Abstract sectional category in model structures on topological spaces

TL;DR

This paper investigates the abstract sectional category within various model structures on topological spaces, demonstrating under certain conditions that it aligns with the classical concept, impacting related invariants.

Contribution

It establishes the equivalence of the abstract sectional category across multiple model structures with the classical notion, extending to related invariants.

Findings

All considered model structures yield the same abstract sectional category under certain conditions.

The classical Lusternik-Schnirelmann category and topological complexity coincide with their abstract counterparts.

Results unify different approaches to categorical invariants in topological spaces.

Abstract

We study the behavior of the abstract sectional category in the Quillen, the Strom and the Mixed proper model structures on topological spaces and prove that, under certain reasonable conditions, all of them coincide with the classical notion. As a result, the same conclusions hold for the abstract Lusternik-Schnirelmann category and the abstract topological complexity of a space.