Factorization Homology as a Fully Extended Topological Field Theory
Factorization Homology as a Fully Extended Topological Field Theory
复制标题
因式分解同调作为完全扩展的拓扑场论
DOI：
10.3929/ethz-a-010399715
复制
发表时间：
2014
期刊：
影响因子：
--
通讯作者：
Claudia I. Scheimbauer
中科院分区：
文献类型：
--
作者：
Claudia I. Scheimbauer
给定一个n-代数A，我们显式地构造了一个完全扩展的n维拓扑场理论，该理论本质上是由分解同调给出的。在协合假设下，这是对应于n-代数A的完全扩展n-TFT，作为合适的Morita-p8， nq-范畴Algn中的对象。我们首先用完全n-fold Segal空间作为p8， nq-范畴的模型给出了完全扩展n维拓扑场论的精确定义。这涉及到建立一个n维边界的n倍西格尔空间边界，并赋予它对称的单轴结构。利用经Lurie证明的en -代数与局部常因子分解代数之间的等价性，利用层空间上的局部常因子分解代数构造了一个以en -代数为对象、（点）双模为1态、双模间（点）双模为2态等的p8， nq范畴，并赋予其对称一元结构。最后，在给定一个n-代数的情况下，通过对因式同调得到的因式代数的适当推进，构造了一个从Bordn到Algn的n-叠西格空间的态射。我们证明了该映射遵循对称单轴结构，因此是一个完全扩展的n-TFT。
Given an En-algebra A we explicitly construct a fully extended n-dimensional topological field theory which is essentially given by factorization homology. Under the cobordism hypothesis, this is the fully extended n-TFT corresponding to the En-algebra A, considered as an object in a suitable Morita-p8, nq-category Algn. We first give a precise definition of a fully extended n-dimensional topological field theory using complete n-fold Segal spaces as a model for p8, nq-categories. This involves developing an n-fold Segal space Bordn of n-dimensional bordisms and endowing it with a symmetric monoidal structure. Exploiting the equivalence between En-algebras and locally constant factorization algebras proven by Lurie we use locally constant factorization algebras on stratified spaces to construct an p8, nq-category with En-algebras as objects, (pointed) bimodules as 1morphisms, (pointed) bimodules between bimodules as 2-morphisms, etc. and endow it with a symmetric monoidal structure. Finally, given an En-algebra we construct a morphism of n-fold Segal spaces from Bordn to Algn given by a suitable pushforward of the factorization algebra obtained by taking factorization homology. We show that this map respects the symmetric monoidal structure and thus is a fully extended n-TFT.