Relative sectional category revisited (Q6617245)

For a fibration \(p \colon E \to B\), the \textit{sectional category} is defined [\textit{S. Schwarz}, Transl., Ser. 2, Am. Math. Soc. 55, 49--140 (1962; Zbl 0178.26202)] as the smallest integer \(k\geq 0\) such that \(B\) admits a cover composed of \(k + 1\) open subsets, each of which has a section of \(p\). It serves as a generalization of the Lusternik--Schnirelmann category.\N\N\textit{J. González} et al. introduced in [Q. J. Math. 70, No. 4, 1209--1252 (2019; Zbl 1496.55003)] the notion of \textit{relative sectional category} of \(p\) with respect to a continuous map \(f\colon X \to B\) as the sectional category of the pull-back \(f^*p\) of \(p\) along \(f\). The relative sectional category unifies several homotopic numerical invariants found in recent literature.\N\NThe paper under review aims to further explore this invariant by employing the homotopical pullback construction to study maps that are not necessarily fibrations. It also provides an axiomatized characterization reminiscent of the Whitehead-Ganea framework, where the join construction [\textit{J.-P. Doeraene}, Bull. Belg. Math. Soc. - Simon Stevin 5, No. 1, 15--37 (1998; Zbl 0918.55004)] plays a central role. Finally, it studies a generalized version of the relative sectional category, considering non-open coverings, and shows that equality holds if all involved spaces have the homotopy type of a CW-complex, or, equivalently, an ANR space.