Find a concept
Start typing to search the mathematical index.
A complex combining the source and target of a chain map and measuring its failure to be an equivalence.
For a chain map , its mapping cone is the complex with graded object
and differential
This formula uses cochain complexes; other grading conventions change the displayed signs. The cone fits into a canonical triangle
Mapping-cone triangles motivate the distinguished triangles and the octahedral axiom of triangulated categories.
Let and be chain complexes of -modules. A chain map is a family of -linear maps
such that for every ,
A chain map induces maps on homology:
see homology module.
A distinguished triangle is a triangle
belonging to a specified class that is required to satisfy the triangle axioms. It abstracts the mapping-cone sequence of a chain map and plays the role that a short exact sequence plays in an abelian category.
The class is closed under isomorphism and rotation, and every morphism must occur as the first map of some distinguished triangle. The precise coherence requirements distinguish a pretriangulated category from an arbitrary shifted preadditive category.
Let be a pretriangulated category, and let be composable morphisms. Complete , , and to distinguished triangles, with third objects , , and . The octahedral axiom asserts that these triangles can be connected by morphisms
so that the standard octahedral diagram commutes and
is a distinguished triangle.