Related papers: Thomason-Type Model Structures on Simplicial Compl…

In this paper we prove that for any simplicial set $B$, there is a Quillen equivalence between the covariant model structure on $\mathbf{S}/B$ and a certain localization of the projective model structure on the category of simplicial…

In this paper we put a cofibrantly generated model category structure on the category of small simplicial categories. The weak equivalences are a simplicial analogue of the notion of equivalence of categories.

This expository article sets forth a self-contained and purely algebraic proof of a deep result of Quillen stating that the category of simplicial commutative algebras over a commutative ring is a model category. This is accomplished by…

We generalize the small object argument in order to allow for its application to proper classes of maps (as opposed to sets of maps in Quillen's small object argument). The necessity of such a generalization arose with appearance of several…

We give a fully constructive proof that there is a proper cartesian $\omega$-combinatorial model structure on the category of simplicial sets, whose generating cofibrations and trivial cofibrations are the usual boundary inclusion and horn…

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 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,…

A model structure on the category of (small) bigroupoids and pseudofunctors is constructed. In this model structure, every object is cofibrant. In order to keep certain calculations of manageable size, a coherence theorem for bigroupoids…

A simplicial set is said to be non-singular if its non-degenerate simplices are embedded. Let $sSet$ denote the category of simplicial sets. We prove that the full subcategory $nsSet$ whose objects are the non-singular simplicial sets…

We construct a category $\OrdFor$ as an arboreal extension of $\Delta_{\mathrm{epi}}\subseteq\Delta$, whose morphisms are ordered forests composed by grafting. We define a full functor $\pi\colon \OrdFor\to\Delta_{\mathrm{epi}}^{op}$…

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…

We construct a model structure on the category of cubical sets with connections whose cofibrations are the monomorphisms and whose fibrant objects are defined by the right lifting property with respect to inner open boxes, the cubical…

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…

We put a model structure on the category of categories internal to simplicial sets whose weak equivalences are reflected by the nerve functor to bisimplicial sets with Rezk's model structure. This model structure is shown to be Quillen…

We provide examples of inductive fibrant replacements in fibrantly generated model categories constructed as Postnikov towers. These provide new types of arguments to compute homotopy limits in model categories. We provide examples for…

We prove, without set theoretic assumptions, that every locally presentable category C endowed with a tractable cofibrantly generated class of cofibrations has a unique minimal (or left induced) Quillen model structure. More generally, for…

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…

The category of small covariant functors from simplicial sets to simplicial sets supports the projective model structure. In this paper we construct various localizations of the projective model structure and also give a variant for…

Diagrammatic sets admit a notion of internal equivalence in the sense of coinductive weak invertibility, with similar properties to its analogue in strict $\omega$-categories. We construct a model structure whose fibrant objects are…

In the first part of the paper, we prove that the category of diffeological spaces does not admit a model structure transferred via the smooth singular complex functor from simplicial sets, resolving in the negative a conjecture of…