相关论文: Homotopy Theory of Non-singular Simplicial Sets

We construct a model structure on the category of ordered simplicial complexes, Quillen equivalent to the standard model structure on simplicial sets. This shows that simplicial complexes, which are fully combinatorial in nature, provide a…

We show that any closed model category of simplicial algebras over an algebraic theory is Quillen equivalent to a proper closed model category. By ``simplicial algebra'' we mean any category of algebras over a simplicial algebraic theory,…

We produce a highly structured way of associating a simplicial category to a model category which improves on work of Dwyer and Kan and answers a question of Hovey. We show that model categories satisfying a certain axiom are Quillen…

A simplicial set is non-singular if the representing map of each non-degenerate simplex is degreewise injective. The simplicial mapping set $X^K$ has $n$-simplices given by the simplicial maps $\Delta[n] \times K \to X$. We prove that $X^K$…

We construct a discrete model of the homotopy theory of $S^1$-spaces. We define a category $\sP$ with objects composed of a simplicial set and a cyclic set along with suitable compatibility data. $\sP$ inherits a model structure from the…

A general method for lifting weak factorization systems in a category S to model category structures on simplicial objects in S is described, analogously to the lifting of cotorsion pairs in Abelian categories to model category structures…

Let $D$ be a large category which is cocomplete. We construct a model structure (in the sense of Quillen) on the category of small functors from $D$ to simplicial sets. As an application we construct homotopy localization functors on the…

We construct a cofibrantly generated Quillen model structure on the category of small n-fold categories and prove that it is Quillen equivalent to the standard model structure on the category of simplicial sets. An n-fold functor is a weak…

As observed recently by various people the topos $\mathbf{sSet}$ of simplicial sets appears as essential subtopos of a topos $\mathbf{cSet}$ of cubical sets, namely presheaves over the category $\mathbf{FL}$ of finite lattices and monotone…

In this article, we construct a cofibrantly generated model structure on the category of spaces stratified over a fixed poset, and show that it is Quillen-equivalent to a category of diagrams of simplicial sets. Then, considering all those…

We study Quillen's model category structure for homotopy of simplicial objects in the context of Janelidze, Marki and Tholen's semi-abelian categories. This model structure exists as soon as the base category A is regular Mal'tsev and has…

In a previous work, by extending the classical Quillen construction to the non-simply connected case, we have built a pair of adjoint functors, 'model' and 'realization', between the categories of simplicial sets and complete differential…

Both simplicial sets and simplicial spaces are used pervasively in homotopy theory as presentations of spaces, where in both cases we extract the "underlying space" by taking geometric realization. We have a good handle on the category of…

Quillen showed that simplicial sets form a model category (with appropriate choices of three classes of morphisms), which organized the homotopy theory of simplicial sets. His proof is very difficult and uses even the classification theory…

We show that a category $\mathscr{M}$ equipped with a model structure defined by a proper, locally small class of orbits $\mathscr{O}$ is Quillen equivalent to the category of small relative presheaves…

In previous work, we showed that there are appropriate model category structures on the category of simplicial categories and on the category of Segal precategories, and that they are Quillen equivalent to one another and to Rezk's complete…

We construct a "diagonal" cofibrantly generated model structre on the category of simplicial objects in the category of topological categories sCat_{Top}, which is the category of diagrams [\Delta^{op}, Cat_{Top}]. Moreover, we prove that…

In Quillen's paper on rational homotopy theory, the category of 1-reduced simplicial sets is endowed with a family of model structures, the most prominent of which is the one in which the weak equivalences are the rational homotopy…

Let $G$ be a discrete group. We prove that the category of $G$-posets admits a model structure that is Quillen equivalent to the standard model structure on $G$-spaces. As is already true nonequivariantly, the three classes of maps defining…

Given any model category, or more generally any category with weak equivalences, its simplicial localization is a simplicial category which can rightfully be called the "homotopy theory" of the model category. There is a model category…