Triangulated Categories: Definitions, Properties and Examples

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.