From Pre-triangulation to Triangulation: Obstructions and Exact Lifting
Abstract
We give a new equivalent formulation of Verdier's octahedral axiom. An initial square in a pre-triangulated category determines an obstruction in a quotient of ordinary Hom groups. Its vanishing is equivalent to the existence of a good completion, and TR4 is equivalent to this vanishing for the squares associated with composable morphisms. As a consequence, if two pre-triangulations have the same underlying additive category and one is triangulated, then the other satisfies TR4 precisely when their relative inverse Heller comparison admits an exact lift along every short exact sequence in the Freyd category. As applications, we prove that the exotic pre-triangulated category of Díaz Cabrera--Muro is in fact a triangulated category over every algebraically closed field. We construct scalar families for type-A3a−1 preprojective algebras over arbitrary fields and for a range of Dynkin preprojective algebras in characteristic two; in each family the zero parameter is the unique triangulated member. The local equations also yield separable descent, including canonical triangulations for non-modular equivariantizations and for the Markman--Mehrotra K3 deformation categories.
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BibTeX
@misc{he2026from,
title = {{From Pre-triangulation to Triangulation: Obstructions and Exact Lifting}},
author = {Ping He and Bo Le},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.08684},
eprint = {2609.08684},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08684},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - From Pre-triangulation to Triangulation: Obstructions and Exact Lifting
AU - Ping He
AU - Bo Le
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.08684
UR - https://arxiv.org/abs/2609.08684
AB - We give a new equivalent formulation of Verdier's octahedral axiom. An initial square in a pre-triangulated category determines an obstruction in a quotient of ordinary Hom groups. Its vanishing is equivalent to the existence of a good completion, and TR4 is equivalent to this vanishing for the squares associated with composable morphisms. As a consequence, if two pre-triangulations have the same underlying additive category and one is triangulated, then the other satisfies TR4 precisely when their relative inverse Heller comparison admits an exact lift along every short exact sequence in the Freyd category. As applications, we prove that the exotic pre-triangulated category of Díaz Cabrera--Muro is in fact a triangulated category over every algebraically closed field. We construct scalar families for type-A_3a−1 preprojective algebras over arbitrary fields and for a range of Dynkin preprojective algebras in characteristic two; in each family the zero parameter is the unique triangulated member. The local equations also yield separable descent, including canonical triangulations for non-modular equivariantizations and for the Markman--Mehrotra K3 deformation categories.
ER -