DUALITY AND TRACES FOR INDEXED MONOIDAL CATEGORIES
DUALITY AND TRACES FOR INDEXED MONOIDAL CATEGORIES
复制标题
索引单曲线类别的对偶性和踪迹
DOI：
10.4171/dm/131
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发表时间：
2012
期刊：
影响因子：
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通讯作者：
Michael Shulman
中科院分区：
文献类型：
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作者：
K. Ponto;Michael Shulman
根据Lefschetz不动点定理，如果拓扑空间的自同态是不动点自由的，则它的Lefschetz数为零。然而，这个必要条件通常是不充分的;为此，我们需要对莱夫谢茨数进行一种改进，称为里德迈斯特迹。抽象地说，Lefschetz数是一个对称monoidal范畴的轨迹，而Reidemeister轨迹是一个双范畴的轨迹，在本文中，我们将这些上下文使用索引对称monoidal范畴。
特别是，我们将表明，任何对称monoidal范畴与相关的索引对称monoidal范畴，有一个相关的双范畴，产生类似于Reidemeister迹的加细迹。这个双范畴也产生了一个新的概念的跟踪参数化空间与对偶纤维，细化明显的“纤维”的痕迹，将行动的基本组的基础空间。我们还提出了索引monoidal范畴的基本理论，包括引入一个字符串图演算，使计算更容易处理。这个抽象的框架为将这些思想推广到其他环境奠定了基础。
By the Lefschetz fixed point theorem, if an endomorphism of a topological space is fixed-point-free, then its Lefschetz number vanishes. This necessary condition is not usually sufficient, however; for that we need a refinement of the Lefschetz number called the Reidemeister trace. Abstractly, the Lefschetz number is a trace in a symmetric monoidal category, while the Reidemeister trace is a trace in a bicategory; in this paper we relate these contexts using indexed symmetric monoidal categories.
In particular, we will show that for any symmetric monoidal category with an associated indexed symmetric monoidal category, there is an associated bicategory which produces refinements of trace analogous to the Reidemeister trace. This bicategory also produces a new notion of trace for parametrized spaces with dualizable fibers, which refines the obvious "fiberwise" traces by incorporating the action of the fundamental group of the base space. We also advance the basic theory of indexed monoidal categories, including introducing a string diagram calculus which makes calculations much more tractable. This abstract framework lays the foundation for generalizations of these ideas to other contexts.