On the $\mathbb{Z} D_\infty$-category

Derived categories were invented by Grothendieck and Verdier around 1960, not very long after the "old" homological algebra (of derived functors between abelian categories) was established. This "new" homological algebra, of derived categories and derived functors between them, provides a significantly richer and more flexible machinery than the "old" homological algebra. For instance, the important concepts of dualizing complex and tilting complex do not exist in the "old" h...

In this paper, we give the introduction to the Hodge-Iwasawa Theory introduced by the author. After that we will give some well-defined extensions to the already shaped framework established in our previous work.

In this informal introduction to dg categories, the slogan is that dg categories are more rudimentary than triangulated categories. We recall some details on the dg quotient category introduced by Bernhard Keller and Vladimir Drinfeld.

This is the first paper in a series. We develop a general deformation theory of objects in homotopy and derived categories of DG categories. Namely, for a DG module $E$ over a DG category we define four deformation functors $\Def ^{\h}(E)$, $\coDef ^{\h}(E)$, $\Def (E)$, $\coDef (E)$. The first two functors describe the deformations (and co-deformations) of $E$ in the homotopy category, and the last two - in the derived category. We study their properties and relations. These...

A survey article for AMS Summer Institute at Seattle in 2005.

These are notes for an advanced course given at Ben Gurion University in Spring 2012.

A new definition for the notion of a (general) $\infty$-category is given.

The goal of this expository article, based on a lecture I gave at the 2016 ICRA, is to explain some recent applications of "categorical symmetries" in topology and algebraic geometry with an eye toward twisted commutative algebras as a unifying framework. The general idea is to find an action of a category on the object of interest, prove some niceness property, like finite generation, and then deduce consequences from the general properties of the category. The key in these ...

The aim of this paper is to prove a generalization of the famous Theorem A of Quillen for strict $\infty$-categories. This result is central to the homotopy theory of strict $\infty$-categories developed by the authors. The proof presented here is of a simplicial nature and uses Steiner's theory of augmented directed complexes. In a subsequent paper, we will prove the same result by purely $\infty$-categorical methods.

An elementary theory of strict $\infty $-categories with application to concrete duality is given. New examples of first and second order concrete duality are presented.