5 - Triangulated Categories and Functors

Summary

We start with the abstract theory of triangulated categories and triangulated functors. Because the octahedral axiom plays no role in our book, we give it minimal attention.

Then we introduce the homotopy category K(A,M), which has the same objects as C(A,M), and its morphisms are the degree 0 cohomology classes of the morphisms of C(A,M). We prove that the homotopy category K(A,M) is triangulated: its distinguished triangles are the images of the standard triangles in Cstr(A,M). A DG functor F as above induces a triangulated functor F : K(A,M) → K(B,N).Finally, we put a triangulated structure on the opposite homotopy category K(A,M)op, and we discuss contravariant triangulated functors.