I am reading the definition of a localizing subcategory $\mathcal{D}$ of a triangulated category $\mathcal{C}$ (Definition 1.1.1 of the book Axiomatic Stable Homotopy Theory). It refers to a 'cofiber sequence' $$X\to Y\to Z\to\Sigma X$$ in $\mathcal{C}$. Does this mean the same as an exact/distinguished triangle or something else? I can't think what else it could mean given the context but I'm not familiar with it being called a 'cofiber sequence'...
2026-04-25 00:11:26.1777075886
What is a 'cofiber sequence' in a triangulated category?
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Yes, these are exact/distinguished triangles. The terminology comes from stable homotopy theory. The stable homotopy category has the structure of a triangulated category, where the exact triangles are given by (co)fiber sequences of spectra.