# The axioms for n–angulated categories

```@article{Bergh2011TheAF,
title={The axioms for n–angulated categories},
author={Petter Andreas Bergh and Marius Thaule},
journal={Algebraic \& Geometric Topology},
year={2011},
volume={13},
pages={2405-2428}
}```
• Published 12 December 2011
• Mathematics
• Algebraic & Geometric Topology
We discuss the axioms for an n-angulated category, recently introduced by Geiss, Keller and Op- permann in (2). 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. Triangulated categories were introduced independently in algebraic geometry by Verdier (9, 10), based on ideas of…

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## References

SHOWING 1-10 OF 17 REFERENCES

### Triangulated categories: Definitions, properties and examples

• Mathematics
• 2010
Triangulated categories were introduced in the mid 1960’s by J.L. Verdier in his thesis, reprinted in [16]. Axioms similar to Verdier’s were independently also suggested in [2]. Having their origins

### n-angulated categories

• Mathematics
• 2012
Abstract We define n-angulated categories by modifying the axioms of triangulated categories in a natural way. We show that Heller's parametrization of pre-triangulations extends to

### The Additivity of Traces in Triangulated Categories

Abstract We explain a fundamental additivity theorem for Euler characteristics and generalized trace maps in triangulated categories. The proof depends on a refined axiomatization of symmetric

### Quelques résultats ( Etat O ) . , Semin . Geom . algebr . Bois - Marie , SGA 4 1 / 2 , Lect

• Topology , Aarhus
• 1962