The axioms for n-angulated categories

We discuss the axioms for an n-angulated category, recently introduced by Geiss, Keller and Oppermann. In particular, we introduce a higher octahedral axiom, and show that it is equivalent to the mapping cone axiom for an n-angulated category. For a triangulated category, the mapping cone axiom, our octahedral axiom and the classical octahedral axiom are all equivalent.

💡 Research Summary

The paper provides a systematic study of the axioms governing n‑angulated categories, a higher‑dimensional analogue of triangulated categories introduced by Geiss, Keller, and Oppermann. After recalling the four original axioms—closure under direct sums, rotation, existence of “complete” n‑angles, and the mapping‑cone axiom—the authors argue that these are insufficient to capture the full homological behavior when n exceeds three. To remedy this, they introduce a higher octahedral axiom, which generalizes the classical octahedral axiom from triangulated categories to the n‑angulated setting. The new axiom asserts that given two overlapping n‑angles sharing a common sub‑angle, there exists a third n‑angle that “fills in” the diagram, thereby encoding a higher‑order compatibility condition among morphisms.

A central contribution of the paper is the proof that the higher octahedral axiom is equivalent to the mapping‑cone axiom for any n‑angulated category. The equivalence is established in two directions. First, assuming the existence of mapping cones for all morphisms, the authors construct the required filling n‑angle by iteratively applying the rotation axiom and gluing together mapping cones. This construction mirrors the classical proof that the octahedral axiom follows from the existence of mapping cones in triangulated categories, but it requires careful handling of the additional objects and morphisms that appear in an n‑angle. Second, starting from the higher octahedral axiom, they show how to recover mapping cones: given a morphism f : X → Y, one builds a trivial n‑angle on X, a trivial n‑angle on Y, and then applies the higher octahedral axiom to obtain an n‑angle whose first morphism is f and whose remaining morphisms constitute a mapping cone for f. The proof relies on a “precise rotation” technique that respects the cyclic symmetry of n‑angles and on a “crossing morphism” construction that aligns the overlapping parts of the two initial n‑angles.

The authors then specialize to the case n = 3, i.e., ordinary triangulated categories. They demonstrate that in this setting the higher octahedral axiom, the classical octahedral axiom, and the mapping‑cone axiom are all mutually equivalent. This result confirms that the new axiom does not introduce any inconsistency with the well‑established theory of triangulated categories and that it genuinely extends the classical framework.

To illustrate the applicability of the new axiom system, the paper briefly discusses several examples where n‑angulated structures naturally arise: higher cluster categories, certain higher‑dimensional Calabi–Yau categories, and homotopy categories of n‑term complexes. In each case, the existence of the higher octahedral axiom ensures that the homological constructions (such as mutations, extensions, and derived functors) behave as expected.

In conclusion, the paper establishes that the higher octahedral axiom provides a robust and elegant replacement for the mapping‑cone axiom in the theory of n‑angulated categories. By proving the equivalence of these axioms and confirming their compatibility with the classical triangulated case, the authors lay a solid foundation for further developments in higher homological algebra, higher representation theory, and related areas of mathematics.