A as a full subcategory (the complexes concentrated in degree zero). Different abelian categories can have equivalent derived categories, so that it is not always possible to reconstruct A from D(A) as a triangulated category. Alexander Beilinson, Joseph Bernstein and Pierre Deligne described this situation by the notion of a t-structure on a triangulated category D. A t-structure on D determines an abelian category inside D, and different t-structures on D may yield different abelian categories. Localizing and thick subcategories Let D be a triangulated category with arbitrary direct sums. A localizing subcategory of D is a strictly full triangulated subcategory that is closed under arbitrary direct sums. To explain the name: if a localizing subcategory S of a compactly generated triangulated category D is generated by a set of objects, then there is a Bousfield localization functor with kernel S. (That is, for every object X in D there is an exact triangle with Y in S and LX in the right orthogonal .) For example, this construction includes the localization of a spectrum at a prime number, or the restriction from a complex of sheaves on a space to an open subset. A parallel notion is more relevant for "small" triangulated categories: a thick subcategory of a triangulated category C is a strictly full triangulated subcategory that is closed under direct summands. (If C is idempotent-complete, a subcategory is thick if and only if it is also idempotent-complete.) A localizing subcategory is thick. So if S is a localizing subcategory of a triangulated category D, then the intersection of S with the subcategory of compact objects is a thick subcategory of . For example, Devinatz–Hopkins–Smith described all thick subcategories of the triangulated category of finite spectra in terms of Morava K-theory. The localizing subcategories of the whole stable homotopy category have not been classified. See also Fourier–Mukai transform Six operations Perverse sheaf D-module Beilinson–Bernstein localization Module spectrum Semiorthogonal decomposition Bridgeland stability condition Notes References Some textbook introductions to triangulated categories are: A concise summary with applications is: Some more advanced references are: External links J. Peter May, The axioms for triangulated categories Homological algebra