves of small categories.