Toward Foundations of Near Sets: (Pre-)Sheaf Theoretic Approach

Abstract

The formal content of near set theory can be summarised in terms of three concepts: a perceptual system, a nearness relation and a near set. Perceptual systems and different forms of nearness relations have been already successfully related to important mathematical structures (e.g., approach spaces) and described in the frameworks of general topology and category theory. However, since near sets actually do not form any regular structure, there is lack of similar results about the concept of a near set. The main goal of this paper is to fill this gap and provide a mathematical basis for near sets on a similar abstract level as it has been already done for nearness relations. However, we take in the paper a down-to-earth approach; that is, instead of seeking the richest mathematical structures which express our intuitions (call it up-to-sky approach), we present in the paper the simplest category theoretic structures which, however, are rich enough to convey our ideas. Thus, although many times we actually deal with sheaves, we speak about them as pre-sheaves. The main reason is that a sheaf (in contrast to a pre-sheaf) embodies the idea of gluing (in the very similar way like manifolds, which are obtainable by gluing open subsets of Euclidean space), which is irrelevant to the study of near sets and rough sets. Furthermore, the concept a pre-sheaf is much simpler than a sheaf and should be easily "digested" by readers who are not very familiar with category theory.

The Bipolar Logic of Feedforward and Feedback Circuits<strong> </strong>

“…The local-to-global machinery of algebraic topology seems poised to provide the perfect tools for gathering disparate neural activities into a coherent whole. To provide an example concerning the human central nervous system, presheaves offer invaluable benefits for the analysis of nervous activities: a) they generalize the local systems that are so ubiquitous in the brain areas and subareas; b) they have powerful applications to the topology of algebraic and the analytic complex varieties of singular spaces such as the nervous phase spaces (Dimca, 2004); c) they make available a suitable notion of "general coefficient systems" that could be useful for the assessment of nonlinear nervous dynamics; d) they stand for common methods of comparison between different cohomology theories of general topological spaces (Bredon, 1997); e) they provide foundations of near sets that can be used to describe brain areas and subareas (Wolski, 2013). Also, globular sets allow to analyze the stability of synchronous states in dynamical systems such as the brain (Papo and Buldú, 2019), shedding new light of the still unknown mechanisms of spike synchronization.…”

“…The orthodromic (i.e., feedforward, bottom-up) and antidromic (i.e., feedback, top-down) brain activity has been tackled via mathematical/topological weapons such as algebraic topology, near set theory and category theory (Tozzi et al, 2017a;Peters et al, 2017). Nevertheless, these theoretical approaches to nervous functions are constrained by soaring abstraction and vagueness in defining regular structures and nearness relations, thus preventing the formulation of sharp experimental previsions (Wolski 2013). The main goal of this paper is to overcome these difficulties, showing how the mathematical apparatus of presheaves /globular sets provides theoretical examples of applications in feedforward/feedback bipolar networks.…”

Measuring the nearness of layered flow graphs: Application to Content Based Image Retrieval

“…The principal difference between rough set theory and near sets is that near sets can be discovered without the approximation of sets [38]. The theory of near sets can be summarized in three simple concepts: a perceptual system, a nearness relation and a near set [46]. The CBIR task can be considered as measuring nearness between two disjoint perceptual systems (i.e., two digital images).…”

Sheaves in the Brain Elucidate the Behavior of Entrained Oscillations<strong> </strong>

“…To make an example, Northoff et al (2019) investigated the relationships (the authors call them "the sameness") between consciousness and its neural correlates in terms of category theory, composition of functions and functors. However, these theoretical approaches to the brain are constrained by soaring abstraction and vagueness in defining regular structures and nearness relations (Wolski 2013), thus preventing the formulation of sharp experimental previsions. The main goal of this paper is to overcome these difficulties, providing a mathematical framework that allows a quantifiable description and assessment of neuroscientific issues.…”

“…Presheaves offer invaluable benefits: a) they generalize the local systems that are so ubiquitous in mathematics; b) have powerful applications to the topology of algebraic and the analytic complex varieties of singular spaces (Dimca, 2004); c) make available a suitable notion of "general coefficient systems"; d) stand for common methods of comparison between different cohomology theories of general topological spaces (Bredon, 1997); provide foundations of near sets (Wolski 2013). Here we show how presheaves give us the possibility to treat neuroscientific issues in terms of category theory, in particular using composition of functions/globular sets, and to draw previsions that can be experimentally tested with the current neuro-technologies.…”