Sectional number of a morphism

TL;DR

This paper generalizes the concept of sectional number from topology to category theory, introducing new invariants and bounds, and explores their properties and applications, including connections to famous conjectures.

Contribution

It extends the classical sectional number and sectional category to morphisms in categories with covers, establishing invariance, bounds, and cohomological methods.

Findings

Introduces a categorical sectional number extending classical notions.

Provides upper and lower bounds using LS category and cohomology.

Demonstrates applications including reformulations of twin prime and Goldbach conjectures.

Abstract

The genus of a fibration was introduced by Schwarz in 1962. Given a continuous map $g:A\to B$, the usual sectional number $\text{sec}_u(g)$ is the least integer~$m$ such that $B$ can be covered by $m$ open subsets, each of which admits a local section of~$g$. Likewise, the sectional category $\text{secat}(g)$ is the least integer~$m$ such that $B$ can be covered by $m$ open subsets, each of which admits a local homotopy section of~$g$. In the case that $g$ is a fibration, the usual sectional number and the sectional category of $g$ coincide with the Schwarz's genus of $g$. In this paper, we introduce a notion of sectional number for a morphism in a category with covers, which extends the usual sectional number and sectional category (and, of course, Schwarz's notion). We study the invariance property, the behaviour under weak pullbacks and continuous functors, and present upper bounds in terms of a notion of LS category and projective covering number for an object. In addition, we introduce a notion of multiplicative cohomology theory on a category with covers, and we use it to present a cohomological lower bound for the sectional number. Several examples of this invariant are presented to support this theory. For instance, our sectional number yields a family of new numerical invariants in Top; we also express the twin prime conjecture and Goldbach's conjecture in terms of sectional numbers in the category of sets. Our unified concept is relevant in this new setting because it provides a foundation for sectional theory in category theory and builds bridges between new areas in mathematics.