Related papers: Rigidification of higher categorical structures

We give a complete and careful proof of Quillen's theorem on the existence of the standard model category structure on the category of topological spaces. We do not assume any familiarity with model categories.

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 this paper we complete a chain of explicit Quillen equivalences between the model category for $\Theta_{n+1}$-spaces and the model category of small categories enriched in $\Theta_n$-spaces. The Quillen equivalences given here connect…

Much research has been done on structures equivalent to topological or simplicial groups. In this paper, we consider instead simplicial monoids. In particular, we show that the usual model category structure on the category of simplicial…

We give a new constructive proof of homotopy canonicity for homotopy type theory (HoTT). Canonicity proofs typically involve gluing constructions over the syntax of type theory. We instead use a gluing construction over a "strict Rezk…

Higher category theory is an exceedingly active area of research, whose rapid growth has been driven by its penetration into a diverse range of scientific fields. Its influence extends through key mathematical disciplines, notably homotopy…

This thesis is devoted to the proof of a theorem showing the existence of a closed model category structure for weakly enriched categories. It requires first of all the definitions of weakly enriched categories and equivalences of weakly…

We give a new construction for rigidifying a quasi-category into a simplicial category, and prove that it is weakly equivalent to the rigidification given by Lurie. Our construction comes from the use of necklaces, which are simplicial sets…

We construct a model category structure on the category of diffeological spaces which is Quillen equivalent to the model structure on the category of topological spaces based on the notions of Serre fibrations and weak homotopy…

Let M be a compact manifold with boundary. In this paper, we discuss some rigidity theorems of metrics in a same conformal class that fixes the boundary and satisfy certain integral conditions on the the scalar curvatures and the mean…

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…

Given a family of model categories $\cal E \to \cal R$ over a Reedy category, we outline a set of conditions which lead to the existence of a Reedy model structure on the category of sections ${\sf Sect}(\cal R, \cal E)$. We prove that for…

In this paper we present a rigidity theorem for locally isometric hypersurfaces with a curvature restriction in de Sitter space. This is an analogue to the case for Riemannian space forms given by Guan and Shen in [5].

Let $\bf C$ be a coreflective subcategory of a cofibrantly generated model category $\bf D$. In this paper we show that under suitable conditions $\bf C$ admits a cofibrantly generated model structure which is left Quillen adjunct to the…

Let $I$ be a small category with finite dimensional nerve, and $X\colon I\to Cat$ a diagram of small categories. We show that, under a "Reedy quasi-fibrancy condition", the homotopy limit of the geometric realization of $X$ is itself the…

We show that for a monoidal model category $\M=(\ul{M}, \otimes, I)$, certain co-Segal $\M$-categories are equivalent to strict ones.

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…

We introduce the notion of a logical model category which is a Quillen model category satisfying some additional conditions. Those conditions provide enough expressive power that one can soundly interpret dependent products and sums in it.…

We produce a direct Quillen equivalence between two models of $(\infty,2)$-categories: the complete Segal $\Theta_2$-spaces due to Rezk and the $2$-complicial sets due to Verity.

We obtain a rigidity phenomena of rational cohomology automorphisms of certain homogeneous spaces, in the presence of external cohomology classes arising from spaces with trivial cup product in rational cohomology algebra. We classify…