Related papers: Fibrations in $\infty$-category theory

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 construct examples of simply connected surfaces with genus 2 fibrations over the projective line which are of "general type" according to the definition of Campana. These fibrations have special fibres such that the minimum of the…

Let $f : X -> B$ be a projective surjective morphism between quasi-projective varieties. The goal of this paper is the study of the Chow groups of $X$ in terms of the Chow groups of $B$ and of the fibers of $f$. One of the applications…

We define and discuss lax and weighted colimits of diagrams in $\infty$-categories and show that the coCartesian fibration associated to a functor is given by its lax colimit. A key ingredient, of independent interest, is a simple…

We construct a left semi-model category of "marked strict $\infty$-categories" for which the fibrant objects are those whose marked arrows satisfy natural closure properties and are weakly invertible. The canonical model structure on strict…

In this work we propose a realization of Lurie's prediction that inner fibrations $p: X \rightarrow A$ are classified by $A$-indexed diagrams in a ``higher category" whose objects are $\infty$-categories, morphisms are correspondences…

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 develop a homotopical framework for small categories that extends classical invarints of algebraic topology to the categorical setting. Our approach is based on the construction of genuine path category, obtained trough a localization…

The purpose of this paper is to give some solutions for the classification problem in fibration theory by using the homotopy sequences of fibrations (sequences of $n$-th homotopy groups $ \pi_{n}(S,s_{o}) $ of total spaces of fibrations).…

This is an exposition of homotopical results on the geometric realization of semi-simplicial spaces. We then use these to derive basic foundational results about classifying spaces of topological categories, possibly without units. The…

We define and study notions of comprehension in $(\infty,1)$-category theory. In essence, we do so by implementing B\'{e}nabou's foundations of naive category theory in a univalent meta-theory. In particular, we develop natural…

Grothendieck's theory of fibred categories establishes an equivalence between fibred categories and pseudo functors. It plays a major role in algebraic geometry and categorical logic. This paper aims to show that fibrations are also very…

Recently Riehl and Verity have introduced $\infty$-cosmoi, which are certain simplicially enriched categories with additional structure. In this paper we investigate those $\infty$-cosmoi which are in fact $2$-categories; we shall refer to…

We define a natural 2-categorical structure on the base category of a large class of Grothendieck fibrations. Given any model category $\mathbf{C}$, we apply this construction to a fibration whose fibers are the homotopy categories of the…

This purpose of this book is twofold: to provide a general introduction to higher category theory (using the formalism of "quasicategories" or "weak Kan complexes"), and to apply this theory to the study of higher versions of Grothendieck…

The main objective of this paper is to show that the homotopy colimit of a diagram of quasi-categories and indexed by a small category is a localization of Lurie's higher Grothendieck construction of the diagram. We thereby generalize…

We define extension $\infty$-categories for exact $\infty$-categories in terms of bifibrations. Extension $\infty$-categories are invariant when passing to the stable hull, and consequently we show that they form an $\Omega$-spectrum,…

We focus on two factorization systems for opfibrations in the 2-category Fib(B) of fibrations over a fixed base category B. The first one is the internal version of the so called comprehensive factorization, where the right orthogonal class…

This is the part II of the series under the same title. In part I, using the approach developed by Catanese--Pignatelli arXiv:math/0503294, we gave a structure theorem for hyperelliptic genus 3 fibrations all of whose fibers are 2-connected…

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…