[Paper Review] Homotopy theory of Well-generated algebraic triangulated categories

Gonçalo Tabuada|ArXiv.org|Mar 6, 2007

Homotopy and Cohomology in Algebraic Topology7 references3 citations

TL;DR

This paper establishes that the category of well-generated algebraic triangulated categories, denoted dgcatex,α, is cocomplete. It achieves this by proving that the adjunction between the category of Tα-algebras and dgcatex,α is monadic, and leveraging the preservation of α-filtered colimits by the relevant functors. The key contribution is the cocompleteness of dgcatex,α, which supports further homotopical and categorical constructions in algebraic triangulated categories.

ABSTRACT

For every regular cardinal $α$, we construct a cofibrantly generated Quillen model structure on a category whose objects are essentially DG categories which are stable under suspensions, cosuspensions, cones and $α$-small sums. Using results of Porta, we show that the category of well-generated (algebraic) triangulated categories in the sense of Neeman is naturally enhanced by our Quillen model category.

Motivation & Objective

• To establish the cocompleteness of the category dgcatex,α of well-generated algebraic triangulated categories.
• To demonstrate that the adjunction between dgcatex,α and Tα-algebras is monadic.
• To verify that the composition of the left adjoint and forgetful functor preserves α-filtered colimits.
• To support the homotopical structure of algebraic triangulated categories by proving cocompleteness.
• To lay categorical foundations for further developments in derived and triangulated category theory.

Proposed method

• Utilizes the monadicity theorem to confirm that the adjunction between dgcatex,α and Tα-algebras is monadic.
• Applies propositions from [Bor] on monadicity and colimit preservation in the context of Tα-algebras.
• Relies on the existence of α-filtered colimits in dgcatex,α, as established by earlier results.
• Uses the fact that left adjoints preserve colimits, and the forgetful functor U1 preserves α-filtered colimits.
• Combines these preservation properties to show that the composite functor U1∘F1 commutes with α-filtered colimits.
• Concludes cocompleteness of dgcatex,α through the monadicity criterion and colimit preservation.

Experimental results

Research questions

• RQ1 Is the category dgcatex,α cocomplete?
• RQ2 Does the adjunction between dgcatex,α and Tα-algebras satisfy the conditions for monadicity?
• RQ3 Do the functors involved preserve α-filtered colimits in dgcatex,α?
• RQ4 Can the cocompleteness of dgcatex,α be established via monadicity and colimit preservation?
• RQ5 What structural properties of dgcatex,α follow from its cocompleteness in the context of algebraic triangulated categories?

Key findings

• The category dgcatex,α is cocomplete, meaning all small colimits exist within it.
• The adjunction between dgcatex,α and Tα-algebras is monadic, as confirmed by the monadicity theorem.
• The composite functor U1∘F1 preserves α-filtered colimits, a critical condition for monadicity.
• α-filtered colimits exist in dgcatex,α and are preserved by the forgetful functor U1.
• The cocompleteness of dgcatex,α is established through the preservation of α-filtered colimits and monadicity.
• The result supports the homotopical coherence of well-generated algebraic triangulated categories by ensuring sufficient colimit structure.