Triangulated categories are important structures lying at the confluence of several exciting areas of mathematics (and even physics). Our notes on the subject are divided into three parts which, if named by the major construction occurring within them, would be titled “Verdier quotients”, “Thomason localisaton” and “Brown representability”. There are many places to learn about triangulated cate...

We review the definition of a Calabi-Yau triangulated category and survey examples coming from the representation theory of quivers and finite-dimensional algebras. Our main motivation comes from the links between quiver representations and Fomin-Zelevinsky’s cluster algebras. Mathematics Subject Classification (2000). Primary 18E30; Secondary 16D90, 18G10.

1 Brown Representability First we introduce a portly abelian category A(S) for any preadditive category S. This will be a certain full subcategory of the category of all contravariant additive functors S −→ Ab. Modulo the fact that S is not assumed to be small, this is precisely what we call a right module over a ringoid in our Rings With Several Objects (RSO) notes. For background on portly ab...

X id → X → 0→ · For any morphism u : X → Y , there is an object Z (called a mapping cone of the morphism u) fitting into a distinguished triangle X u − → Y → Z → · Any triangle isomorphic to a distinguished triangle is distinguished. This means that if X u − → Y v − → Z w −→ X[1] is a distinguished triangle, and f : X → X, g : Y → Y , and h : Z → Z are isomorphisms, then X′ gu f −1 −−−−→ Y ′ hv...