The notion of Hochschild homology of a dg algebra admits a natural dualization, the coHochschild homology of a dg coalgebra, introduced in arXiv:0711.1023 by Hess, Parent, and Scott as a tool to study free loop spaces. In this article we prove "agreement" for coHochschild homology, i.e., that the coHochschild homology of a dg coalgebra $C$ is isomorphic to the Hochschild homology of the dg category of appropriately compact $C$-comodules, from which Morita invariance of coHochschild homology follows. Generalizing the dg case, we define the topological coHochschild homology (coTHH) of coalgebra spectra, of which suspension spectra are the canonical examples, and show that coTHH of the suspension spectrum of a space $X$ is equivalent to the suspension spectrum of the free loop space on $X$, as long as $X$ is a nice enough space (for example, simply connected.) Based on this result and on a Quillen equivalence established by the authors in arXiv:1402.4719, we prove that "agreement" holds for coTHH as well.

中文翻译：

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coHochschild同源性的不变性

dg 代数的 Hochschild 同调的概念承认自然二元化，即 dg 代数的 coHochschild 同调，由 Hess、Parent 和 Scott 在 arXiv:0711.1023 中引入，作为研究自由环空间的工具。在本文中，我们证明了 coHochschild 同源性的“一致性”，即 dg 余代数 $C$ 的 coHochschild 同源性与适当紧致的 $C$-comodules 的 dg 范畴的 Hochschild 同构同构，从中得出 coHochschild 的 Morita 不变性同源性如下。概括dg情况，我们定义了余代数谱的拓扑coHochschild同源性（coTHH），其中悬浮谱是典型例子，并表明空间$X$的悬浮谱的coTHH等价于自由的悬浮谱。 $X$ 上的循环空间，只要 $X$ 是一个足够好的空间（例如，