Triangulated structures and mapping cones

For a cochain complex, the shifted term $C[1]^n$ is $C^{n+1}$ and its differential is the negative of the original differential. Given a cochain map $f:C\to D$, the cone term is $D^n\oplus C^{n+1}$. Apply the proposed differential twice: the first component contains $d_Df(x)-f(d_Cx)$, which vanishes by the cochain-map identity, while the remaining square terms vanish separately. This calculation both checks the sign and explains why a map of graded modules alone would not suffice.

The inclusion $i:D\to\operatorname{Cone}(f)$ and projection $p:\operatorname{Cone}(f)\to C[1]$ are cochain maps with these conventions. To match the positive connecting maps obtained by lifting and differentiating a short exact sequence, our distinguished cone triangle is $(f,i,-p)$. Other textbooks may choose a different last-map convention. Either convention can be used consistently, but borrowing one displayed arrow from the other convention without a sign translation changes the triangle.

In the homotopy category, distinguished triangles are those isomorphic to the specified signed cone triangles; they pass to distinguished triangles after derived localization. Their cohomology gives a long exact sequence. A map is a quasi-isomorphism exactly when its cone is acyclic, by exactness of that sequence. Being acyclic need not make the cone contractible, so the same criterion does not characterize homotopy equivalences without replacing acyclicity by the stronger chain-level condition.

For a degreewise short exact sequence of complexes, its associated triangle is distinguished in the derived category. A degreewise split sequence also has a corresponding triangle in the homotopy category, because the chosen graded splitting exposes the off-diagonal differential. A general nonsplit short exact sequence need not give that homotopy-category conclusion directly. Distinguishing these statements avoids silently assuming that every quotient admits a compatible splitting before the derived category has been formed.

The triangle axioms include completing every morphism to a distinguished triangle, closure under the signed rotation, completing a compatible pair of maps to a map of triangles, and an octahedral compatibility for composable morphisms. These are structural requirements, not consequences of adjacent composites being zero. The octahedral axiom relates cones of two maps and their composite, expressing how successive failures of equivalence fit together. Proving the full axioms is beyond this lesson; the cone construction provides their concrete source.

At the level of strict cochain maps and strictly commuting squares, cone maps can be constructed by block matrices. For a square commuting only up to homotopy, a chosen homotopy contributes an extra block; changing that choice can change the resulting map. Thus a triangulated category does not, from its axioms alone, provide a canonical functorial cone for every arrow. Keeping the explicit complex or a richer enhancement retains choices that the triangulated description intentionally suppresses.