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We prove that the marked triangulation functor from the category of marked cubical sets equipped with a model structure for ($n$-trivial, saturated) comical sets to the category of marked simplicial set equipped with a model structure for…

In a previous paper, we showed that a discrete version of the $S_\bullet$-construction gives an equivalence of categories between unital 2-Segal sets and augmented stable double categories. Here, we generalize this result to the homotopical…

This paper is the third paper of a series devoted to higher dimensional transition systems. The preceding paper proved the existence of a left determined model structure on the category of cubical transition systems. In this sequel, it is…

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…

We construct a cocycle model for complex analytic equivariant elliptic cohomology that refines Grojnowski's theory when the group is connected and Devoto's when the group is finite. We then construct Mathai--Quillen type cocycles for…

We give a new construction of the model structure on the category of simplicial sets for homotopy $n$-types, originally due to Elvira-Donazar and Hernandez-Paricio, using a right transfer along the coskeleton functor. We observe that an…

We construct a model of type theory enjoying parametricity from an arbitrary one. A type in the new model is a semi-cubical type in the old one, illustrating the correspondence between parametricity and cubes. Our construction works not…

We show that weak monoidal Quillen equivalences induce equivalences of symmetric monoidal $\infty$-categories with respect to the Dwyer-Kan localization of the symmetric monoidal model categories. The result will induce a Dold-Kan…

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 prove, under mild assumptions, that a Quillen equivalence between symmetric monoidal model categories gives rise to a Quillen equivalence between their model categories of (non-symmetric) operads, and also between model categories of…

We define a computational type theory combining the contentful equality structure of cartesian cubical type theory with internal parametricity primitives. The combined theory supports both univalence and its relational equivalent, which we…

We extend Schwede's work on the unstable global homotopy theory of orthogonal spaces and $\mathcal{L}$-spaces to the category of $*$-modules (i.e., unstable $S$-modules). We prove a theorem which transports model structures and their…

We study properties of the cubical Joyal model structures on cubical sets by means of a combinatorial construction which allows for convenient comparisons between categories of cubical sets with and without symmetries. In particular, we…

Let $I$ be a small category, $\mathcal{C}$ be the category $\mathbf{Cat}$, $\mathbf{Ac}$ or $\mathbf{Pos}$ of small categories, acyclic categories, or posets, respectively. Let $\mathcal{O}$ be a locally small class of objects in…

In this paper we construct new categorical models for the identity types of Martin-L\"of type theory, in the categories Top of topological spaces and SSet of simplicial sets. We do so building on earlier work of Awodey and Warren, which has…

A Quillen model structure on the category Gray-Cat of Gray-categories is described, for which the weak equivalences are the triequivalences. It is shown to restrict to the full subcategory Gray-Gpd of Gray-groupoids. This is used to provide…

In this paper, we construct a model structure for $(\infty,1)$-categories on the category of simplicial spaces, whose fibrant objects are the Segal spaces. In particular, we show that it is Quillen equivalent to the models of…

We obtain a class of locally symmetric Kaehler Einstein structures on the cotangent bundle of a Riemannian manifold of negative sectional curvature. Similar results are obtained in the case of a Riemannian manifold of positive sectional…

Let $R$ be a left-Gorenstein ring. We show that there is a Quillen equivalence between singular contraderived model category and singular coderived model category. Consequently, an equivalence between the homotopy category of exact…

By using homotopy transfer techniques in the context of rational homotopy theory, we show that if $C$ is a coalgebra model of a space $X$, then the $A_\infty$-coalgebra structure in $H_*(X;\mathbb{Q})\cong H_*(C)$ induced by the higher…