Stable Homotopy Seminar, 11: Stable Model Categories and Triangulated Categories

(Note: I messed up the first recording and had to re-record the first 20 minutes of this.) I show that cofiber sequences agree with fiber sequences in Spectra, or indeed in any pointed model category where suspension is invertible. The homotopy category of such a model category is a highly structured category -- it's enriched in graded abelian groups and comes equipped with a notion of cofiber sequence. This motivates the notion of triangulated category, generalizing both the homotopy category of spectra, and homotopy categories of chain complexes over a ring.

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