Related papers: Hochschild cohomology in toposes
For the field $\mathbb{K} = \mathbb{R}$ or $\mathbb{C}$, and an integrable distribution $F \subseteq T_M \otimes_{\mathbb{R}} \mathbb{K}$ on a smooth manifold $M$, we study the Hochschild cohomology of the dg manifold $(F[1],d_F)$ and…
This chapter sets out preliminaries for the duality theory in later chapters. An underlying idea is that local cohomology functors are higher derived functors of colocalizations (a.k.a.~coreflections). Predominantly well-known facts about…
A general theory of topological classification of defects is introduced. We illustrate the application of tools from algebraic topology, including homotopy and cohomology groups, to classify defects including several explicit calculations…
In the context of relative topos theory via stacks, we introduce the notion of existential fibred site and of existential topos of such a site. These notions allow us to develop relative topos theory in a way which naturally generalizes the…
Topological Hochschild homology is a topological analogue of classical Hochschild homology of algebras and bimodules. Beliakova, Putyra, and Wehrli introduced quantum Hochschild homology (qHH) and used it to define a quantization of annular…
By Rickard's work, two rings are derived equivalent if there is a tilting complex, constructed from projective modules over the first ring such that the second ring is the endomorphism ring of this tilting complex. In this work I describe,…
We show that the secondary Hochschild cohomology associated to a triple $(A,B,\varepsilon)$ has several of the properties of the usual Hochschild cohomology. Among others, we prove the existence of the cup and Lie products, discuss the…
For almost finite groupoids, we study how their homology groups reflect dynamical properties of their topological full groups. It is shown that two clopen subsets of the unit space has the same class in H_0 if and only if there exists an…
We view difference algebra as the study of algebraic objects in the topos of difference sets. The methods of topos theory and categorical logic enable us to develop difference homological algebra, identify a solid foundation for difference…
We give an overview of differential cohomology from the point of view of algebraic topology. This includes a survey of several different definitions of differential cohomology groups, a discussion of differential characteristic classes, an…
We introduce and study the category of Hodge microsheaves which is a Hodge-version of the category of microsheaves for a certain class of holomorphic exact symplectic manifolds. We then study Hodge-theoretic version of wrapped sheaves and…
We prove that the integral cohomology ring modulo torsion $H^*(\mathrm{Sym}^n X;\mathbb{Z})/\mathrm{Tor}$ for the symmetric product of a connected CW-complex $X$ of finite homology type is a functor of $H^*(X;\mathbb{Z})/\mathrm{Tor}$ (see…
In this paper, we study the ring structure of the integral cohomology of the Peterson variety of type $\text{A}_{n-1}$. We give two kinds of descriptions: (1) we show that it is isomorphic to the $\mathfrak{S}_n$-invariant subring of the…
For $k$ a field of positive characteristic and $X$ a smooth variety over $k$, we compute the Hochschild cohomology of Grothendieck's differential operators on $X$. The answer involves the derived inverse limit of the Frobenius acting on the…
We develop a theory of R-module Thom spectra for a commutative symmetric ring spectrum R and we analyze their multiplicative properties. As an interesting source of examples, we show that R-algebra Thom spectra associated to the special…
We define $H$-Galois extensions for $k$-linear categories and a Hopf algebra $H$ and prove the existence of a Grothendieck spectral sequence for Hochschild-Mitchell cohomology, related to this situation. This spectral sequence is…
We introduce a formalism of Hochschild (co)-homology for $\mathcal{D}$-cap modules on smooth rigid analytic spaces based on the homological tools of Ind-Banach $\mathcal{D}$-cap modules. We introduce several categories of $\mathcal{D}$-cap…
The technique developed in Hochschild's works "Lie algebra kernels and cohomology " and "Cohomology classes of finite dimensional kernels for Lie algebras " [2],[3] can be applied to the special case of transitive Lie algebroids. Our task…
We transport Steenrod's cup-i products from the singular cochains on the free loop space Maps(S^1, BG) to Hochschild's original cochain complex Hom (k[G]^*, k[G]) defining Hochschild cohomology. Here G is a discrete group, k an arbitrary…
Bergman has given the following abstract characterisation of the inner automorphisms of a group $G$: they are exactly those automorphisms of $G$ which can be extended functorially along any homomorphism $G \rightarrow H$ to an automorphism…