Explore: Triangulated Categories

manifold. Shift operator is a decategorified analogue of triangulated category. Triangulated categories were introduced independently by Dieter Puppe (1962)

paper of Bridgeland, for arbitrary triangulated categories. Let D {\displaystyle {\mathcal {D}}} be a triangulated category. A slicing P {\displaystyle {\mathcal

0})^{\text{op}})} . Let D ′ {\displaystyle {\mathcal {D}}'} be a triangulated subcategory of a triangulated category D {\displaystyle {\mathcal {D}}} . If ( D ≤ 0 , D

information. Since the (bounded) derived category is triangulated, there is a Grothendieck group for derived categories too. This has applications in representation

homotopy theory. The homotopy category of spectra and the derived category of a ring are both examples of triangulated categories. Let A {\displaystyle {\mathcal

K-groups; Voevodsky, Triangulated categories of motives over a field, section 2.2 and Proposition 4.2.9. Voevodsky, Triangulated categories of motives over

between triangulated categories satisfying certain technical conditions to have a right adjoint functor. Namely, if C and D are triangulated categories with

CT/0608040, ISBN 978-0-691-14049-0, MR 2522659 Neeman, Amnon (2001a), Triangulated Categories, Annals of Mathematics Studies, vol. 148, Princeton University

enhancements of triangulated categories or functors between them, for example stable homotopy category can be shown not to arise from a dg category in this way

prototypical example of an abelian category is the category of abelian groups, Ab. Abelian categories are very stable categories; for example they are regular