Model theory in compactly generated (tensor-)triangulated categories

Abstract

We give an account of model theory in the context of compactly generated triangulated and tensor-triangulated categories T . We describe pp formulas, pp-types and free realisations in such categories and we prove elimination of quantifiers and elimination of imaginaries. We compare the ways in which definable subcategories of T may be specified. Then we link definable subcategories of T and finite-type torsion theories on the category of modules over the compact objects of T . We briefly consider spectra and dualities. If T is tensor-triangulated then new features appear, in particular there is an internal duality in rigidly-compactly generated tensor-triangulated categories. 1. Introduction and background 147 2. Model theory in compactly generated triangulated categories 155 3. Definable subcategories 168 4. Tensor-triangulated categories 187 References 194 MSC2020: primary 18E45; secondary 03C60, 18G80.