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Eilenberg and Ganea proved the equality \(\textrm{cd}(\Gamma )=\textrm{cat}\,\Gamma \) for discrete groups \(\Gamma \) . We explore the similar equality \(\textrm{cat}(\phi )=\textrm{cd}(\phi )\) for group homomorphisms. In particular, we prove it for homomorphisms \(\phi :\Gamma \rightarrow \Lambda \) of any torsion free finitely generated nilpotent group \(\Gamma \) to an arbitrary group \(\Lambda \) . We construct a counterexample \(\psi :G\rightarrow H\) , \(\textrm{cat}(\psi )> \textrm{cd}(\psi )\) , with geometrically finite groups.
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