Related papers: Simplicial sets with a notion of smallness
We propose a sequential topology on the space of sub-$\sigma$-algebras of a separable probability space $(\Omega,\mathcal{F},\mathbb{P})$ by linking conditional expectations on $L^{2}$ along sequences of sub-$\sigma$-algebras. The varying…
The notions of null-sets and nullity are present in all discourses of mathematics. They are based on the dual-pair of notions of "almost-every" and "almost none". A notion of nullity corresponds to a choice of subsets that one interprets as…
We introduce the notion of topological entropy of a formal languages as the topological entropy of the minimal topological automaton accepting it. Using a characterization of this notion in terms of approximations of the Myhill-Nerode…
A co-valuation is, essentially, a minimal finite cover. We introduce a logic based on co-valuations, which play the role of valuations of free variables in classical first-order logic, and show that the fundamental tools of model theory --…
A simplicial complex is a set equipped with a down-closed family of distinguished finite subsets. This structure, usually viewed as codifying a triangulated space, is used here directly, to describe "spaces" whose geometric realisation can…
In the spirit of topological entropy we introduce new complexity functions for general dynamical systems (namely groups and semigroups acting on closed manifolds) but with an emphasis on the dynamics induced on simplicial complexes. For…
Frobenius algebras in the category of sets and relations ($\mathbf{Rel}$) serve as a unifying framework for various algebraic and combinatorial structures, including groupoids, effect algebras, and abstract circles. Recently, a nerve…
We relativize the notion of a compact object in an abelian category with respect to a fixed subclass of objects. We show that the standard closure properties persist to hold in this case. Furthermore, we describe categorical and…
Basic pairs and their morphisms are the most elementary framework in which standard topological notions can be defined. We present here a new interpretation of topological concepts as those which can be communicated faithfully between the…
The essential subtoposes of a fixed topos form a complete lattice, which gives rise to the notion of a level in a topos. In the familiar example of simplicial sets, levels coincide with dimensions and give rise to the usual notions of…
The compactness phenomenon is one of the featured aspects of structuralism in mathematics. In simple and broad words, a compactness property holds in a structure if a related property is satisfied by sufficiently many substructures of that…
We provide a random simplicial complex by applying standard constructions to a Poisson point process in Euclidean space. It is gigantic in the sense that - up to homotopy equivalence - it almost surely contains infinitely many copies of…
Tilings and point sets arising from substitutions are classical mathematical models of quasicrystals. Their hierarchical structure allows one to obtain concrete answers regarding spectral questions tied to the underlying measures and…
Let $X$ be a locally compact topological space, $(Y,d)$ be a boundedly compact metric space and $LB(X,Y)$ be the space of all locally bounded functions from $X$ to $Y$. We characterize compact sets in $LB(X,Y)$ equipped with the topology of…
The simplicial volume is a homotopy invariant of oriented closed connected manifolds measuring the efficiency of representing the fundamental class by singular chains with real coefficients. Despite of its topological nature, the simplicial…
This paper provides a self-contained exploration of subdivisions of simplicial complexes, with emphasis on barycentric subdivision. We present formal definitions of subdivisions, show how the realization of a complex is preserved under…
In order to provide a good categorical setting to the many different spaces of fields arising in the description of physical theories, a pedagogical introduction to the categorical notion of smooth sets is provided and some simple…
This survey contains a selection of topics unified by the concept of positive semi-definiteness (of matrices or kernels), reflecting natural constraints imposed on discrete data (graphs or networks) or continuous objects (probability or…
We give a self-contained introduction to accessible categories and how they shed light on both model- and set-theoretic questions. We survey for example recent developments on the study of presentability ranks, a notion of cardinality…
We introduce the notion of doubling and r-tupling for simplicial complexes, a notion reminiscent to that of matching complexes in graph theory. We prove a connectivity result for such complexes and relate r-tupling to stabilizing r times…