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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 built from the differentials of and together with , with signs determined by the grading convention. It 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.
The octahedral axiom concerns composable morphisms in a pretriangulated category. Choose distinguished triangles for , , and . The axiom requires connecting morphisms between their third objects so that the resulting octahedral diagram commutes and those third objects themselves form a distinguished triangle.
Conceptually, it states that forming cones is coherent with composition. It is often labeled TR4. In mathlib, IsTriangulated is precisely the assertion that the required octahedron exists for every such composable pair and choices of distinguished triangles.