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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 ofGrothendieck. This implies that presheaves on \tilde\Theta model homotopy typesin 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 usto compare weak equivalences of presheaves on \tilde\Theta to weak equivalencesof presheaves on \Theta. Our proofs apply to other categories analogous to\Theta.

The groupoidal analogue Theta~ to Joyal's category Theta is a test category