Baues-Wirsching Cohomology and Svarc Genus in Small Categories

TL;DR

This paper establishes a lower bound for the homotopic sectional category in small categories using Baues-Wirsching cohomology, extending classical inequalities and providing a new computational approach.

Contribution

It introduces a categorical extension of Svarc inequalities and a simplified cochain complex for Baues-Wirsching cohomology to improve computational efficiency.

Findings

Length of cup product bounds homotopic sectional category

Extension of classical Svarc inequalities to categories

New reduced cochain complex for Baues-Wirsching cohomology

Abstract

We prove that for a bifibration P between small categories, the lenght of the cup product in the kernel of the induced morphism in the Baues-Wirsching cohomology with coefficients in any natural system is a lower bound for the homotopic sectional category (also called Svarc genus). Our results extend classical Svarc type inequalties to the categorical setting and introduce a computationally efficient method via a reduced cochain complex for Baues-Wirching cohomology.