相对可对偶性与缺陷的配边假设
Relative dualizability and the cobordism hypothesis for defects
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中文总结 AI 辅助
本文证明完全局部拓扑场论中缺陷的两种描述（基于余维数缺陷与基于对称幺半lax自然变换）等价，并给出判据及应用于更高Morita范畴的可约性定理。
中文摘要 AI 辅助
本文比较了完全局部拓扑场论中两种候选的态射概念。第一种描述可以用余维数为$k$的缺陷来表述，这些缺陷分隔一对余维数为$(k-1)$的缺陷。Lurie的带奇点配边假设将此类缺陷等同于目标范畴中适当可对偶的$k$-态射。第二种描述可以用源理论和目标理论的截断之间的对称幺半（弱）lax自然变换来表述。我们证明了这两种描述是一致的。该比较依赖于一个判据，用于判断一个$k$-态射何时确定一个余维数为$k$的缺陷，该判据表达为单侧迭代伴随性条件。假设普通配边假设成立，这一比较为缺陷的配边假设提供了用oplax自然变换系统表述的重构。我们还将该判据应用于更高Morita范畴，在那里它产生了关于$E_n$-代数上模的可约性定理：一个在$(n+1)$-可对偶的$E_n$-代数$A$上的模，在$A$上是$n$-可对偶的，当且仅当其底层的$E_{n-1}$-代数是$n$-可对偶的。
英文摘要
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.
