  We introduce pseudocubical objects with pseudoconnections in an arbitrary
category, obtained from the Brown-Higgins structure of a cubical object with
connections by suitably relaxing their identities, and construct a cubical
analog of the Tierney-Vogel theory of simplicial derived functors. The crucial
point in the construction is that projective precubical resolutions which are
naturally used to define our cubical derived functors possess
pseudodegeneracies and pseudoconnections. The same fact is essentially used for
proving that in the case of an additive functor between abelian categories, our
theory coincides with the classical relative theory of derived functors by
Eilenberg-Moore.
