Related papers: A Course on Derived Categories
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…
This is a review article on modular categories, extending an invited talk given at the workshop "Categorical (co)algebraic methods in quantum informatics and linguistics", Oxford, October 29-31, 2010. To appear in C. Heunen, M. Sadrzadeh,…
These are notes of a graduate course on representations of non-compact semisimple Lie groups given by the author at MIT.
Differential graded categories enhance our understanding of triangulated categories appearing in algebra and geometry. In this survey, we review their foundations and report on recent work by Drinfeld, Dugger-Shipley, ..., Toen and…
This is an extended version of my earlier articel "Projective and injective objects in symmetric categorical groups. arXiv:1007.0121v1." Several new facts added, including the material on the derived 2-functors and the proof of the…
This paper is a sequel to "t-structures and twisted complexes on derived injectives" by the same authors. We develop the foundations of the infinitesimal derived deformation theory of pretriangulated dg-categories endowed with t-structures.…
In this paper, we introduced a generalization of the derived category, which is called the $n$-derived category and denoted by $\D_{n}(R)$, of a given ring $R$ for each $n\in\mathbb{N}\cup\{\infty\}$. The $n$-derived category of a ring is…
In these notes we provide the foundation for the deformation theoretic parts of arXiv:0807.3753 and arXiv:math/0102005.
The aim of these notes is to provide a succinct, accessible introduction to some of the basic ideas of category theory and categorical logic. The notes are based on a lecture course given at Oxford over the past few years. They contain…
For an exact category having enough projective objects, we establish a bijection between thick subcategories containing the projective objects and thick subcategories of the stable derived category. Using this bijection we classify thick…
A working mathematician's summary of many results on the derived category, perverse sheaves, and vanishing cycles. This is the August 2025 version, with a completely revised section on vanishing cycles.
This is a short textbook on Category Theory for Russian speaking students. It consists of three chapters: Categories and Functors, Representable Functors (including Adjoint Functors and (Co)limits) and Tensor Categories.
Differential categories provide the categorical foundations for the algebraic approaches to differentiation. They have been successful in formalizing various important concepts related to differentiation, such as, in particular,…
These are lecture notes for a 1-semester undergraduate course (in computer science, mathematics, physics, engineering, chemistry or biology) in applied categorical meta-language. The only necessary background for comprehensive reading of…
We prove a Noether-Deuring theorem for the derived category of bounded complexes of modules over a Noetherian algebra.
We study $N$-differential graded ($NDG$) categories and their the derived categories. First, we introduce $N$-differential modules over an $NDG$ category $\mathcal{A}$. Then we show that the category $\mathsf{C}_{Ndg}(\mathcal{A})$ of…
We classify the module categories over the double (possibly twisted) of a finite group.
These notes are based on a series of five lectures given during the summer school ``Interactions between Homotopy Theory and Algebra'' held at the University of Chicago in 2004.
These notes contain a survey of some aspects of the theory of differential modules and complexes as well as of their generalization, that is, the theory of $N$-differential modules and $N$-complexes. Several applications and examples coming…
These are extended notes of the course given by the author at RIMS, Kyoto, in October 2016. The aim is to give a self-contained overview on the recently developed approach to differential calculus on metric measure spaces. The effort is…