  We consider a Weyssenhoff fluid assuming that the spacetime is homogeneous
and isotropic, therefore being relevant for cosmological considerations of
gravity theories with torsion. In this paper, it is explicitely shown that the
Weyssenhoff fluids obeying the Frenkel condition or the Papapetrou-Corinaldesi
condition are incompatible with the cosmological principle, which restricts the
torsion tensor to have only a vector and an axial vector component. Moreover it
turns out that the Weyssenhoff fluid obeying the Tulczyjew condition is also
incompatible with the cosmological principle. Based on this result we propose
to reconsider a number of previous works that analysed cosmological solutions
of Einstein-Cartan theory, since their spin fluids usually did not obey the
cosmological principle.
