Higher orthogonality and truncation in canonical pretriangulated quotients
Abstract
Canonical ideal quotients of extriangulated categories admit natural one-sided triangulated or pretriangulated structures. A fundamental question is whether these quotient structures still retain enough information to recover higher orthogonality properties of the subcategory being factored out. We show that, for a strongly functorially finite n-rigid subcategory, such information is encoded by the nilpotency of the canonical suspension and, equivalently, of the canonical loop. More precisely, this nilpotency characterizes two-sided maximal n-orthogonality. We further establish an objectwise recognition criterion that combines one-sided higher orthogonality with the vanishing of the nth suspension or loop. In the triangulated setting, our results give a converse to the known truncation construction and show that the quotient is n-truncated if and only if the subcategory is (n+1)-cluster tilting. Moreover, in the nonsplit case, the nilpotency index is exactly n. These results turn truncation from a consequence of higher orthogonality into a sharp criterion for recognizing it.
Related
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BibTeX
@misc{zhang2026higher,
title = {{Higher orthogonality and truncation in canonical pretriangulated quotients}},
author = {Yixia Zhang and Panyue Zhou},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.34289},
eprint = {2609.34289},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.34289},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Higher orthogonality and truncation in canonical pretriangulated quotients
AU - Yixia Zhang
AU - Panyue Zhou
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.34289
UR - https://arxiv.org/abs/2609.34289
AB - Canonical ideal quotients of extriangulated categories admit natural one-sided triangulated or pretriangulated structures. A fundamental question is whether these quotient structures still retain enough information to recover higher orthogonality properties of the subcategory being factored out. We show that, for a strongly functorially finite n-rigid subcategory, such information is encoded by the nilpotency of the canonical suspension and, equivalently, of the canonical loop. More precisely, this nilpotency characterizes two-sided maximal n-orthogonality. We further establish an objectwise recognition criterion that combines one-sided higher orthogonality with the vanishing of the nth suspension or loop. In the triangulated setting, our results give a converse to the known truncation construction and show that the quotient is n-truncated if and only if the subcategory is (n+1)-cluster tilting. Moreover, in the nonsplit case, the nilpotency index is exactly n. These results turn truncation from a consequence of higher orthogonality into a sharp criterion for recognizing it.
ER -