论文标题
关于同型Segal $ e_ \ infty $ -HOPF COOPERAD的概念
On a notion of homotopy Segal $ E_\infty $-Hopf cooperad
论文作者
论文摘要
我们在$ e_ \ infty $ -Algebras类别中定义了同型Segal Cooperad的概念。我们在论文中定义的Segal Cooperad模型，我们称之为同型Segal $ e_ \ Infty $ -HOPF Cooperad，涵盖了拓扑作业的Cochain Complex给出的例子，并为研究此类物体的同质性研究提供了一个框架。第一步，我们考虑了Segal $ e_ \ infty $ -HOPF Cooperads的类别，该类别由$ e_ \ eftty $ -slgebras组成，由树木索引并配备了与facet oterators相对应的coproduct Operators，并配备了与subtree completors相对应。相关操作员模型在树内操作的索引。假定刻面操作员满足Segal条件。我们旨在定义的同质性Segal Cooperads是通过将同型组合在共同算子的组成方案中而形成的。为此，我们替换了通过同型函数的结构来替换控制副操作员组成的功能结构，该结构我们在$ e_ \ infty $ -Algebras的类别的立方富集上塑造。我们证明，在我们意义上，每个同型segal $ e_ \ infty $ -HOPF COOPERAD与严格的segal $ e_ \ infty $ -HOPF COOPERAD相同。我们还定义了同型同质型segal $ e_ \ infty $ -HOPF COOPERADS的概念。 We prove that every homotopy Segal $ E_\infty $-Hopf cooperad admits a cobar construction and that every homotopy morphism of homotopy Segal $ E_\infty $-Hopf cooperads induces a morphism on this cobar construction, so that our approach provides a lifting to the context of $ E_\infty $-algebras of classical homotopy cooperad structures that are当我们以差异分级模块类别工作时，以操作的条二元性进行建模。
We define a notion of homotopy Segal cooperad in the category of $ E_\infty $-algebras. This model of Segal cooperad that we define in the paper, which we call homotopy Segal $ E_\infty $-Hopf cooperad, covers examples given by the cochain complex of topological operads and provides a framework for the study of the homotopy of such objects. In a first step, we consider a category of Segal $ E_\infty $-Hopf cooperads, which consists of collections of $ E_\infty $-algebras indexed by trees and equipped with coproduct operators, corresponding to tree morphisms, together with facet operators, corresponding to subtree inclusions. The coproduct operators model coproducts of operations inside a tree. The facet operators are assumed to satisfy a Segal condition. The homotopy Segal cooperads that we aim to define are formed by integrating homotopies in the composition schemes of the coproduct operators. For this purpose, we replace the functorial structure that governs the composition of the coproduct operators by the structure of a homotopy functor which we shape on a cubical enrichment of the category of $ E_\infty $-algebras. We prove that every homotopy Segal $ E_\infty $-Hopf cooperad in our sense is weakly-equivalent to a strict Segal $ E_\infty $-Hopf cooperad. We also define a notion of homotopy morphism of homotopy Segal $ E_\infty $-Hopf cooperads. We prove that every homotopy Segal $ E_\infty $-Hopf cooperad admits a cobar construction and that every homotopy morphism of homotopy Segal $ E_\infty $-Hopf cooperads induces a morphism on this cobar construction, so that our approach provides a lifting to the context of $ E_\infty $-algebras of classical homotopy cooperad structures that are modeled on the bar duality of operads when we work in a category of differential graded modules.
