Related papers: Fibrations of $\infty$-categories
We prove properness of (co)Cartesian fibrations as well as a straightening and unstraightening equivalence, which is compatible with cartesian products, when the base is the nerve of a small category.
We set the foundations of a theory of Grothendieck $(\infty,2)$-topoi based on the notion of fibrational descent, which axiomatizes both the existence of a classifying object for fibrations internal to an $(\infty,2)$-category as well as…
The goal of this paper is to provide the last equivalence needed in order to identify all known models for $(\infty,2)$-categories. We do this by showing that Verity's model of saturated $2$-trivial complicial sets is equivalent to Lurie's…
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…
We use Lurie's symmetric monoidal envelope functor to give two new descriptions of $\infty$-operads: as certain symmetric monoidal $\infty$-categories whose underlying symmetric monoidal $\infty$-groupoids are free, and as certain symmetric…
We provide a straightening-unstraightening adjunction for $\infty$-operads in Lurie's formalism, and show it establishes an equivalence between the $\infty$-category of operadic left fibrations over an $\infty$-operad $\mathcal{O}^\otimes$…
We study $\infty$-categories in the synthetic simplicial type theory developed by Riehl and Shulman. In particular, we define cocartesian fibrations and prove their closure properties using a novel equivalence between LARI adjunctions and…
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…
A standard result from the theory of Grothendieck fibrations states that if $p : E \to B$ is a fibration, then $E$ has limits of shape $\mathcal{J}$ if $B$ has limits of shape $\mathcal{J}$ the fibers of $\mathcal{E}$ have limits of shape…
In an abelian category, the (bi)fibration of subobjects is isomorphic to the (bi)fibration of quotients. This property captures substantial information about the exactness structure of a category. Indeed, as it was shown by the second…
We define the notion of Cartesian 2-fibrations, and prove a weak analogue of straightening. Using Barwick's notion of operator categories and the notion of a Cartesian 2-fibration, we extend the notion of $\infty$-operads to the…
We study the interaction between the notions of filteredness, fractions and fibrations in the theory of bicategories, generalizing classical results for categories. We give an explicit formula for filtered pseudo-colimits of categories…
Initiated by Gromov, the study of holomorphic curves in symplectic manifolds has been a powerfull tool in symplectic topology, however the moduli space of holomorphic curves is often very difficult to find. A common technique is to study…
Various models of $(\infty,1)$-categories, including quasi-categories, complete Segal spaces, Segal categories, and naturally marked simplicial sets can be considered as the objects of an $\infty$-cosmos. In a generic $\infty$-cosmos, whose…
We show that any functor between $\infty$-categories can be straightened. More precisely, we show that for any $\infty$-category $\mathcal{C}$, there is an equivalence between the $\infty$-category $(\mathrm{Cat}_{\infty})_{/\mathcal{C}}$…
Categories of lenses/optics and Dialectica categories are both comprised of bidirectional morphisms of basically the same form. In this work we show how they can be considered a special case of an overarching fibrational construction,…
We establish that a category of fibrant objects (in the sense of Brown) admits a Dwyer-Kan homotopical calculus of right fractions. This is done using a homotopical calculus of cocycles, which is an auxiliary structure that can be defined…
We develop a categorical framework for reasoning about abstract properties of differentiation, based on the theory of fibrations. Our work encompasses the first-order fragments of several existing categorical structures for differentiation,…
In this paper we start by pointing out that Yoneda's notion of a regular span $S \colon \mathcal{X} \to \mathcal{A} \times \mathcal{B}$ can be interpreted as a special kind of morphism, that we call fiberwise opfibration, in the 2-category…
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…