论文标题
绝对代数，禁用和二元广场
Absolute algebras, contramodules, and duality squares
论文作者
论文摘要
绝对代数是一种新型的代数结构，并具有有意义的无限操作概念，而无需假设任何潜在的拓扑结构。与通常对经营微积分的定义相反，它们被定义为库珀的代数。本文的目的是发展这一新理论。首先，我们将绝对代数的同义理论与通过二元方形的常规代数合物理论联系起来。它将条形辅助的条件与线性二元套合纠缠在一起。特别是，我们表明，山地类型和代数类型之间的线性二元函数是Quillen函子，并且它们在其同源性上具有有限性条件的对象之间诱导等价。我们在代数的绝对类型及其经典对应物之间给出一般比较结果。我们列出了该理论的例子，例如绝对的联想代数和绝对谎言代数，并表明它包括违规理论。坎波斯 - 彼得森 - 罗伯特·尼科德（Robert-Nicoud） - Wierstra表明，两个nilpotent Lie代数的代数，其通用包围代数是同构的，因为缔合代数必须是同构的。作为结果的应用，我们将其定理推广到绝对谎言代数和绝对$ \ Mathcal {l} _ \ infty $ -Algebras的设置。
Absolute algebras are a new type of algebraic structures, endowed with a meaningful notion of infinite sums of operations without supposing any underlying topology. Opposite to the usual definition of operadic calculus, they are defined as algebras over cooperads. The goal of this article is to develop this new theory. First, we relate the homotopy theory of absolute algebras to the homotopy theory of usual algebras via a duality square. It intertwines bar-cobar adjunctions with linear duality adjunctions. In particular, we show that linear duality functors between types of coalgebras and types of algebras are Quillen functors and that they induce equivalences between objects with finiteness conditions on their homology. We give general comparison results between absolute types of algebras and their classical counterparts. We work out examples of this theory such as absolute associative algebras and absolute Lie algebras, and show that it includes the theory of contramodules. Campos--Petersen--Robert-Nicoud--Wierstra showed that two nilpotent Lie algebras whose universal enveloping algebras are isomorphic as associative algebras must be isomorphic. As an application of our results, we generalize their theorem to the setting of absolute Lie algebras and absolute $\mathcal{L}_\infty$-algebras.
