A Shadow Perspective on Equivariant Hochschild Homologies
A Shadow Perspective on Equivariant Hochschild Homologies
复制标题
等变 Hochschild 同调的影子视角
DOI：
10.1093/imrn/rnac250
复制
发表时间：
2022
影响因子：
1
通讯作者：
Kong, Hana Jia
中科院分区：
文献类型：
--
作者：
Adamyk, Katharine;Gerhardt, Teena;Hess, Kathryn;Klang, Inbar;Kong, Hana Jia
阴影的bicategories，庞托定义，提供了一个有用的框架，概括了经典和拓扑Hochschild同源。在本文中，我们定义了Hochschild型不变量的幺半群在一个对称的么半群，单纯模型范畴，以及小范畴。我们表明，这些结构中的每一个延伸到一个适当的双范畴的阴影，这意味着特别是，他们是森田不变。我们还定义了一个广义的Hochschild同调理论扭曲的自同构，并表明它是森田不变量。绿色函子的Hochschild同调和-扭曲拓扑Hochschild同调符合这个框架，这使我们能够得出结论，这些理论是森田不变的。我们还研究了与拓扑理论和代数理论相关的线性化映射，证明了拓扑Hochschild同调的线性化映射是作为一个松弛阴影函子产生的，并构造了一个新的与拓扑限制同调和代数限制同调相关的线性化映射.最后，我们构造了一个从等变代数理论的不动点到扭曲拓扑Hochschild同调的扭曲Dennis迹映射。
Shadows for bicategories, defined by Ponto, provide a useful framework that generalizes classical and topological Hochschild homology. In this paper, we define Hochschild-type invariants for monoids in a symmetric monoidal, simplicial model category, as well as for small-categories. We show that each of these constructions extends to a shadow on an appropriate bicategory, which implies in particular that they are Morita invariant. We also define a generalized theory of Hochschild homology twisted by an automorphism and show that it is Morita invariant. Hochschild homology of Green functors and-twisted topological Hochschild homology fit into this framework, which allows us to conclude that these theories are Morita invariant. We also study linearization maps relating the topological and algebraic theories, proving that the linearization map for topological Hochschild homology arises as a lax shadow functor, and constructing a new linearization map relating topological restriction homology and algebraic restriction homology. Finally, we construct a twisted Dennis trace map from the fixed points of equivariant algebraic-theory to twisted topological Hochschild homology.