  The following inequality
  \cat X\le \cat Y+\lceil\frac{hd(X)-r}{r+1}\rceil
  holds for every locally trivial fibration between $ANE$ spaces $f:X\to Y$
which admits a section and has the $r$-connected fiber where $hd(X)$ is the
homotopical dimension of $X$. We apply this inequality to prove that
  \cat X\le \lceil\frac{\dim X-1}{2}\rceil+cd(\pi_1(X))
  for every complex $X$ with $cd(\pi_1(X))\le 2$.
