[Paper Review] Idempotent completion of n-angulated categories

Zengqiang Lin|arXiv (Cornell University)|Jan 16, 2017

Homotopy and Cohomology in Algebraic Topology3 references4 citations

TL;DR

This paper establishes that the idempotent completion of an n-angulated category uniquely inherits an n-angulated structure making the inclusion functor n-angulated, satisfying a universal property. The construction extends Balmer and Schlichting's result on triangulated categories (n=3) to higher n-angulated categories, using lifting of idempotents and mapping cone techniques in pre-n-angulated categories to prove axioms (N1)(c) and (N4).

ABSTRACT

We show that the idempotent completion of an n-angulated category admits a unique n-angulated structure such that the inclusion is an n-angulated functor, which satisfies a universal property.

Motivation & Objective

• To extend Balmer and Schlichting's result on idempotent completion of triangulated categories (n=3) to n-angulated categories for n ≥ 3.
• To construct a canonical n-angulated structure on the idempotent completion of an n-angulated category.
• To show that the inclusion functor into the idempotent completion is n-angulated and satisfies a universal property.
• To overcome the failure of standard lifting and uniqueness properties in higher n-angles by developing new techniques in pre-n-angulated categories.

Proposed method

• Define the idempotent completion eC of an additive category C, consisting of pairs (A,e) where A is an object and e is an idempotent endomorphism.
• Lift the n-angulated structure from C to eC by defining eΘ as the idempotent completion of the class Θ of n-angles in C.
• Prove that (eC, eΣ, eΘ) satisfies the axioms (N1)-(N4) of an n-angulated category using properties of mapping cones and weak isomorphisms.
• Use Lemma 2.1 to decompose n-angles when certain components vanish or are sections/retractions, enabling direct sum decomposition.
• Apply Lemma 2.2 to show that mapping cones of weak isomorphisms preserve n-angles, crucial for proving (N4).
• Use Lemma 2.4 to relate the mapping cone of a composite to the direct sum of cones, ensuring closure under the octahedral axiom (N4).

Experimental results

Research questions

• RQ1 Can the idempotent completion of an n-angulated category be endowed with a unique n-angulated structure such that the inclusion functor is n-angulated?
• RQ2 How can the failure of standard lifting and uniqueness properties in n-angles (n > 3) be overcome to prove the existence of the n-angulated structure?
• RQ3 What role do mapping cones and weak isomorphisms play in generalizing the octahedral axiom (N4) to higher n-angulated categories?
• RQ4 Is the n-angulated structure on the idempotent completion universal among all n-angulated functors from the original category?

Key findings

• The idempotent completion eC of an n-angulated category C admits a unique n-angulated structure (eC, eΣ, eΘ) such that the inclusion functor ι: C → eC is n-angulated.
• The n-angulated structure on eC satisfies a universal property: for any n-angulated functor F: C → D with D idempotent complete, there exists a unique n-angulated functor G: eC → D such that F = G∘ι.
• The proof establishes that (eC, eΣ, eΘ) satisfies all axioms (N1)-(N4), with (N1)(c) and (N4) proven via decomposition lemmas and mapping cone techniques.
• The class eΘ is closed under isomorphisms, direct sums, and direct summands, and contains all contractible n-angles.
• The uniqueness of the n-angulated structure on eC follows from the fact that any alternative n-angulation eΘ' must contain eΘ and, by [5, Proposition 2.5(c)], must equal eΘ.
• The natural isomorphism α: FΣ → Σ'F for an n-angulated functor F: C → D lifts to a natural isomorphism eα: GeΣ → Σ'G, ensuring G is n-angulated.