Related papers: Decidable objects and molecular toposes

Let $\pi$ be a group satisfying the Farrell-Jones conjecture and assume that $B\pi$ is a 4-dimensional Poincar\'e duality space. We consider topological, closed, connected manifolds with fundamental group $\pi$ whose canonical map to $B\pi$…

The monadic theory of $(\mathbb R,\le)$ with quantification restricted to Borel sets is decidable. The Boolean combinations of $F_\sigma$ sets form an elementary substructure of the Borel sets. Under determinacy hypotheses, the proof…

We discuss the problem of characterizing the property of a Grothendieck topos to satisfy a given 'geometric' invariant as a property of its sites of definition, and indicate a set of general techniques for establishing such criteria. We…

Graphons $W$ can be used as stochastic models to sample graphs $G_n$ on $n$ nodes for $n$ arbitrarily large. A graphon $W$ is said to have the $H$-property if $G_n$ admits a decomposition into disjoint cycles with probability one as $n$…

We prove that a set of finite perimeter is indecomposable if and only if it is, up to a choice of suitable representative, connected in the 1-fine topology. This gives a topological characterization of indecomposability which is new even in…

Every pseudo-Anosov homeomorphism $f$ admits infinitely many Markov partitions. A \textit{geometric Markov partition} is a Markov partition $\mathcal{R}$ in which each rectangle is equipped with a vertical orientation. To each pair $(f,…

In the setting of relative topos theory, we show that the pullback of a relative presheaf topos on an arbitrary fibration is the relative presheaf topos on its inverse image. To this end, we develop and exploit a notion of extension with…

We study morphisms of internal locales of Grothendieck toposes externally: treating internal locales and their morphisms as sheaves and natural transformations. We characterise those morphisms of internal locales that induce surjective…

We study geometric variations of the discriminating code problem. In the \emph{discrete version} of the problem, a finite set of points $P$ and a finite set of objects $S$ are given in $\mathbb{R}^d$. The objective is to choose a subset…

We prove that any diffeomorphism of a compact manifold can be approximated in topology C1 by another diffeomorphism exhibiting a homoclinic bifurcation (a homoclinic tangency or a heterodimensional cycle) or by one which is essentially…

We prove a number of results to the general effect that, under obviously necessary numerical and determinant constraints, "most" morphisms between fixed bundles on a complex elliptic curve produce (co)kernels which can either be specified…

We propose a conservative algorithm to test the geometrical validity of simplicial (triangles, tetrahedra), tensor product (quadrilaterals, hexahedra), and mixed (prisms) elements of arbitrary polynomial order as they deform over a…

Suppose a finitely generated group $G$ is hyperbolic relative to $\mathcal P$ a set of proper finitely generated subgroups of $G$. Established results in the literature imply that a "visual" metric on $\partial (G,\mathcal P)$ is "linearly…

The aim of this paper is to study some variants of nowhere dense sublocales called maximal nowhere dense and homogeneous maximal nowhere dense sublocales. These concepts were initially introduced by Veksler in classical topology. We give…

We prove the existence and uniqueness of geometric models of local isometry classes of locally homogeneous spaces with sectional curvature $|\operatorname{sec}|\leq 1$. Moreover, we show that the set of geometric models is compact in the…

We prove that a Hom-finite additive category having determined morphisms on both sides is a dualizing variety. This complements a result by Krause. We prove that in a Hom-finite abelian category having Serre duality, a morphism is right…

Let M be a meromorphic connection with poles along a smooth divisor D in a smooth algebraic variety. Let Sol M be the solution complex of M. We prove that the good formal decomposition locus of M coincides with the locus where the…

Defining P* to be the complete lattice of upsets (ordered by reverse inclusion) of a poset P we give necessary and sufficient conditions on a subset S of P* for P to admit a meet-completion e from P to Q where e preserves the infimum of an…

We relate the topology of the Morse boundary of a group to geometric and algorithmic properties of the group. In particular, we show that a group has $\sigma$-compact Morse boundary if and only if it is Morse local-to-global. We also…

Topologies $\tau , \sigma \in \mathop{{\mathrm{Top}}}\nolimits _X$ are bijectively related, in notation $\tau \sim \sigma$, if there are continuous bijections $f: (X, \tau )\rightarrow (X, \sigma )$ and $g: (X, \sigma)\rightarrow (X,…