Triangulated Categories: Definitions, Properties and Examples
نویسندگان
چکیده
Triangulated categories were introduced in the mid 1960’s by J.L. Verdier in his thesis, reprinted in [15]. Axioms similar to Verdier’s were independently also suggested in [2]. Having their origins in algebraic geometry and algebraic topology, triangulated categories have by now become indispensable in many different areas of mathematics. Although the axioms might seem a bit opaque at first sight it turned out that very many different objects actually do carry a triangulated structure. Nowadays there are important applications of triangulated categories in areas like algebraic geometry (derived categories of coherent sheaves, theory of motives) algebraic topology (stable homotopy theory), commutative algebra, differential geometry (Fukaya categories), microlocal analysis or representation theory (derived and stable module categories). It seems that the importance of triangulated categories in modern mathematics is growing even further in recent years, with many new applications only recently found; see B. Keller’s article in this volume for one striking example, namely the cluster categories occurring in the context of S. Fomin and A. Zelevinsky’s cluster algebras which have been introduced only around 2000. In this chapter we aim at setting the scene for the survey articles in this volume by providing the relevant basic definitions, deducing some elementary general properties of triangulated categories and providing a few examples. Certainly, this cannot be a comprehensive introduction to the subject. For more details we refer to one of the well-written textbooks on triangulated categories, e.g. [4], [5], [7], [11], [16], and for further topics also to the surveys in this volume. This introductory chapter should be accessible for a reader with a good background in algebra and some basic knowledge of category theory and homological algebra.
منابع مشابه
Triangulated categories without models
We exhibit examples of triangulated categories which are neither the stable category of a Frobenius category nor a full triangulated subcategory of the homotopy category of a stable model category. Even more drastically, our examples do not admit any non-trivial exact functors to or from these algebraic respectively topological triangulated categories. Introduction. Triangulated categories are ...
متن کاملCLUSTER ALGEBRAS AND CLUSTER CATEGORIES
These are notes from introductory survey lectures given at the Institute for Studies in Theoretical Physics and Mathematics (IPM), Teheran, in 2008 and 2010. We present the definition and the fundamental properties of Fomin-Zelevinsky’s cluster algebras. Then, we introduce quiver representations and show how they can be used to construct cluster variables, which are the canonical generator...
متن کاملOn Triangulated Orbit Categories
We show that the category of orbits of the bounded derived category of a hereditary category under a well-behaved autoequivalence is canonically triangulated. This answers a question by Aslak Buan, Robert Marsh and Idun Reiten which appeared in their study [8] with M. Reineke and G. Todorov of the link between tilting theory and cluster algebras (cf. also [16]) and a question by Hideto Asashiba...
متن کاملOn Neeman’s Well Generated Triangulated Categories
We characterize Neeman’s well generated triangulated categories and discuss some of its basic properties. 2000 Mathematics Subject Classification: 18E30 (55P42, 55U35).
متن کاملTriangulated Categories of Rational Equivariant
This article is designed to provide an introduction to some examples of triangulated categories that arise in the study of G-equivariant cohomology theories for a compact Lie group G. We focus on cohomology theories whose values are rational vector spaces since one may often give explicit algebraic constructions of the triangulated category in that case. As general references for equivariant co...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2010