Fully Triangulated Categories

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 $2$-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 $2$-triangles are distinguished if and only if the maps are distinguished and a strong version of the $3$x$3$ lemma. These illustrate some key proof methods. We also show that maps of faces of distinguished triangles are distinguished. We construct some useful distinguished $5$-triangles.