Abstract / Description of output
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 3x3 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.
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 3x3 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.
Original language | English |
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Publisher | ArXiv |
Number of pages | 24 |
Publication status | Published - 13 Dec 2023 |
Keywords / Materials (for Non-textual outputs)
- Triangulated categories
- Derived category