Related papers: 2-Cartesian fibrations II: A Grothendieck construc…

We discuss right fibrations in the $\infty$-categorical context of Segal objects in a category V and prove some basic results about these.

We will give quiver presentations of the Grothendieck constructions of functors from a small category to the 2-category of $\Bbbk$-categories for a commutative ring $\Bbbk$.

Classification questions are often about understanding components of a category. It is much more desirable however to be able to understand the entire homotopy type of this category and not just the set of its components. In this paper we…

We provide a calculus of mates for functors to the $\infty$-category of $\infty$-categories and extend Lurie's unstraightening equivalences to show that (op)lax natural transformations correspond to maps of (co)cartesian fibrations that do…

We provide bicategorical analogs of several aspects of the notion of geometry in the sense of the theory of spectrum. We first introduce a notion of local right biadjoint, and prove it to be equivalent to a notion of bistable pseudofunctor,…

Every Grothendieck fibration gives rise to a vertical/cartesian orthogonal factorization system on its domain. We define a cartesian factorization system to be an orthogonal factorization in which the left class satisfies 2-of-3 and is…

In the first part of this paper we study fibrations of $(\infty,2)$-categories. We give a simple characterization of such fibrations in terms of a certain square being a pullback, and apply this to show that in some cases…

For a small category A, we prove that the homotopy colimit functor from the category of simplicial diagrams on A to the category of simplicial sets over the nerve of A establishes a left Quillen equivalence between the projective (or Reedy)…

We prove a universal property for $\infty$-categories of spans in the generality of Barwick's adequate triples, explicitly describe the cocartesian fibration corresponding to the span functor, and show that the latter restricts to a…

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 study fppf descent for enhanced derived categories. We revisit the work of [HS] and [TV08] in a lax context. More precisely, we construct a Cartesian and coCartesian fibration ${}^{\mathrm{op}}\mathscr D^+_S\rightarrow…

We develop a notion of an algebra over an infinity-operad with values in infinity-categories which is completely intrinsic to the formalism of dendroidal sets. Its definition involves the notion of a coCartesian fibration of dendroidal sets…

Consider a locally cartesian closed category with an object I and a class of trivial fibrations, which admit sections and are stable under pushforward and retract as arrows. Define the fibrations to be those maps whose Leibniz exponential…

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 calculate the Lusternik-Schnirelmann category of the k-th ordered configuration spaces F(R^n,k) of R^n and give bounds for the category of the corresponding unordered configuration spaces B(R^n,k) and the sectional category of the…

An important result in quasi-category theory due to Lurie is the that cocartesian fibrations are exponentiable, in the sense that pullback along a cocartesian fibration admits a right Quillen right adjoint that moreover preserves cartesian…

In this article, we develop a general technique for gluing subcategories of $\infty$-categories. We obtain categorical equivalences between simplicial sets associated to certain multisimplicial sets. Such equivalences can be used to…

We construct a nerve from double categories into double $(\infty,1)$-categories and show that it gives a right Quillen and homotopically fully faithful functor between the model structure for weakly horizontally invariant double categories…

We construct a theory of 2-vector bundles over a Lie groupoid, with fibers modeled by the bicategory of super algebras, bimodules and intertwiners. We demonstrate that these 2-vector bundles form a symmetric monoidal 2-stack. From this…

We (re)consider how the Fukaya category of a Lefschetz fibration is related to that of the fibre. The distinguishing feature of the approach here is a more direct identification of the bimodule homomorphism involved.