Related papers: A model theory of topology
The theory of persistence modules is an emerging field of algebraic topology which originated in topological data analysis. In these notes we provide a concise introduction into this field and give an account on some of its interactions…
We define a class of subsets of a topological space that coincides with the class of compact saturated subsets when the space is sober, and with enough good properties when the space is not sober. This class is introduced especially in view…
In \cite{To} S. Todorcevic introduced the notion of a topological Ramsey space and a list of axioms required to be satisfied by any such a space. Here we show that any topological Ramsey space that satisfies a strengthened version of one of…
A new methodological approach for the study of topology for shapes made of arrangements of lines, planes or solids is presented. Topologies for shapes are traditionally built on the classical theory of point-sets. In this paper, topologies…
A framework is developed to describe the Zariski topologies on the prime and primitive spectra of a quantum algebra $A$ in terms of the (known) topologies on strata of these spaces and maps between the collections of closed sets of…
This paper introduces and develops M\"obius homology, a homology theory for representations of finite posets into abelian categories. Although the connection between poset topology and M\"obius functions is classical, we go further by…
Various spaces of symmetries of a structure are naturally endowed with both an algebraic and a topological structure. For example, the automorphism group of a structure is, on top of being a group, a topological group when equipped with the…
In the first half of twentieth century the theory of complex analytic functions and of their zerosets was fully developed. The definition of holomorphic function has a local nature. Germs of holomorphic functions form a distinguished…
In this paper we introduce a new kind of topological space, called 'structured space', which locally resembles various kinds of algebraic structures. This can be useful, for instance, to locally study a space that cannot be globally endowed…
The commutative and homological algebra of modules over posets is developed, as closely parallel as possible to the algebra of finitely generated modules over noetherian commutative rings, in the direction of finite presentations, primary…
This chapter develops the concept of \textbf{meekly $SC^*$-normality}, a novel generalization of the classical notion of normality in topology. The proposed framework simultaneously broadens $SC^*$-normality and other established forms of…
A topos theoretic generalisation of the category of sets allows for modelling spaces which vary according to time intervals. Persistent homology, or more generally, persistence is a central tool in topological data analysis, which examines…
We develop a categorical and algebro-geometric treatment of localization for cohomological theories endowed with an open--closed recollement. Starting from a class on a space whose restriction to the open complement vanishes, we show that…
Through the subsequent discussion we consider a certain particular sort of (topological) algebras, which may substitute the `` structure sheaf algebras'' in many--in point of fact, in all--the situations of a geometrical character that…
We present a matrix-theoretic approach for studying and enumerating finite posets through their incidence representations, referred to as poset matrices. Naturally labelled posets are encoded as Boolean lower triangular matrices, allowing a…
The context of this paper is the use of formal methods for topology-based geometric modelling. Topology-based geometric modelling deals with objects of various dimensions and shapes. Usually, objects are defined by a graph-based topological…
We introduce the concept of a prenormed model of a particular kind of finitary single-sorted first-order theories, interpreted over a category with finite products. These are referred to as prealgebraic theories, for the fact that their…
We investigate a certain class of posets arising from semilattice actions. Let $S$ be a semilattice with identity. Let $S$ act on a set $C$. For $c,d\in C$ put $c\leq d$ iff there is some $s\in S$ with $ds=c$. Then $(C,\leq)$ is a poset.…
This paper establishes a link between the theory of cluster algebras and the theory of representations of partially ordered sets. We introduce a class of posets by requiring avoidance of certain types of peak-subposets and show that these…
Topos theory occupies a singular place in contemporary mathematics: born from Grothendieck's algebraic geometry, it has emerged as a unifying language for geometry, topology, algebra, and logic. This book offers a progressive introduction…