Related papers: A characterization of fibrant Segal categories
We use fibrations of complete Segal spaces to construct four complete Segal spaces: Reedy fibrant simplicial spaces, Segal spaces, complete Segal spaces, and spaces. Moreover, we show each one comes with a universal fibration that…
In this paper, we study properties of maps between fibrant objects in model categories. We give a characterization of weak equivalences between fibrant object. If every object of a model category is fibrant, then we give a simple…
We discuss right fibrations in the $\infty$-categorical context of Segal objects in a category V and prove some basic results about these.
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…
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 introduce the dendroidal analogs of the notions of complete Segal space and of Segal category, and construct two appropriate model categories for which each of these notions corresponds to the property of being fibrant. We prove that…
A relative category is a category with a chosen class of weak equivalences. Barwick and Kan produced a model structure on the category of all relative categories, which is Quillen equivalent to the Joyal model structure on simplicial sets…
We define a model category structure on a slice category of simplicial spaces, called the "Segal group action" structure whose fibrant-cofibrant objects may be viewed as representing spaces $X$ with a coherent action of a given Segal group…
We show that the category of categories fibred over a site is a generalized Quillen model category in which the weak equivalences are the local equivalences and the fibrant objects are the stacks, as they were defined by J. Giraud. The…
We define complete Segal objects, which play the role of internal higher category objects. Then we study them using representable Cartesian fibrations, in particular defining adjunctions and limits of complete Segal objects. Finally we use…
We prove that the category of (strictly unital) A$_\infty$-categories, linear over a commutative ring $R$, with strict A$_\infty$-morphisms has a cofibrantly generated model structure. In this model structure every object is fibrant and the…
We define a new model structure on the category of small categories, which is intimately related to the notion of coverings and fundamental groups of small categories. Fibrant objects in the model structure coincide with groupoids, and the…
We use the complete Segal approach to the theory of Cartesian fibrations to define and study representable Cartesian fibrations, generalizing representable right fibrations which have played a key role in $\infty$-category theory. In…
In this article, we develop a new model for the category of dg-categories. Following Rezk's example in the case of classic Segal spaces, we define dg-Segal spaces: functors between free dg-categories of finite type and simplicial spaces to…
We show that the fibrant objects in the minimal model structure on the category of simplicial sets are characterized by a lifting condition with respect to maps which resemble the horn inclusions that define Kan complexes.
In this document, we develop a new model for the category of dg-categories. Following Rezk's example in the case of classic Segal spaces, we define dg-Segal spaces: functors between free dg-categories of finite type and simplicial spaces to…
In this note we describe conditions under which the algebras for a monad on a presheaf category equipped with some additional structure are fibrant objects in a model structure. We also prove that when these conditions are satisfied the…
We introduce a new model structure on the category of dendroidal spaces, designed to provide a further model for the homotopy theory of $\infty$-operads. This model is directly analogous to a recent construction on the category of…
In this note we prove the existence, in the category of (strictly unital) A$_{\infty}$categories, of the pullback of a (strictly unital) A$_{\infty}$functor, satisfying a particular property (denoted by F1), along any A$_{\infty}$functor.…
In this survey article, we review some conceptual approaches to the cyclic category $\Lambda$, as well as its description as a crossed simplicial group. We then give a new proof of the model structure on cyclic sets, work through the…