We give conditions which determine if $\cat$ of a map go up when extending over a cofibre. We apply this to reprove a result of Roitberg giving an example of a CW complex $Z$ such that $\cat(Z)=2$ but every skeleton of $Z$ is of category $1$. We also find conditions when $\cat (f\times g) < \cat(f) + \cat(g)$. We apply our result to show that under suitable conditions for rational maps $f$, $\mcat(f) < \cat(f)$ is equivalent to $\cat(f) = \cat (f\times \id_{S^n})$. Many examples with $\mcat(f) < \cat(f)$ satisfying our conditions are constructed. We also answer a question of Iwase by constructing $p$-local spaces $X$ such that $\cat (X\times S^1) = \cat(X) = 2$. In fact for our spaces and every $Y \not\simeq *$, $\cat (X\times Y) \leq \cat(Y) +1 < \cat(Y) + \cat(X)$. We show that this same $X$ has the property $\cat(X) = \cat (X\times X) = \cl (X\times X) = 2$.