标题： 从非半单模张量范畴得到的上同调值拓扑量子场论 显示英文标题

标题： Cochain valued TQFTs from nonsemisimple modular tensor categories

摘要： 我们证明了在De Renzi等人的工作[DGGPR23]中构造的向量空间值的TQFT自然地扩展为一个取值于线性上链的对称单子范畴的拓扑场理论。具体来说，我们考虑一个边流形范畴，其对象是来自给定模张量范畴（如小量子群表示的范畴）上的上链范畴Ch(A)的标记曲面，其态射是带有嵌入缎带图的三维边流形，这些缎带图在这样的标记曲面之间传播。我们从上述的缎带边流形范畴构造了一个对称单子函子到线性上链的范畴。该理论在曲面上的值被识别为Ch(A)的Hom复形，而3-流形不变量是[DGGPR23]中归一化的Lyubashenko不变量的交替和。我们证明了我们的上链值TQFT进一步保持同伦，因此局部化为一个取值于微分分次向量空间的导出$\infty$-范畴的理论。该$\infty$-范畴理论的定义域，经过一些近似，是带有在同伦$\infty$-范畴K(A)中的标签的$\infty$-范畴。我们建议将我们的局部化理论作为构造导出量子群表示的$\infty$-范畴的“导出TQFT”的起点。 显示英文摘要

摘要： We show that a vector space valued TQFT constructed in work of De Renzi et al. [DGGPR23] extends naturally to a topological field theory which takes values in the symmetric monoidal category of linear cochains. Specifically, we consider a bordism category whose objects are surfaces with markings from the category of cochains Ch(A) over a given modular tensor category (such as the category of small quantum group representations), and whose morphisms are 3-dimensional bordisms with embedded ribbon graphs traveling between such marked surfaces. We construct a symmetric monoidal functor from the aforementioned ribbon bordism category to the category of linear cochains. The values of this theory on surfaces are identified with Hom complexes for Ch(A), and the 3-manifold invariants are alternating sums of the renormalized Lyubashenko invariant from [DGGPR23]. We show that our cochain valued TQFT furthermore preserves homotopies, and hence localizes to a theory which takes values in the derived $\infty$-category of dg vector spaces. The domain for this $\infty$-categorical theory is, up to some approximation, an $\infty$-category of ribbon bordism with labels in the homotopy $\infty$-category K(A). We suggest our localized theory as a starting point for the construction of a "derived TQFT" for the $\infty$-category of derived quantum group representations.