Triangulated category

Triangulated category
Name	Triangulated category
Discipline	Mathematics
Subdiscipline	Category Theory, Homological algebra, Algebraic geometry
Introduced	1960s
Introduced by	Jean-Louis Verdier, Alexander Grothendieck

Triangulated category

A triangulated category is an additive category equipped with an autoequivalence and a class of distinguished triangles satisfying axioms that codify the formal properties of exact sequences in contexts such as derived categories, homotopy categories, and stable homotopy theory. It arose in foundational work of Jean-Louis Verdier in the setting of Grothendieck's homological methods and plays a central role in modern developments linking Alexander Grothendieck, Pierre Deligne, Alexandre Grothendieck-inspired cohomological techniques, and later interactions with Maxim Kontsevich, Paul Seidel, Richard Thomas in Mirror symmetry and Derived algebraic geometry.

Definition

A triangulated category consists of an additive category C, an autoequivalence T: C → C (often denoted by shift or suspension), and a class of distinguished triangles X → Y → Z → T(X) subject to axioms modeled on the properties of exact triangles in derived categories and mapping cones in homotopy categories. The formalism was introduced by Jean-Louis Verdier in the study of derived functors in the work of Alexander Grothendieck and Jean-Pierre Serre; it formalizes constructions used by Henri Cartan and Samuel Eilenberg in earlier homological algebra. Fundamental examples include the bounded derived category D^b of Sir Michael Atiyah-relevant coherent sheaves on a Calabi–Yau manifold studied by Maxim Kontsevich and categories arising in Stable homotopy category studied by J. Peter May and Adams spectral sequence developers.

Basic properties and axioms

The axioms TR1–TR4 (or variants) specify existence of distinguished triangles for identities and mapping cones, rotations of triangles, functoriality under the shift, and octahedral behavior. These axioms echo the classical machinery developed by Henri Cartan, Samuel Eilenberg, Alexander Grothendieck, and later systematized by Jean-Louis Verdier; they are essential in comparisons with abelian categories such as those appearing in Serre duality contexts considered by Jean-Pierre Serre and Alexander Grothendieck. The octahedral axiom, connecting composites of morphisms to successive cones, features in constructions used by Pierre Deligne and appears in categorical approaches by Maxim Kontsevich in Homological mirror symmetry contexts and in work of Gelfand and Manin.

Exact triangles and morphisms of triangles

Distinguished triangles function as the analogue of short exact sequences: given a morphism f: X → Y one forms a cone object Cone(f) fitting into X → Y → Cone(f) → T(X). Morphisms of triangles and the resulting long exact sequences on Hom-sets parallel constructions in derived categories, Ext-groups studied by Samuel Eilenberg and Samuel Eilenberg-collaborators, and spectral sequence techniques developed by Jean Leray and J. Peter May. Compatibility of triangles with functors is central in studies by Pierre Deligne and Grothendieck-school techniques such as six operations formalism used by Alexander Beilinson, Joseph Bernstein, and Pierre Deligne in Perverse sheaves theory.

Derived and homotopy categories (examples)

Key examples include the homotopy category K(A) of complexes over an additive category A and the derived category D(A) obtained by localizing quasi-isomorphisms, constructions foundational to Alexander Grothendieck’s work and Verdier’s thesis. Important specific instances appear in the bounded derived category D^b(Coh(X)) of coherent sheaves on a scheme X studied by Alexander Grothendieck, Jean-Pierre Serre, and later by Maxim Kontsevich in Mirror symmetry, and in the stable module category for group algebras pivotal in representation theory developed by John Green and Dade; the stable homotopy category central to algebraic topology is the arena of work by J. H. Conway-era topologists including J. Peter May and Frank Adams.

t-structures and heart

A t-structure on a triangulated category gives an abelian heart, recovering abelian categories like those studied by Alexander Grothendieck and Jean-Pierre Serre inside triangulated settings. t-structures were formalized in work of Alexander Beilinson, Joseph Bernstein, and Pierre Deligne leading to the theory of perverse sheaves and connections to Hodge theory investigated by Pierre Deligne and Wilfried Schmid. Hearts of t-structures provide contexts for comparing derived categories with module categories arising in representation theory such as in work of Bernstein–Gelfand–Gelfand and Dmitri Piontkovsky-style developments in noncommutative geometry pursued by Maxim Kontsevich and Michel Van den Bergh.

Localization and Verdier quotients

Verdier localization constructs quotients of triangulated categories by triangulated subcategories, producing Verdier quotients central to the foundations by Jean-Louis Verdier and applied in Algebraic K-theory work by Daniel Quillen. Localization procedures play roles in the development of Motivic homotopy theory by Vladimir Voevodsky, in the construction of categories for Stable homotopy category methods of J. P. May, and in categorical approaches to Birational geometry explored by Shinichi Mochizuki and Maxim Kontsevich.

Applications and variants

Triangulated categories appear across algebraic geometry, representation theory, algebraic topology, and mathematical physics: in Homological mirror symmetry by Maxim Kontsevich, in categorification programs influenced by Louis Crane and Mikhail Khovanov, and in Donald'son–Thomas theory developments by Richard Thomas and Simon Donaldson. Variants include differential graded enhancements (DG categories) advanced by Bernhard Keller and A-infinity categories studied by Jim Stasheff and Kenji Fukaya in Fukaya category contexts, as well as stable infinity-categories as developed by Jacob Lurie and explored in derived algebraic geometry by Dennis Gaitsgory and Vladimir Drinfeld.

Category:Mathematics