Fully Triangulated Categories
arXiv:2312.06843
Abstract
We modify the axioms of triangulated categories to include both higher triangles and distinguished maps of higher triangles. The distinguished maps are specializations of Neeman's ``good'' maps of -triangles. The axioms both simplify Neeman's axioms in his 1991 paper and generalize them to higher triangles in the way proposed by Balmer et al. We provide a geometric formulation via directed truncated simplices to enable a more concrete approach. The axioms are modelled by homotopy and derived categories. We look at a number of key theorems including the fact that sums of maps of -triangles are distinguished if and only if the maps are distinguished and a strong version of the x lemma. These illustrate some key proof methods. We also show that maps of faces of distinguished triangles are distinguished. We construct some useful distinguished -triangles.
23 pages