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Triangulated category
In mathematics, a triangulated category is a category together with additional structure, a "translation functor" and a class of "distinguished triangles". Prominent examples are the derived category of an abelian category (more generally, the homotopy category of a stable ∞-category) and the stable homotopy category of spectra, both of which carry the structure of a triangulated category in a natural fashion. The distinguished triangles are reminiscent of the long exact sequences of homology; they play a role akin to that of short exact sequences in abelian categories.
A t-category is a triangulated category with a t-structure.
== History ==
The notion of a derived category was introduced by in his Ph.D. thesis, based on the ideas of Grothendieck. He also defined the notion of a triangulated category, based upon the observation that a derived category had some special "triangles", by writing down axioms for the basic properties of these triangles. A very similar set of axioms was written down at about the same time by .
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