Related papers: Asphericity structures, smooth functors, and fibra…
This is an introduction to Grothendieck's descent theory, with some stress on the general machinery of fibered categories and stacks.
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…
We show how to treat families of $\infty$-categories fibered in categorical patterns (e.g., $\infty$-operads and monoidal $\infty$-categories) in terms of fibrations by relativizing the Grothendieck construction. As applications, we…
In this paper we try to introduce a good smoothness notion for a functor. We consider properties and conditions from geometry and algebraic geometry which we expect a smooth functor should to have.
In this paper we develop the theory of topological categories over a base category, that is, a theory of topological functors. Our notion of topological functor is similar to (but not the same) the existing notions in the literature (see…
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…
A category of FI type is one which is sufficiently similar to finite sets and injections so as to admit nice representation stability results. Several common examples admit a Grothendieck fibration to finite sets and injections. We begin by…
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,…
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 main objective of this paper is to construct a homotopy colimit functor on a category of functors taking values in the model category of quasi-categories.
The Grothendieck construction is a classical correspondence between diagrams of categories and coCartesian fibrations over the indexing category. In this paper we consider the analogous correspondence in the setting of model categories. As…
This paper has been withdrawn and replaced by arXiv:1309.5035. In this paper we describe some examples of so called spherical functors between triangulated categories, which generalize the notion of a spherical object. We also give…
We introduce and develop the notion of *displayed categories*. A displayed category over a category C is equivalent to "a category D and functor F : D --> C", but instead of having a single collection of "objects of D" with a map to the…
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…
We give an account, in terms of fibered categories and their fibrewise duals, of aspects of the theory of bundle functors and star-bundle functors in differential geometry.
Layered monoidal theories provide a categorical framework for studying scientific theories at different levels of abstraction, via string diagrammatic algebra. We introduce models for three closely related classes of layered monoidal…
The Grothendieck construction establishes an equivalence between fibrations, a.k.a. fibred categories, and indexed categories, and is one of the fundamental results of category theory. Cockett and Cruttwell introduced the notion of…
We study the framework of $\infty$-equipments which is designed to produce well-behaved theories for different generalizations of $\infty$-categories in a synthetic and uniform fashion. We consider notions of (lax) functors between these…
The main aim of the present work is to arrive at a mathematical theory close to the historically original conception of generalized functions, i.e. set theoretical functions defined on, and with values in, a suitable ring of scalars and…
We develop the theory of (op)fibrations of 2-multicategories and use it to define abstract six-functor-formalisms. We also give axioms for Wirthm\"uller and Grothendieck formalisms (where either $f^!=f^*$ or $f_!=f_*$) or intermediate…