nLab sectional category

Definition

Definition

A Hurewicz fibration $f \colon X \to Y$ has sectional category $\le n$ if there is an open cover of $Y$ by at most $n$ sets $U\subseteq Y$ equipped with local sections $s\colon U\to X$ of $f$. The sectional category or Schwarz genus of $f$ is the least $n$ such that $X$ has sectional category $\le n$.

Remark

As with the Lusternik–Schnirelmann category, parts of the literature use an alternative definition, according to which the sectional category is one less than the definition used here.

Remark

There are several inequivalent definitions of sectional category for arbitrary maps: James (1978: 342) simply extends the definition to any space $f \colon X \to Y$ over $Y$, whereas e.g. Moraschini–Murillo (2016: 23) replace local sections in the definition by local sections in the homotopy category (which is equivalent for Hurewicz fibrations by the homotopy lifting property).

Properties

• If $f \colon X \to Y$ is a fibration with $Y$ paracompact Hausdorff (so that every open cover of $Y$ is numerable), then $f$ has sectional category $\le n$ iff the $n$-fold fiberwise join $f^{\star n} \colon X \star_Y \dots \star_Y X \to Y$ (with Milnor’s topology) has a global section.

• For path-connected spaces $X$ and $x_0\in X$, the Lusternik–Schnirelmann category of $X$ is exactly the sectional category of the path fibration $ev_1 \colon \{\gamma \in X^I : \gamma(0)=x_0\}\to X$.

References

• I.M. James, On category, in the sense of Lusternik–Schnirelmann, Topology 17 4 (1978) 331–348 [doi:10.1016/0040-9383(78)90002-2]

• Marco Moraschini, Aniceto Murillo, Abstract sectional category in model structures on topological spaces, Topology and its Applications 199 (2019) 23–31 [doi:10.1016/j.topol.2015.12.002]