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
by induction on its
CW skeleta. The
term in the categorical sequence of a CW complex
,
, is the least
integer
for which
. We show that
is a well-defined
homotopy invariant of
.
We prove that
,
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
is a member of the Mislin genus of the Lie group
, then
.
