论文标题
通过广义轨道的凝结反转和威特等价
Condensation inversion and Witt equivalence via generalised orbifolds
论文作者
论文摘要
在Arxiv的Mulevičius-runkel中，在给定的模块化融合类别（MFC）$ \ Mathcal {C C}中，所谓的Orbifold Datum $ \ Mathbb {a} $如何显示出了所谓的Orbifold Datum $ \ Mathbb {a} $如何产生新的MFC $ \ MathCAL {C c} $ native。这些关联的MFC的示例包括冷凝，即类别$ \ Mathcal {C} _b^\ Circ $的本地模块的circ $ in \ Mathcal {C} $的可分离交换代数$ b \。在本文中，我们证明了MFCS上的关系$ \ MATHCAL {C} \ MATHCAL {C} _ {\ MATHBB {A}} $与Witt等价相同。这部分是通过为倒置冷凝的明确结构来实现的，即在$ \ m varycal {c} _b^\ circ $中找到一个Orbifold datum $ \ mathbb {a} $，其关联的mfc等于$ \ m nathcal {c} $。作为该构造中使用的工具，我们还探索了哪些类型的函数$ f \ colon \ Mathcal {C} \ RightArrow \ Mathcal \ Mathcal {D} $之间的MFCS Preserve orbifold数据。事实证明，$ f $不一定是强大的单体，而是一个“丝带Frobenius”函子，该功能子具有弱和弱的共同体结构，与Frobenius样属性相关。
In Mulevičius-Runkel, arXiv:2002.00663, it was shown how a so-called orbifold datum $\mathbb{A}$ in a given modular fusion category (MFC) $\mathcal{C}$ produces a new MFC $\mathcal{C}_{\mathbb{A}}$. Examples of these associated MFCs include condensations, i.e. the categories $\mathcal{C}_B^\circ$ of local modules of a separable commutative algebra $B\in\mathcal{C}$. In this paper we prove that the relation $\mathcal{C} \sim \mathcal{C}_{\mathbb{A}}$ on MFCs is the same as Witt equivalence. This is achieved in part by providing one with an explicit construction for inverting condensations, i.e. finding an orbifold datum $\mathbb{A}$ in $\mathcal{C}_B^\circ$ whose associated MFC is equivalent to $\mathcal{C}$. As a tool used in this construction we also explore what kinds of functors $F\colon\mathcal{C}\rightarrow\mathcal{D}$ between MFCs preserve orbifold data. It turns out that $F$ need not necessarily be strong monoidal, but rather a `ribbon Frobenius' functor, which has weak monoidal and weak comonoidal structures, related by a Frobenius-like property.
