on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
For $\Theta_n$ the $n$-truncation of the Theta-category, an $n$-cellular set, hence a presheaf on $\Theta_n$, may be viewed as a collection of n-morphisms of an “$n$-graph” underlying an (∞,n)-category. The model structure on cellular sets (Ara) models $(\infty,n)$-categories this way. This model is referred to as n-quasicategories.
Let $\Theta_n$ be the Theta category restricted to $n$-cells. The model structure on cellular sets is the Cisinski model structure on the category of presheaves $PSh(\Theta_n)$ defined by the following localizer: (…)
In (Ara) the fibrant objects in this model structure are called $n$-quasi-categories, see Relation to quasi-categories below.
For $n = 1$ we have $\Theta_1 \simeq \Delta$ is the simplex category; and the model structure on 1-cellular sets reproduces the model structure for quasi-categories. (Ara, theorem 5.26)
The model structure on $\Theta_n$-cellular sets is Quillen equivalent to that ∞-n spaces. (Ara, theorem 7.4).
Last revised on October 8, 2013 at 20:35:15. See the history of this page for a list of all contributions to it.