Categorical sequences

Abstract

We define and study the categorical sequence of a space, which is a new formalism that streamlines the computation of the Lusternik–Schnirelmann category of a space $X$ by induction on its CW skeleta. The $k^{th}$ term in the categorical sequence of a CW complex $X$, ${\sigma }_X\left(k\right)$, is the least integer $n$ for which ${cat}_X\left(X_n\right)\ge k$. We show that ${\sigma }_X$ is a well-defined homotopy invariant of $X$. We prove that ${\sigma }_X\left(k+l\right)\ge {\sigma }_X\left(k\right)+{\sigma }_X\left(l\right)$, which is one of three keys to the power of categorical sequences. In addition to this formula, we provide formulas relating the categorical sequences of spaces and some of their algebraic invariants, including their cohomology algebras and their rational models; we also find relations between the categorical sequences of the spaces in a fibration sequence and give a preliminary result on the categorical sequence of a product of two spaces in the rational case. We completely characterize the sequences which can arise as categorical sequences of formal rational spaces. The most important of the many examples that we offer is a simple proof of a theorem of Ghienne: if $X$ is a member of the Mislin genus of the Lie group $Sp\left(3\right)$, then $cat\left(X\right)=cat\left(Sp\left(3\right)\right)$.