nLab algebraic triangulated category

A triangulated category is called algebraic (in the sense of B. Keller) if it is equivalent to the stable category of a Quillen exact category of a Frobenius category (a Quillen exact category is Frobenius if it has enough injectives and enough projectives and the two classes coincide).

• B. Keller, On differential graded categories, In: Proc. ICM, Madrid, 2006. vol. II, pp. 151–190, Eur. Math. Soc., Zürich (2006) pdf
• Stefan Schwede, Algebraic versus topological triangulated categories, in Triangulated categories, 389–407, London Mathematical Society Lecture Notes 375,

  Cambridge Univ. Press 2010, MR2681714, pdf.

• Fernando Muro, Stefan Schwede, Neil Strickland, Triangulated categories without models, Invent. math. 170, 231–241 (2007) doi, pdf

Every algebraic triangulated category which is well generated in the sense of Amnon Neeman is triangle equivalent to a localization of the derived category of a small pretriangulated dg-category by a localizing subcategory generated by a set of objects:

• M. Porta, The Popescu-Gabriel theorem for triangulated categories, Adv. Math. 225 (2010) 1669-1715 doi