Probing L-S category with maps

For any map $X\xrightarrow{f}Y$, we introduce two new homotopy invariants, ${\text{dcat}}\;f$ and ${\text{rcat}}\;f$. The classical category ${\text{cat}}\;f$ is a lower bound for both, while ${\text{dcat}}\;f \leq {\text{cat}}\;X$ and ${\text{rcat}}\;f \leq {\text{cat}}\;Y$. When $Y$ is an Eilenberg-Mac Lane space, $f$ represents a cohomology class and ${\text{dcat}}\;f$ often gives a good estimate for ${\text{cat}}\;X$. We prove that if $\Omega \in {H^n}(M;\mathbb{Z})$ is the fundamental class of a compact, simply connected $n$-manifold, then ${\text{dcat}}\;\Omega = {\text{cat}}\;M$. Similarly, when $X$ is sphere, then $f$ is a homotopy class and while ${\text{cat}}\;f = 1$, ${\text{rcat}}\;f$ can be a good approximation to ${\text{cat}}\;Y$. We show that if $\alpha \in {\pi _2}(\mathbb{C}{P^n})$ is nonzero, then ${\text{rcat}}\;\alpha = n$. Rational analogues are introduced and we prove that for $u \in {H^\ast}(X;\mathbb{Q})$, ${\text{dcat}_0}\;u = 1 \Leftrightarrow {u^2} = 0$ and $u$ is spherical.