nLab
algebraic triangulated category

A triangulated category is called algebraic 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