  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(k), is the least integer n for
which cat_X(X_n) >= k. We show that \sigma_X is a well-defined homotopy
invariant of X.
  We prove that \sigma_X(k+l) >= \sigma_X(k) + \sigma_X(l), 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(3), then cat(X) = cat(Sp(3)).
