Related papers: Rigidification and the Coherent Nerve for Enriched…

We collect in one place a variety of known and folklore results in enriched model category theory and add a few new twists. The central theme is a general procedure for constructing a Quillen adjunction, often a Quillen equivalence, between…

In this note we consider partial model categories, by which we mean relative categories that satisfy a weakened version of the model category axioms involving only the weak equivalences. More precisely, a partial model category will be a…

In this paper we establish Koszul duality between dg categories and a class of curved coalgebras, generalizing the corresponding result for dg algebras and conilpotent curved coalgebras. We show that the normalized chain complex functor…

We prove the compatibility between the suspension construction and the complicial nerve of $\omega$-categories. As a motivating application, we produce a Quillen pair between the models of $(\infty,n)$-categories given by Rezk's complete…

For any category ${\mathcal E}$ and monad $T$ thereon, we introduce the notion of $T$-simplicial object in ${\mathcal E}$. Any $T$-category in the sense of Burroni induces a $T$-simplicial object as its nerve. This nerve construction…

We give sufficient conditions for the existence of a Quillen model structure on small categories enriched in a given monoidal model category. This yields a unified treatment for the known model structures on simplicial, topological, dg- and…

We show that the homotopy theory of strict 2-categories embeds in that of $(\infty,2)$-categories in the form of 2-precomplicial sets. More precisely, we construct a nerve-categorification adjunction that is a Quillen pair between Lack's…

We study presheaves on semicategories enriched in a quantaloid: this gives rise to the notion of regular presheaf. A semicategory is regular when its representable presheaves are regular, and its regular presheaves then constitute an…

Triangulated categories arising in algebra can often be described as the homotopy category of a pretriangulated dg-category, a category enriched in chain complexes with a natural notion of shifts and cones that is accessible with all the…

We establish a Quillen equivalence between the homotopy theories of equivariant Segal operads and equivariant simplicial operads with norm maps. Together with previous work, we further conclude that the homotopy coherent nerve is a…

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…

For a discrete colored operad $P$, we construct an adjunction between the category of dendroidal sets over the nerve of $P$ and the category of simplicial $P$-algebras, and prove that when $P$ is $\Sigma$-free it establishes a Quillen…

A modern insight due to Quillen, which is further developed by Lurie, asserts that many cohomology theories of interest are particular cases of a single construction, which allows one to define cohomology groups in an abstract setting using…

We introduce the notion of an exact dg category, which is a simultaneous generalization of the notions of exact category in the sense of Quillen and of pretriangulated dg category in the sense of Bondal--Kapranov. It is also a differential…

We introduce the notion of weighted limit in an arbitrary quasi-category, suitably generalizing ordinary limits in a quasi-category, and classical weighted limits in an ordinary category. This is accomplished by generalizing Joyal's…

We prove that four different ways of defining Cartesian fibrations and the Cartesian model structure are all Quillen equivalent: On marked simplicial sets, on bisimplicial spaces, on bisimplicial sets, on marked simplicial spaces. The main…

We establish a Quillen equivalence relating the homotopy theory of Segal operads and the homotopy theory of simplicial operads, from which we deduce that the homotopy coherent nerve functor is a right Quillen equivalence from the model…

We construct a new model structure on the category of dg presheaves over a topological space $X$, obtained through the right Bousfield localization of the local projective model structure. The motivation for this construction arises from…

We continue the study of enriched infinity categories, using a definition equivalent to that of Gepner and Haugseng. In our approach enriched infinity categories are associative monoids in an especially designed monoidal category of…

We prove the existence of a model structure on the category of stratified simplicial sets whose fibrant objects are precisely $n$-complicial sets, which are a proposed model for $(\infty,n)$-categories, based on previous work of Verity and…