Related papers: On de Rham Cohomology of Linear Categories

The purpose of this paper is to present a ``Cech-De Rham'' model for the cohomology of leaf spaces. This model lends itself to the construction of characteristic classes (in the cohomology of classifying spaces) by explicit geometrical…

In this paper, we continue our program of systematic categorification of the Noncommutative Differential Geometry of Connes. We replace a ring with a small $\mathbb C$-linear category, seen as a ring with several objects in the sense of…

We replace a ring with a small $\mathbb C$-linear category $\mathcal{C}$, seen as a ring with several objects in the sense of Mitchell. We introduce Fredholm modules over this category and construct a Chern character taking values in the…

We study the cohomology theory of sheaf complexes for open embeddings of topological spaces and related subjects. The theory is situated in the intersection of the general Cech theory and the theory of derived categories. That is to say, on…

We consider the category of comodules over a smash coproduct coalgebra $C\smashco H$. We show that there is a Grothendieck spectral sequence connecting the derived functors of the Hom functors coming from $C\smashco H$-colinear,…

The construction of characteristic classes via the curvature form of a connection is one motivation for the refinement of integral cohomology by de Rham cocycles -- known as differential cohomology. We will discuss the analog in the case of…

To each second-order ordinary differential equation $\sigma $ on a smooth manifold $M$ a $G$-structure $P^\sigma $ on $J^1(\mathbb{R},M)$ is associated and the Chern connection $\nabla ^\sigma $ attached to $\sigma $ is proved to be…

In this note, introducing notions of CH module, CH morphism and CH connection, we define a meromorphic connection in the "$z$-direction" on periodic cyclic homology of an $A_\infty$ category as a connection on cohomology of a CH module.…

Motivated by problems in which data are given over covering generating families, we suggest a new cohomology theory for diffeological spaces, called diffeological \v{C}ech cohomology, which is an exact $ \partial $-functor of the section…

To a coarse structure we associate a Grothendieck topology which is determined by coarse covers. A coarse map between coarse spaces gives rise to a morphism of Grothendieck topologies. This way we define sheaves and sheaf cohomology on…

In "Chern classes for coherent sheaves", H.I. Green constructs Chern classes in de Rham cohomology of coherent analytic sheaves. We construct here a formal $(\infty,1)$-categorical framework into which we can place Green's work and…

Let $H$ be a Hopf algebra. We consider $H$-equivariant modules over a Hopf module category $\mathcal C$ as modules over the smash extension $\mathcal C\# H$. We construct Grothendieck spectral sequences for the cohomologies as well as the…

Let $\mathbb{X}=[X_1\rightrightarrows X_0]$ be a Lie groupoid equipped with a connection, given by a smooth distribution $\mathcal{H} \subset T X_1$ transversal to the fibers of the source map. Under the assumption that the distribution…

We define a cohomology for an arbitrary $K$-linear semistrict semigroupal 2-category $(\mathfrak{C},\otimes)$ (called in the paper a Gray semigroup) and show that its first order (unitary) deformations, up to the suitable notion of…

We define the rigid homology. The trace morphism in rigid cohomology define by duality the cycle class in rigid homology. We verify the compatibility of this classes with rationnal equivalence and intersection theory. We deduce some formal…

A cocycle $\Omega: P \times G \to H$ taking values in a Lie group $H$ for a free right action of $G$ on $P$ defines a principal bundle $Q$ with the structure group $H$ over $P/G.$ The Chern character of a vector bundle associated to $Q$…

Using the concept of a cohesive module defined by Block, we use the theory of superconnections in the sense of Quillen to construct natural superconnections on Hermitian cohesive modules. By the Chern-Weil construction, we obtain…

Classical Chern-Weil theory proves existence of a ring homomorphism - the Chern character - from the Grothendieck ring of locally free finite rank A-modules to the algebraic de Rham cohomology of A. In this paper we prove existence of a…

A relative derived category for the category of modules over a presheaf of algebras is constructed to identify the relative Yoneda and Hochschild cohomologies with its homomorphism groups. The properties of a functor between this category…

We study the additivity of various geometric invariants involved in Reimann-Roch type formulas and defined via the trace map. To do so in a general context we prove that given any Grothendieck category A, the derived category D(A) has a…