For any map X -> Y , we introduce two new homotopy invariants, dcat/ and rcat/. The classical category cat/ is a lower bound for both, while dcat / < cat X and rcat f < cat Y . When Y is an Eilenberg-Mac Lane space, / represents a cohomology class and dcat / often gives a good estimate for cat X . We prove that if SI e H" (M ; Z) is the fundamental class of a compact, simply connected «-manifold, then dcat Í2 = cat M . Similarly, when X is sphere, then / is a homotopy class and while cat / = 1

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... rcat / can be a good approximation to cat Y . We show that if a e 712 (CP") is nonzero, then rcat a = n . Rational analogues are introduced and we prove that for u e H*(X ; Q), dcat0 «=lft»2 = 0 and u is spherical.