Realization functors in algebraic triangulated categories

TL;DR

This paper completes the proof of the existence of a realization functor from the bounded derived category of a certain subcategory to an algebraic triangulated category, clarifying foundational aspects of algebraic triangulated categories.

Contribution

It provides the missing details for the construction of realization functors in algebraic triangulated categories, extending previous partial results.

Findings

Complete proof of the existence of realization functors

Clarification of the structure of extension-closed subcategories

Validation of Keller and Vossieck's claim

Abstract

Let $\mathcal{T}$ be an algebraic triangulated category and $\mathcal{C}$ an extension-closed subcategory with $\operatorname{Hom}(\mathcal{C}, \Sigma^{<0} \mathcal{C})=0$. Then $\mathcal{C}$ has an exact structure induced from exact triangles in $\mathcal{T}$. Keller and Vossieck say that there exists a triangle functor $\operatorname{D}^b(\mathcal{C}) \to \mathcal{T}$ extending the inclusion $\mathcal{C} \subseteq \mathcal{T}$. We provide the missing details for a complete proof.