About sectional category of the Ganea maps

TL;DR

This paper computes the sectional and relative categories of Ganea maps in terms of the original map's categories, establishing inequalities and providing examples of strictness.

Contribution

It introduces the relative category of order k and derives formulas and inequalities relating secat and relcat for Ganea maps.

Findings

secatt(g_k) = floor(secat(f)/(k+1))

relcat(g_k) = floor(relcat_k(f)/(k+1))

established inequalities linking secat and relcat of various orders

Abstract

We first compute the James' sectional category (secat) of the Ganea map g_k of any map f in terms of the sectional category of f: We show that secat(g_k) is the integer part of secat(f)/(k+1). Next we compute the relative category (relcat) of g_k. In order to do this, we introduce the relative category of order k (relcat_k) of a map and show that relcat(g_k) is the integer part of relcat_k(f)/(k+1). Then we establish some inequalities linking secat and relcat of any order: We show that secat(f) <= relcat_k(f) <= secat(f) + k + 1 and relcat_k(f) <= relcat_(k+1)(f) <= relcat_k(f) + 1. We give examples that show that these inequalities may be strict.