The axioms for n-angulated categories

TL;DR

This paper explores the foundational axioms of n-angulated categories, introducing a higher octahedral axiom and demonstrating its equivalence to existing axioms, thereby clarifying the structure of these categories.

Contribution

It introduces a higher octahedral axiom for n-angulated categories and proves its equivalence to the mapping cone axiom, enhancing the theoretical framework.

Findings

Higher octahedral axiom is equivalent to the mapping cone axiom.

For triangulated categories, various octahedral axioms are equivalent.

Clarifies the axiomatic structure of n-angulated categories.

Abstract

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.