Related papers: On the derived DG functors

In this thesis we study two main topics which culminate in a proof that four distinct definitions of the equivariant derived category of a smooth algebraic group $G$ acting on a variety $X$ are in fact equivalent. In the first part of this…

Let $\mathcal{X}$ be a resolving and contravariantly finite subcategory of $\rm{mod}\mbox{-}\Lambda$, the category of finitely generated right $\Lambda$-modules. We associate to $\mathcal{X}$ the subcategory…

Consider a finite group $G$ acting on a triangulated category $\mathcal T$. In this paper we investigate triangulated structure on the category $\mathcal T^G$ of $G$-equivariant objects in $\mathcal T$. We prove (under some technical…

In the paper "Deformation theory of abelian categories", the last two authors proved that an abelian category with enough injectives can be reconstructed as the category of finitely presented modules over the category of its injective…

We show that for the path algebra $A$ of an acyclic quiver, the singularity category of the derived category $\mathsf{D}^{\rm b}(\mathsf{mod}\,A)$ is triangle equivalent to the derived category of the functor category of…

We introduce the notion of a quasi DG category, generalizing that of a DG category. To a quasi DG category satisfying certain additional conditions, we associate another quasi DG category, the quasi DG category of $C$-diagrams. We then show…

We prove in ZFC that if $G$ is a (right) $R$-module such that the groups $\Hom_R(\prod_{i\in I}G_i,G)$ and $\prod_{i\in I}\Hom_R(G_i,G)$ are naturally isomorphic for all families of $R$-modules $(G_i)_{i\in I}$ then G=0. The result is valid…

We uncover several general phenomenas governing functor homology over additive categories. In particular, we generalize the strong comparison theorem of Franjou Friedlander Scorichenko and Suslin to the setting of Fp-linear additive…

A dg-natural transformation between dg-functors is called an objectwise homotopy equivalence if its induced morphism on each object admits a homotopy inverse. In general an objectwise homotopy equivalence does not have a dg-inverse but has…

We consider "Hopfological" techniques as in \cite{Ko} but for infinite dimensional Hopf algebras, under the assumption of being co-Frobenius. In particular, $H=k[{\mathbb Z}]\#k[x]/x^2$ is the first example, whose corepresentations category…

Bivariant (equivariant) K-theory is the standard setting for non-commutative topology. We may carry over various techniques from homotopy theory and homological algebra to this setting. Here we do this for some basic notions from…

We construct a category $\OrdFor$ as an arboreal extension of $\Delta_{\mathrm{epi}}\subseteq\Delta$, whose morphisms are ordered forests composed by grafting. We define a full functor $\pi\colon \OrdFor\to\Delta_{\mathrm{epi}}^{op}$…

The natural problem we approach in the present paper is to show how the notion of formally smooth (co)algebra inside monoidal categories can substitute that of (co)separable (co)algebra in the study of splitting bialgebra homomorphisms.…

We consider several types of non-existence theorems for functors. For example, there are no nontrivial functors from the category of groups (or the category of pointed sets, or vector spaces) to any small category. Another type of questions…

Let $\Sigma$ be a small category and $\mathcal{A}$ be a $\Sigma$-co-complete (resp. $\Sigma$-complete) abelian category. It is a well-known fact that the category $\operatorname{Fun}(\Sigma,\mathcal{A})$ of functors of $\Sigma$ in…

We develop the theory of exact completions of regular $\infty$-categories, and show that the $\infty$-categorical exact completion (resp. hypercompletion) of an abelian category recovers the connective half of its bounded (resp. unbounded)…

We prove that the notion of Drinfeld center defines a functor from the category of indecomposable multi-tensor categories with morphisms given by bimodules to that of braided tensor categories with morphisms given by monoidal bimodules.…

Drinfeld showed that any finite dimensional Hopf algebra \G extends to a quasitriangular Hopf algebra \D(\G), the quantum double of \G. Based on the construction of a so--called diagonal crossed product developed by the authors, we…

We study higher Hochschild homology evaluated on wedges of circles, viewed as a functor on the category of free groups. The main results use coefficients arising from square-zero extensions; this is motivated by work of Turchin and…

Given a non-positive DG-ring $A$, associated to it are the reduction and coreduction functors $F(-) = \mathrm{H}^0(A)\otimes^{\mathrm{L}}_A -$ and $G(-) = \mathrm{R}\operatorname{Hom}_A(\mathrm{H}^0(A),-)$, considered as functors…