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

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

2010-12-20

arxiv.org/abs/1012.4319v3

Mathematics

Algebraic Topology

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

Scientific paper

10.1007/s10485-011-9255-2

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.

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