Related papers: Simplicial sets in topology, category theory, and …

We introduce a notion of discrete topological complexity in the setting of simplicial complexes, using only the combinatorial structure of the complex by means of the concept of contiguous simplicial maps. We study the links of this new…

This paper makes some preliminary observations towards an extension of current work on graphs defined on groups to simplicial complexes. I define a variety of simplicial complexes on a group which are preserved by automorphisms of the…

We provide a formal introduction into the classic theorems of general topology and its axiomatic foundations in set theory. Starting from ZFC, the exposition in this first part includes relation and order theory as well as a construction of…

Various simplicial complexes can be associated with a graph. Box complexes form an important families of such simplicial complexes and are especially useful for providing lower bounds on the chromatic number of the graph via some of their…

We begin with a context more general than set theory. The basic ingredients are essentially the object and functor primitives of category theory, and the logic is weak, requiring neither the Law of Excluded Middle nor quantification. Inside…

We give a new characterization of partial groups as a subcategory of symmetric (simplicial) sets. This subcategory has an explicit reflection, which permits one to compute colimits in the category of partial groups. We also introduce the…

We show how the framework of crossed simplicial groups may be used to provide a classification of topological field theories on open cobordism categories defined by reductions of the structure group to a planar Lie group. Such theories are…

The paper is a naive introduction to descriptive set theory. It is aimed mathematicians without a background in logic. The goal is to provide the basic facts used for applications of descriptive set theory to other areas of mathematics,…

The concept of typed topological space is introduced, for which open sets in a topology on a finite set will be assigned types (from lattice). The neighborhood system of a point, the closure and the connectedness can be defined according to…

In this article we consider the homotopy theory of stratified spaces through a simplicial point of view. We first consider a model category of filtered simplicial sets over some fixed poset $P$, and show that it is a simplicial…

We define a monoidal semantics for algebraic theories. The basis for the definition is provided by the analysis of the structural rules in the term calculus of algebraic languages. Models are described both explicitly, in a form that…

We introduce a method to reduce the study of the topology of a simplicial complex to that of a simpler one. We give some applications of this method to complexes arising from graphs. As a consequence, we answer some questions raised in…

We observe that the category of topological space, uniform spaces, and simplicial sets are all, in a natural way, full subcategories of the same larger category, namely the simplicial category of filters; this is, moreover, implicit in the…

This paper introduces a model that identifies spatial relationships for a structural analysis based on the concept of simplicial complex. The spatial relationships are identified through overlapping two map layers, namely a primary layer…

We show that any closed model category of simplicial algebras over an algebraic theory is Quillen equivalent to a proper closed model category. By ``simplicial algebra'' we mean any category of algebras over a simplicial algebraic theory,…

This article is an introduction to the basic generalized category theory used in recent work on an extension of the theory of categories and categorical logic, including parts of topos theory. We discuss functors, equivalences, natural…

We define a class of probability distributions that we call simplicial mixture models, inspired by simplicial complexes from algebraic topology. The parameters of these distributions represent their topology and we show that it is possible…

This paper describes the homology of various simplicial complexes associated to set families from combinatorial number theory, including primitive sets, pairwise coprime sets, product-free sets, and coprime-free sets. We present a condition…

Higher category theory is an exceedingly active area of research, whose rapid growth has been driven by its penetration into a diverse range of scientific fields. Its influence extends through key mathematical disciplines, notably homotopy…

The Catalan numbers are well-known to be the answer to many different counting problems, and so there are many different families of sets whose cardinalities are the Catalan numbers. We show how such a family can be given the structure of a…