相关论文: On the $\infty$-stack of complexes over a scheme
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 work, we conclude our study of fibred $\infty$-bicategories by providing a Grothendieck construction in this setting. Given a scaled simplicial set $S$ (which need not be fibrant) we construct a 2-categorical version of Lurie's…
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.
An $F$-zip over a scheme $S$ over a finite field is a certain object of semi-linear algebra consisting of a locally free module with a descending filtration and an ascending filtration and a $\Frob_q$-twisted isomorphism between the…
We provide a semiorthogonal decomposition for the derived category of fibrations of quintic del Pezzo surfaces with rational Gorenstein singularities. There are three components, two of which are equivalent to the derived categories of the…
In this paper, we provide a notion of $\infty$-bicategories fibred in $\infty$-bicategories which we call 2-Cartesian fibrations. Our definition is formulated using the language of marked biscaled simplicial sets: Those are scaled…
We construct a flagged $\infty$-category ${\sf Corr}$ of $\infty$-categories and bimodules among them. We prove that ${\sf Corr}$ classifies exponentiable fibrations. This representability of exponentiable fibrations extends that…
In this article we introduce four variance flavours of cartesian 2-fibrations of $\infty$-bicategories with $\infty$-bicategorical fibres, in the framework of scaled simplicial sets. Given a map $p\colon \mathcal{E} \rightarrow\mathcal{B}$…
Latent fibrations are an adaptation, appropriate for categories of partial maps (as presented by restriction categories), of the usual notion of fibration. The paper initiates the development of the basic theory of latent fibrations and…
Kuznetsov showed that for a flat quadric fibration $\mathcal{Q}$ over a smooth base $S$, $\mathrm{D}^b(\mathcal{Q})$ admits a semiorthogonal decomposition where one of the components is the derived category of the sheaf of even parts of a…
We show that the conditions in Steimle's 'additivity theorem for cobordism categories' can be weakened to only require \emph{locally} (co)Cartesian fibrations, making it applicable to a larger class of functors. As an application we compute…
We formulate a model-independent theory of co/cartesian morphisms and co/cartesian fibrations: that is, one which resides entirely *within the $\infty$-category of $\infty$-categories*. We prove this is suitably compatible with the…
Our work shows forms of descent, in the fppf, h and \'{e}tale topologies, for strong generation of the bounded derived category of a noncommutative coherent algebra over a scheme. Even for (commutative) schemes this yields new perspectives.…
We prove that for $X$ a quasi-compact $\mathbb{F}_p$-scheme with affine diagonal (e.g.\ $X$ quasi-compact and separated) there is a t-exact equivalence $\mathcal D(\mathrm{Frob}(\mathrm{QCoh}(X),F_*)) \to \mathrm{Frob}(\mathcal…
We give a model-independent construction of directed univalent cocartesian fibrations of $(\infty,1)$-categories, and prove a straightening equivalence against such fibrations. The key step is showing that cocartesian fibrations descend…
The theory of derivators enhances and simplifies the theory of triangulated categories. In this article a notion of fibered (multi-)derivator is developed, which similarly enhances fibrations of (monoidal) triangulated categories. We…
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…
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 define and study cartesian and cocartesian fibrations between categories internal to an $\infty$-topos and prove a straightening equivalence in this context.
We construct a fully-faithful functor of $\infty$-categories from complexes of D-cap modules with Fr\'echet cohomology to quasi-coherent sheaves on an analytic stack. We prove various descent results for $\infty$-categories of D-cap modules…