  We introduce the groupoidal analogue \tilde\Theta to Joyal's cell category
\Theta and we prove that \tilde\Theta is a strict test category in the sense of
Grothendieck. This implies that presheaves on \tilde\Theta model homotopy types
in a canonical way. We also prove that the canonical functor from \Theta to
\tilde\Theta is aspherical, again in the sense of Grothendieck. This allows us
to compare weak equivalences of presheaves on \tilde\Theta to weak equivalences
of presheaves on \Theta. Our proofs apply to other categories analogous to
\Theta.
