相关论文: Simplicial sets in topology, category theory, and …

We consider simplicial sets equipped with a notion of smallness, and observe that this slight "topological" extension of the "algebraic" simplicial language allows a concise reformulation of a number of classical notions in topology, e.g.…

This is an expository introduction to simplicial sets and simplicial homotopy theory with particular focus on relating the combinatorial aspects of the theory to their geometric/topological origins. It is intended to be accessible to…

In these self-contained low prerequisite introductory notes we first present (in part 1) basic concepts of set theory and algebra without explicit category theory. We then present (in part 2) basic category theory involving a somewhat…

In the field of mathematics, a purely combinatorial equivalent to a simplicial complex, or more generally, a down-set, is an abstract structure known as a family of sets. This family is closed under the operation of taking subsets, meaning…

The development of mathematics has been characterized by the increasing interconnectivity of seemingly separate disciplines. Such interplay has been facilitated by a massive development in formalism; category theory has provided a common…

We provide a short introduction to the field of topological data analysis and discuss its possible relevance for the study of complex systems. Topological data analysis provides a set of tools to characterise the shape of data, in terms of…

The aim of these notes is to introduce the intuition motivating the notion of a "complicial set", a simplicial set with certain marked "thin" simplices that witness a composition relation between the simplices on their boundary. By varying…

Both simplicial sets and simplicial spaces are used pervasively in homotopy theory as presentations of spaces, where in both cases we extract the "underlying space" by taking geometric realization. We have a good handle on the category of…

These notes were originally developed as lecture notes for a category theory course. They should be well-suited to anyone that wants to learn category theory from scratch and has a scientific mind. There is no need to know advanced…

Using ideas from synthetic topology, a new approach to descriptive set theory is suggested. Synthetic descriptive set theory promises elegant explanations for various phenomena in both classic and effective descriptive set theory.…

We provide a formal introduction into the classic theorems of general topology and its axiomatic foundations in set theory. In this second part we introduce the fundamental concepts of topological spaces, convergence, and continuity, as…

Simplicial complexes are a versatile and convenient paradigm on which to build all the tools and techniques of the logic of knowledge, on the assumption that initial epistemic models can be described in a distributed fashion. Thus, we can…

Questions of set-theoretic size play an essential role in category theory, especially the distinction between sets and proper classes (or small sets and large sets). There are many different ways to formalize this, and which choice is made…

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…

The paper is devoted to an approach to the bounded cohomology theory based on the theories of simplicial sets and Postnikov systems. In particular, the main results of the bounded cohomology theory of topological spaces are extended to…

We introduce a notion of complexity of diagrams (and in particular of objects and morphisms) in an arbitrary category, as well as a notion of complexity of functors between categories equipped with complexity functions. We discuss several…

We review a collection of models of random simplicial complexes together with some of the most exciting phenomena related to them. We do not attempt to cover all existing models, but try to focus on those for which many important results…

This paper provides a bundle perspective to contextuality by introducing new categories of contextuality scenarios based on bundles of simplicial complexes and simplicial sets. The former approach generalizes earlier work on the…

In this paper, a quantum computational framework for algebraic topology based on simplicial set theory is presented. This extends previous work, which was limited to simplicial complexes and aimed mostly to topological data analysis. The…

Metaphysical interpretations of set theory are either inconsistent or incoherent. The uses of sets in mathematics actually involve three distinct kinds of collections (surveyable, definite, and heuristic), which are governed by three…