Relative dualizability and the cobordism hypothesis for defects
Abstract
This paper compares two candidate notions of morphism in the setting of fully local topological field theory. A first description can be formulated in terms of defects of codimension k separating a pair of defects of codimension (k−1). Lurie's cobordism hypothesis with singularities identifies such defects with suitably dualizable k-morphisms in the target category. A second description can be formulated in terms of symmetric monoidal (op)lax natural transformations between the truncations of the source and target theories. We show that these two descriptions agree. The comparison rests on a criterion for when a k-morphism determines a codimension-k defect, expressed as a one-sided iterated adjunctibility condition. Assuming the ordinary cobordism hypothesis, this comparison provides a reformulation of the cobordism hypothesis for defects in terms of systems of oplax natural transformations. We also apply the criterion to the higher Morita category, where it yields a reducibility theorem for modules over En-algebras: a module over an (n+1)-dualizable En-algebra A is n-dualizable over A if and only if its underlying En−1-algebra is n-dualizable.
Related
- Defect TQFTs, the Generalized Cobordism Hypothesis, and Algebraic K-Theory
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- Cobordism of algebraic knots defined by Brieskorn polynomials, II}
- The homology cobordism group is not generated by graph homology $3$-spheres
- Localization, cobordism, and twisted χy-genera of ℳ̅0,n
BibTeX
@misc{stewart2026relative,
title = {{Relative dualizability and the cobordism hypothesis for defects}},
author = {William Stewart},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.32723},
eprint = {2609.32723},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.32723},
}
RIS
TY - UNPB
TI - Relative dualizability and the cobordism hypothesis for defects
AU - William Stewart
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2609.32723
UR - https://arxiv.org/abs/2609.32723
AB - This paper compares two candidate notions of morphism in the setting of fully local topological field theory. A first description can be formulated in terms of defects of codimension k separating a pair of defects of codimension (k−1). Lurie's cobordism hypothesis with singularities identifies such defects with suitably dualizable k-morphisms in the target category. A second description can be formulated in terms of symmetric monoidal (op)lax natural transformations between the truncations of the source and target theories. We show that these two descriptions agree. The comparison rests on a criterion for when a k-morphism determines a codimension-k defect, expressed as a one-sided iterated adjunctibility condition. Assuming the ordinary cobordism hypothesis, this comparison provides a reformulation of the cobordism hypothesis for defects in terms of systems of oplax natural transformations. We also apply the criterion to the higher Morita category, where it yields a reducibility theorem for modules over E_n-algebras: a module over an (n+1)-dualizable E_n-algebra A is n-dualizable over A if and only if its underlying E_n−1-algebra is n-dualizable.
ER -