For a suitable collection D of small categories, we define the D-accessible categories, generalizing the λ-accessible categories of Lair, Makkai, and Paré; here the λ-accessible categories are seen as the D-accessible categories where D consists of the λ-small categories. A small category C is called D-filtered when C-colimits commute with D-limits in the category of sets. An object of a catego...

We prove that the minimal Euler characteristic of a closed symplectic four-manifold with given fundamental group is often much larger than the minimal Euler characteristic of almost complex closed four-manifolds with the same fundamental group. In fact, the difference between the two is arbitrarily large for certain groups. It was first proved by Dehn [2] that every finitely presentable group Γ...

We suggest a linear non-conforming triangular element for Maxwell’s equations and test it in the context of the vector Helmholtz equation for the electric field. The element uses discontinuous normal fields and tangential fields with continuity at the midpoint of the element sides, an approximation related to the Crouzeix-Raviart element for Stokes. The element is stabilized using the jump of t...

We show that the category of algebraically cofibrant objects in a combinatorial and simplicial model category A has a model structure that is left-induced from that on A. In particular it follows that any presentable model category is Quillen equivalent (via a single Quillen equivalence) to one in which all objects are cofibrant. Let A be a cofibrantly generated model category. Garner’s version...

Hilbert’s Tenth Problem over the rationals is one of the biggest open problems in the area of undecidability in number theory. In this paper we construct new, computably presentable subrings R ⊆ Q having the property that Hilbert’s Tenth Problem for R, denoted HTP(R), is Turing equivalent to HTP(Q). We are able to put several additional constraints on the rings R that we construct. Given any co...

Abstract A left orderable monster is a finitely generated group all of whose fixed point-free actions on the line are proximal : action semiconjugate to minimal so that for every bounded interval I and open J , there element sends into . In his 2018 ICM address, Navas asked about existence monsters. By now several examples, which but not presentable. We provide first examples monsters presentab...

We propose a convenient category for directed homotopy consisting of “directed” topological spaces generated by “directed” cubes. Its main advantage is that, like the category of topological spaces generated by simplices suggested by J. H. Smith, it is locally presentable.

Finitary endofunctors of locally presentable categories are proved to have equational presentations. Special attention is paid to the Hausdorff functor of non-empty compact subsets of a complete metric space.

We propose a convenient category for directed homotopy consisting of preordered topological spaces generated by cubes. Its main advantage is that, like the category of topological spaces generated by simplices suggested by J. H. Smith, it is locally presentable.