# Tibor Beke's research while affiliated with University of Massachusetts Lowell and other places

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## Publications (17)

In each characteristic, there is a canonical homomorphism from the Grothendieck ring of varieties to the Grothendieck ring of sets definable in the theory of algebraically closed fields. We prove that this homomorphism is an isomorphism in characteristic zero. In positive characteristics, we exhibit specific elements in the kernel of the correspond...

We introduce the notion of $\lambda$-equivalence and $\lambda$-embeddings of objects in suitable categories. This notion specializes to $L_{\infty\lambda}$-equivalence and $L_{\infty\lambda}$-elementary embedding for categories of structures in a language of arity less than $\lambda$, and interacts well with functors and $\lambda$-directed colimits...

The best-known version of Shelah's celebrated singular cardinal compactness
theorem states that if the cardinality of an abelian group is singular, and all
its subgroups of lesser cardinality are free, then the group itself is free.
The proof can be adapted to cover a number of analogous situations in the
setting of non-abelian groups, modules, gra...

An axiomatization of a finitary, equational universal algebra by a convergent term rewrite system gives rise to a finite, coherent categorification of the algebra.

We prove the rationality of the generating function associated to the number of equivalence classes of Fqk-points of a constructible equivalence relation defined over the finite field Fq. This is a consequence of the rationality of Weil zeta functions and of first-order formulas, together with the existence of a suitable parameter space for constru...

We prove the topological invariance of the combinatorial
Euler characteristic with the help of a canonical, topologically
defined stratification of tame spaces by locally compact, tame
strata.

We show that metric abstract elementary classes (mAECs) are, in the sense of
[LR] (i.e. arXiv:1404.2528), coherent accessible categories with directed
colimits, with concrete $\aleph_1$-directed colimits and concrete
monomorphisms. More broadly, we define a possible notion of $\kappa$-AEC---an
AEC-like category in which only the $\kappa$-directed c...

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 f...

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.

Let us say that a geometric theory T is of presheaf type if its classifying topos is (equivalent to) a presheaf topos. (We adhere to the convention that geometric logic allows arbitrary disjunctions, while coherent logic means geometric and finitary.) Write Mod( T ) for the category of Set -models and homomorphisms of T . The next proposition is we...

We introduce a notion of cover of level n for a topological space, or more generally any Grothendieck site, with the key property that simplicial homotopy classes computed along the filtered diagram of n-covers biject with global homotopy classes when the target is an n-type. When the target is an Eilenberg–MacLane sheaf, this specializes to comput...

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 o...

If a Quillen model category is defined via a suitable right adjoint over a sheafifiable homotopy model category (in the sense of part I of this paper), it is sheafifiable as well; that is, it gives rise to a functor from the category of topoi and geometric morphisms to Quillen model categories and Quillen adjunctions. This is chiefly useful in deal...

The interpretation by Duskin and Glenn of abelian sheaf cohomology as connected components of a category of torsors is extended to homotopy classes. This is simultaneously an extension of Verdier's version of Cech cohomology to homotopy.

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 Quillen's axioms over any site is a purely formal consequence of their being satisfied over the category of sets. Such data give rise to a func...

We prove the topological invariance of the combinational Euler characteristics of o- minimal sets with the help of a canonical, topologically defined stratification of o-minimal sets by locally compact ones.

The question that serves as the title is to be understood in the context of Quillen model categories: having fixed the category of models and the subcategory of weak equiva-lences, how much ambiguity is there in finding suitable fibration and cofibration classes? Already for simplicial sets, infinitely many different subclasses of the monomorphisms...

## Citations

... We will need a criterion for isomorphism in the Grothendieck ring. [1] is correct in characteristic zero, but in characteristic p there is a gap in the proof, since separable maps do not necessarily remain separable when restricted to closed subsets. In our case this is guaranteed by the assumption that all of our fibers are reduced points. ...

Reference: On degenerate sections of vector bundles

... It is known that the Euler characteristics of homeomorphic semialgebraic sets coincide [3]. Therefore, the equality of Euler characteristics (7) in Theorem 1.2 can be obtained from Corollary 5.1. ...

... Anyway, this property holds more generally for any atomic complete theory which is Morita-equivalent to a coherent (atomic and complete) theory with a special model M; for example, T might be the theory of homogeneous S-models where S is a theory of presheaf type such that its category of finitely presentable models satisfies AP and JEP and has all fc finite colimits (notice that the latter condition is automatically satisfied if S is coherent, cf. [4]) and such that there exists a ultrahomogeneous S-model. ...

Reference: Topological Galois Theory

... Then there exists a combinatorial model structure on C with cofibrations given by cof(I), fibrant objects given by the naive fibrant objects, and such that the class of weak equivalences between fibrant objects is precisely W f . This result is similar in flavor to Smith's theorem [Bek00] and to a result by Stanculescu [Sta14], but unlike these, it restricts the conditions to be verified to the morphisms between fibrant objects which is where we assume the user has the most control. In practice, this restriction significantly reduces the difficulty for the user, and we expect this theorem to have a wide range of applications. ...

Reference: Fibrantly-induced model structures

... (Said differently: the identity functor may fail to be a Quillen equivalence between the same category of models, same class of weak equivalences, but with two different choices of cofibrations!) The note [1] proves a kind of obverse to Thm. 2.2, namely that in any model category satisfying mild set-theoretic assumptions (ones satisfied by all the examples appearing above), if one fixes the weak equivalences, and restricts cofibrations to those that can be generated by some set, then the collection of possible cofibration classes, partially ordered by inclusion, has least upper bounds for any set of elements. It follows that the possible set-generated cofibration classes on these (combinatorial, in the sense of Jeff Smith) model categories all yield Quillen-equivalent model structures, the equivalence of any two arising, possibly, via a "zig-zag" -not by a direct Quillen adjunction but by comparison with a third model structure. ...

Reference: Fibrations of Simplicial Sets

... Programming Languages, such as the -calculus, can be viewed as algebraic structures with variable-binding operators, which can be formalised using second-order algebraic theories [Fiore and Mahmoud 2010], or algebraic theories with closed structure [Hyland 2017], called -theories, making the -calculus the presentation of the initial -theory Λ. Our family of reversible languages Π have been presented as first-order algebraic 2theories [Beke 2011;Cohen 2009;Yanofsky 2000], which are a categorification of algebraic theories. The types 0 and 1 are nullary function symbols, the type formers + and × are binary function symbols, the 1-combinators are invertible reduction rules, and the 2-combinators are equations or coherence diagrams of compositions of reduction rules. ...

... Thus, the theory of the Deligne groupoid does not apply, and in fact the formal deformation theory is modeled by a 2-groupoid. (This 2-groupoid was constructed by Deligne [Del94], and, independently, by Getzler [Get02,2].) The functor .g/ ...

Reference: Lie theory for nilpotent L ∞ -algebras

... Joyal's original proof, dating from the early 80's [13], was never published, but see Jardine [10], [12] for careful expositions. Beke [2] shows just how automatic the reduction to Set can be made for other homotopy model categories too. All that is needed for our present purposes, however, is an easy 2.4. ...

Reference: Simplicial torsors

... This is a close variant on work of K. Brown [5] and Jardine [14]. See [3] for a proof. ...

Reference: Higher Čech Theory

... Proof of Theorem 1.1: Assume we have an accessible category with directed colimits K which is categorical for some λ ≥ ℶ (2 κ ) + in S. By Theorem 4.5 in [BR12], K is a reflective subcategory of a finitely accessible category, K ′ , which must be equivalent to the category of models of a ω-geometric theory; without loss of generality we can restrict our accessible category to monomorphisms and so we can assume the morphisms of K ′ are embeddings. Both categories are axiomatizable in µcoherent logic for µ = LS(K) + . ...