Generation time for biexact functors and Koszul objects in triangulated categories
Janina C. Letz, Marc Stephan
TL;DR
The paper studies generation time, or level, in triangulated categories and extends level inequalities to enhanced (topological) tensor triangulated settings via biexact functors and Koszul objects. It introduces strong Verdier structures to guarantee coherent interaction of homotopy pushouts and triangles, proving a central inequality $\operatorname{level}_{\mathcal{U}}^{\mathsf{F}(X,X')} (\mathsf{F}(Y,Y')) \leq \operatorname{level}_{\mathcal{S}}^{X}(Y) + \operatorname{level}_{\mathcal{T}}^{X'}(Y') - 1$, and establishes level bounds for Koszul objects arising from center elements and monoidal actions. The work situates these results inside stable cofibration categories, showing that their homotopy categories are strongly triangulated and that biexact functors (including monoidal products and bimodule tensoring) admit strong Verdier structures, with broad examples spanning derived categories, stable module categories, spectra, and motivic contexts. These findings connect generation time with Rouquier dimension and provide practical tools for estimating bounds in a wide range of tensor-triangulated settings. The results have implications for rank and dimension inequalities, and they unify several strands of triangulated-category theory under the umbrella of strong Verdier-bifunctional behavior.
Abstract
This paper concerns the generation time that measures the number of cones necessary to obtain an object in a triangulated category from another object. This invariant is called level. We establish level inequalities for enhanced triangulated categories: One inequality concerns biexact functors of topological triangulated categories, another Koszul objects. In particular, this extends inequalities for the derived tensor product from commutative algebra to enhanced tensor triangulated categories. We include many examples.