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

Abstract

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

41 pages, v2: references added, Remark 7.3 added, v3: metadata updated