Related papers: The dual fibration in elementary terms
In this paper, we consider an equivalence problem of second order partially differential equations (PDE) and a duality of the flat differential equation. For the equivalence problem, explicit form of invariants (curvatures) are given. We…
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…
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 investigate fibrancy conditions in the Thomason model structure on the category of small categories. In particular, we show that the category of weak equivalences of a partial model category is fibrant. Furthermore, we describe…
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…
(Pseudo) double categories have two sorts of morphisms: tight ones which compose strictly, and loose ones which compose up to coherent isomorphism. In this paper, we consider bimodules between double categories in the loose direction. We…
We introduce a notion of Poincar\'e duality for pairs of $\infty$-categories, extending Poincar\'e-Lefschetz duality for pairs of spaces. This categorical extension yields an efficient book-keeping device that affords, among other things, a…
String backgrounds with a local torus fibration such as T-folds are naturally formulated in a doubled formalism in which the torus fibres are doubled to include dual coordinates conjugate to winding number. Here we formulate and explore a…
We remark that forcing on fiber bundles of structures of first order languages is not a compatible semantics with the pullback (of fiber bundles) and we describe a semantics which behaves well with respect to it. This new semantics uses…
The purpose of this article is threefold: Firstly, we propose some enhancements to the existing definition of 6-functor formalisms. Secondly, we systematically study the category of kernels, which is a certain 2-category attached to every…
Many structures of interest in two-dimensional category theory have aspects that are inherently strict. This strictness is not a limitation, but rather plays a fundamental role in the theory of such structures. For instance, a monoidal…
In this paper we define a functor-- leveled sub-cohomology. (It bears no relation with the level of elliptic curves). It is based on leveled cycles on a smooth projective variety, and will be expected to reveal a structure in the level.
We prove that Ringel duality in the category of strict polynomial functors can be interpreted as derived functors of non-additive functors (in the sense of Dold and Puppe). We give applications of this fact for both theories.
We give a natural notion of (non-exact) integral functor in the context of k-linear and graded categories. In this broader sense, we prove that every k-linear and graded functor is integral.
We prove fibration theorems \`a la Milnor for differentiable real maps with non isolated critical values. We study the situation for maps with linear discriminant, and prove that the concept of d-regularity is the key point for the…
We present a construction of a non-hermitian fermionic Lagrangian which has a second-order kinetic term. Despite the non-hermicity of the latter, the theory is unitary and the perturbation theory that can be derived is equivalent to the…
This is the final part of the work started in math.DG/0611281 and math.DG/0703916. Here the question of double fibration ois adressed both for relative k-theory and free multiplicative K-theory. In the case of relative and ``nonfree''…
Given a fibration in groupoids d : D -> I, we define a fibered multicategory as a particular functor p : M -> I, where M has the same objects as D, and its arrows a : X -> Y should be thought of as families of arrows in the multicategory,…
The goal of this paper is to prove an equivalence between the $(\infty,2)$-category of cartesian factorization systems of $\infty$-categories and that of pointed cartesian fibrations of $\infty$-categories. This generalizes a similar result…
By regarding the classical non abelian cohomology of groups from a 2-dimensional categorical viewpoint, we are led to a non abelian cohomology of groupoids which continues to satisfy classification, interpretation and representation…