Lusternik–Schnirelmann category of skeleta

In this paper we make explicit the relationship between the category of a connected CW complex and the category of its skeleta. In particular we prove that if Y is a connected non-contractible CW complex, then catYk⩽catY for each k-skeleton Yk. On the other hand, we prove that cat0(Y)=limk→∞ cat0(Yk), where cat0 denotes the rational category of the space.