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

TL;DR

This paper introduces a groupoidal analogue of Joyal's category Theta, proving it is a strict test category that models homotopy types, and compares it to the original category in terms of weak equivalences.

Contribution

The paper defines the groupoidal analogue tetildeTheta of Joyal's Theta and proves it is a strict test category, establishing a canonical model for homotopy types.

Findings

tetildeTheta is a strict test category.

The canonical functor from Theta to tetildeTheta is aspherical.

Weak equivalences on tetildeTheta and Theta are comparable.

Abstract

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.