The axioms for n–angulated categories
By Petter Andreas Bergh, Marius Thaule. 2011
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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 Grothendieck, and in algebraic topology by Puppe (8). These constructions have since played a crucial role in representation theory, algebraic geometry, commutative algebra, algebraic topology and other areas of mathematics (and even theoretical physics). Recently, Geiss, Keller and Oppermann intro- duced in (2) a new type of categories, called n-angulated categories, which generalize triangulated cate- gories: the classical triangulated categories are the special case n =3. These categories appear for instance when considering certain (n−2)-cluster tilting subcategories of triangulated categories. Conversely, certain n-angulated Calabi-Yau categories yield triangulated Calabi-Yau categories of higher Calabi-Yau dimen- sion. The four axioms for n-angulated categories are generalizations of the axioms for triangulated categories. Inthis paper, wediscuss these axioms, inspiredbyworksofNeeman(6,7). First, weshowthatthefirsttwoof the original axioms canbe replaced by two alternative axioms. One of these alternative axioms requires that the collection of n-angles be closed under so-called weak isomorphisms, but not under direct sums and summands. The other axiom requires that the collection of n-angles be closed only under left rotations, but not right rotations. Second, we discuss the axioms that enable us to complete certain diagrams to morphisms of n-angles. The last of these axioms says that we can complete diagrams to morphisms of n- angles in such a way that the mapping cone is itself an n-angle. For triangulated categories (that is, when n = 3), this axiom is equivalent to the octahedral axiom, which was one of Verdier's original axioms. We show that this generalizes to n-angulated categories. Namely, we introduce a higher "octahedral axiom" for n-angulated categories, and show that this is equivalent to the mapping cone axiom. For n = 3, that is, for triangulated categories, our new axiom is almost the same as the classical octahedral axiom. In fact, it is apparently a bit weaker, but we show that they are equivalent. Therefore, for a triangulated category, the mapping cone axiom, our octahedral axiom and the classical octahedral axiom are all equivalent. This paper is organized as follows. In Section 2, we recall the definition of n-angulated categories from (2). Then, in Section 3, we discuss the first two axioms, and in Section 4 we introduce the higher octahedral axiom. Finally, in Section 5, we look at an example, namely the n-angulated categories originating from (n −2)-cluster tilting subcategories of triangulated categories. We verify the higher octahedral axiom for these categories, in the case when n =4.
Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/e2d6bc3769232d11fadd730bd95acca3372e4cf8
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Citations in the corpus
- n-Abelian and n-exact categories (2014, methodology): “The properties of (n + 2)-angulated categories have been investigated by Bergh-Thaule in [12, 13, 14].”