Triangulated categories were invented about 50 years ago as a convenient tool to do homological algebra. Yet it has been known for some time now that the notion of a triangulated category is not satisfactory: morphisms between objects in such a category are usually given by cohomology groups of certain complexes, and you forget too much by passing to cohomology. The problem is that the cone of a morphism is not functorial in triangulated categories. Let us give a couple of "frustrating" examples.
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