The principal aim of this article is to prove that the cellular category Θ of Joyal is a strict test category in Grothendieck's sense. In particular, this implies that cellular sets, presheaves on Θ, modelize homotopy types in a very precise sense, and this in a way compatible with finite cartesian products. To achieve this, we develop a formalism of "décalages" on a small category, or on a functor, of independent interest and having other applications. © 2010 Elsevier B.V.
Cisinski, D. C., & Maltsiniotis, G. (2011). La catégorie Θ de Joyal est une catégorie test. Journal of Pure and Applied Algebra, 215(5), 962–982. https://doi.org/10.1016/j.jpaa.2010.07.003