1 Distinguished triangles in triangulated categories
1.1 Triangulated categories: basic framework
A triangulated category is an additive category equipped with an autoequivalence (often denoted by a shift functor) and a specified class of “distinguished” triangles. The axioms are designed so that triangles behave like the homological data produced by taking mapping cones in homotopy-theoretic constructions. This structure abstracts the relationship between exact sequences and cones, allowing one to study derived and homotopical phenomena in a unified language.
1.2 Distinguished triangles: definition and axioms
A distinguished triangle is a diagram of the form \[ A \xrightarrow{f} B \xrightarrow{g} C \xrightarrow{h} A[1], \] where \(A[1]\) is the shift of \(A\). The class of distinguished triangles is required to satisfy several axioms, typically including:
- closure under isomorphism,
- existence of a distinguished triangle for any morphism,
- stability under rotation,
- a compatibility condition expressing how morphisms of triangles extend to a larger commutative diagram (often called the octahedral axiom in one of its standard formulations).
Together these axioms encode the essential formal properties of cone constructions.
1.3 Morphisms of triangles and isomorphisms
A morphism of triangles is a triple of morphisms making the obvious squares commute. Two distinguished triangles are considered isomorphic if there are morphisms between them inducing isomorphisms on each vertex object and respecting the structure maps. Because the axioms demand invariance under isomorphism, one can replace a distinguished triangle by an isomorphic one without changing any intrinsic “triangulated” information.
1.4 Exactness properties and rotation of triangles
Distinguished triangles support a notion analogous to exactness: applying a suitable cohomological or homological functor turns triangles into long exact sequences. In particular, if a functor sends distinguished triangles to long exact sequences, then the connecting morphisms in the long sequence correspond to the structure maps in the triangle. Rotation of triangles reflects the interplay between shifting and cone-like constructions; rotating changes the placement of objects and maps while preserving distinguished-ness.
2 What it means for a triangle to split
2.1 Splitting via sections and retractions
A distinguished triangle \[ A \xrightarrow{f} B \xrightarrow{g} C \xrightarrow{h} A[1] \] is said to split if the morphism \(f\) admits a retraction or, equivalently, if \(g\) admits a section in the additive sense. Concretely, if there exists \(r:B\to A\) with \(r\circ f=\mathrm{id}_A\), then \(A\) is a direct summand of \(B\), and the triangulated extension encoded by the triangle collapses to a decomposable form. Symmetrically, if there exists \(s:C\to B\) with \(g\circ s=\mathrm{id}_C\), then \(C\) is also a direct summand of \(B\), again forcing the triangle to behave as though the “extension part” is trivial.
2.2 Equivalent formulations of “split”
2.2.1 Section/retraction criteria
The defining intuition for splitting is that \(B\) decomposes as a direct sum of the pieces contributed by \(A\) and \(C\). This can be expressed by the existence of morphisms exhibiting \(f\) as a split monomorphism or \(g\) as a split epimorphism. In additive categories, the presence of such sections/retractions is precisely what guarantees that certain triangles become equivalent to triangles arising from direct sum decompositions.
2.2.2 Direct sum characterizations
A split distinguished triangle is isomorphic to a canonical triangle associated to a direct sum decomposition. That is, if \(B \cong A\oplus C\) in a way compatible with the triangle maps, then the triangle can be modeled by the standard cone triangle for an inclusion/projection pair inside \(A\oplus C\). This characterization is central: “splitting” is not merely about the existence of certain maps, but about the triangle being realizable from direct-sum algebra.
2.2.3 Vanishing of connecting morphisms
Another common viewpoint is that splitting corresponds to the disappearance of the “connecting” data. In practice, one can formulate conditions saying that the structure morphism \(h:C\to A[1]\) factors through something trivial or is forced to be zero in appropriate equivalent formulations. When the extension encoded by the triangle is trivial, the connecting morphism in derived-type constructions becomes null, and long exact sequences produced by functors break into shorter pieces compatible with direct sums.
2.3 Relationship to idempotent splittings
Direct sum decompositions are closely tied to idempotent endomorphisms. If an idempotent splits in the category, then it produces a decomposition of objects, and triangles that reflect these decompositions are more likely to be split. Conversely, when a distinguished triangle splits, it often induces an idempotent on the middle object that separates \(B\) into summands representing \(A\) and \(C\). Thus split triangles can be understood through how the category handles idempotent completion and additive decompositions.
3 Canonical form of a split distinguished triangle
3.1 Decomposition into direct summands
For a split distinguished triangle, the middle object decomposes as a direct sum of the other two vertices. In the most conceptual form, there exists an isomorphism that identifies \(f\) with an inclusion of one summand and \(g\) with the corresponding projection onto the complementary summand. Under such identifications, the “triangle” contains no hidden extension information beyond the direct sum structure.
3.2 Standard model: triangle associated to a direct sum
Given objects \(A\) and \(C\) in a triangulated category, one can form a standard distinguished triangle associated to the direct sum \(A\oplus C\). In this model, the map \(A\to A\oplus C\) is the canonical inclusion and the map \(A\oplus C\to C\) is the canonical projection. The remaining morphism \(C\to A[1]\) is determined by the triangulated structure and, for the split case, is compatible with the fact that no genuine extension occurs. The result is that any split distinguished triangle is isomorphic to this standard form.
3.3 Behavior under rotation
Because distinguished triangles are stable under rotation, the split property persists across rotated versions. Rotating a split triangle produces another distinguished triangle in which the relevant section/retraction data is transported through the shift functor. In canonical terms, the rotation corresponds to rearranging which summand is viewed as the “left” or “right” object, while maintaining direct sum decomposability.
3.4 Functoriality and preservation under equivalences
Triangulated equivalences preserve distinguished triangles up to isomorphism and commute with the shift functor up to specified natural isomorphisms. As a consequence, splitting properties are invariant under triangulated equivalences: if a triangle splits in one triangulated category, its image under an equivalence splits in the target. This invariance allows one to transfer split computations between equivalent models (for example, between different presentations of derived categories).
4 Morphisms involving split triangles
4.1 Maps between split triangles
Morphisms between split distinguished triangles can often be analyzed componentwise using the direct sum decomposition. Once one replaces each split triangle by its canonical form, a morphism of triangles typically corresponds to a pair of morphisms on the summands satisfying compatibility with the inclusion and projection maps. This reduces diagram-chasing in triangulated categories to more familiar additive-category reasoning.
4.2 Chain-level intuition (mapping cone viewpoint)
Although the discussion is categorical, it is helpful to keep the mapping cone picture in mind. A distinguished triangle is formally the abstraction of a cone construction. When the triangle splits, the cone is chain-homotopy equivalent to a direct sum of simpler complexes, reflecting that the corresponding attaching map is homotopically trivial. This interpretation explains why splitting makes homological calculations easier: it removes the need to manage nontrivial cone differentials.
4.3 Compatibility with long exact sequences
Applying a cohomological (or homological) functor to a distinguished triangle yields a long exact sequence. For a split triangle, this long exact sequence decomposes: connecting morphisms that measure the “extension” either vanish or become trivial in the relevant segments, producing exact sequences that effectively split into pieces. Thus functorial images of split triangles behave as though they arose from a short exact sequence that is already split.
4.4 Homological consequences of splitting
Homological invariants derived from triangulated structures simplify dramatically for split triangles. Typical consequences include:
- additivity of the resulting homology objects,
- reduction of extension classes to zero,
- and the absence of nontrivial connecting maps in long exact sequences.
In effect, splitting turns what would be an “extension-controlled” problem into a direct-sum problem.
5 Examples and typical use cases
5.1 Split triangles in derived categories
In derived categories, distinguished triangles arise from short exact sequences, cone constructions, or mapping cones of chain maps. A split short exact sequence produces a distinguished triangle whose extension class is trivial, yielding a split distinguished triangle. This is a common mechanism: when complexes decompose as direct sums in a way compatible with differentials and degrees, the corresponding triangles split, and cohomology computations become straightforward.
5.2 Additive categories and triangulated structures
Some triangulated categories are built from additive foundations where direct sum behavior is prominent. In such settings, split triangles frequently correspond to situations where a morphism factors through a direct summand or where idempotents split cleanly. These triangles serve as a bridge between purely additive decompositions and the more elaborate triangulated formalism.
5.3 Triangulated categories from stable homotopy perspectives (light overview)
In stable homotopy theory, triangulated structures model phenomena of spectra up to stable equivalence. Distinguished triangles correspond to cofiber sequences, and splitting corresponds to a cofiber sequence that is equivalent to a wedge (direct sum) decomposition. While the details depend on the model of spectra and the formalism used, the guiding idea remains: splitting reflects that the attaching information is trivial in the stable category.
5.4 Computational advantages in proofs
Split triangles often appear as intermediate steps in proofs. When one can identify that a triangle splits—perhaps by exhibiting a section or retraction, or by showing a connecting morphism vanishes—one can avoid heavy triangulated arguments and replace them by direct sum computations. This is particularly useful for inductive arguments, where splitting clarifies the structure of successive approximations.
6 Connections to exact sequences and extensions
6.1 From short exact sequences to distinguished triangles
Short exact sequences in abelian categories give rise to distinguished triangles in derived categories. If the short exact sequence is split, then the associated distinguished triangle is split. Thus split distinguished triangles can be viewed as the triangulated-category analogue of split short exact sequences: both represent “extension classes” that are trivial and yield direct-sum decompositions.
6.2 Ext-groups viewpoint on splitting
In derived and triangulated settings, extension groups often classify triangles up to an appropriate equivalence. A split triangle corresponds to the zero element in the relevant extension group, reflecting that the triangle encodes no nontrivial extension. Consequently, Ext-theoretic computations can detect splitting: if an extension class vanishes, then the corresponding distinguished triangle is expected to split.
6.3 When “split” implies homotopy triviality
A guiding principle is that splitting in the triangulated category often corresponds to homotopy-trivial attachment data at the level of models. Translating back to chain complexes or homotopy-theoretic constructions, the mapping cone representing the triangle becomes homotopy equivalent to a direct sum, which means the “gluing” that would create a nontrivial cone is absent up to homotopy.
6.4 Detecting splitting in practice
In concrete computations, splitting is typically detected by one of several equivalent signals:
- exhibiting a section or retraction for one of the triangle morphisms,
- showing that the connecting morphism in the long exact sequence is zero in the relevant portion,
- or producing an explicit isomorphism to the canonical direct-sum triangle.
Often, combining these approaches is effective: for instance, one may first infer vanishing of connecting maps from functorial computations, then reconstruct a splitting morphism using the triangulated axioms.