Realization functors in algebraic triangulated categories
Janina C. Letz, Julia Sauter
TL;DR
The paper completes a detailed proof of the Keller–Vossieck realization result for algebraic triangulated categories: given an extension-closed subcategory ${\mathcal C}$ with ${\mathrm{Hom}}_{\mathcal T}({\mathcal C}, {\Sigma}^{<0}{\mathcal C})=0$, ${\mathcal C}$ inherits an exact structure and yields a realization functor ${\operatorname{D}^{b}}({\mathcal C})\to {\mathcal T}$. It introduces weak realization functors from ${\operatorname{K}^{b}}({\mathcal C})$ and uses non-negativity plus Frobenius-model techniques to upgrade to actual realizations, culminating in a criterion for when the realization is fully faithful or an equivalence via Ext-groups and thick subcategories. The work clarifies when the bounded derived category of an exact category can be realized inside a given triangulated category, and it connects to tilting theory while offering a robust tool for classifying exact subcategories up to derived equivalence. It also provides explicit constructive methods (via chain complexes, quotients, and total complexes) to realize morphisms and to identify when the realization functor preserves triangulated structure, with non-negativity shown to be essential by counterexamples.
Abstract
Let $\mathcal{T}$ be an algebraic triangulated category and $\mathcal{C}$ an extension-closed subcategory with $\operatorname{Hom}(\mathcal{C}, Σ^{<0} \mathcal{C})=0$. 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.