弯曲 operad 演算
Curved operadic calculus
AI总结：
本文建立弯曲 operad 的演算与同伦理论，提出弯曲绝对 operad、完备 Bar–Cobar 伴随及模型结构，并据此得到弯曲泛包络代数、最小余纤维分解与弯曲代数的同伦转移定理。
摘要：
弯曲代数是微分分次代数的一种推广，近年来已得到大量应用。这篇基础性文章的目标是引入弯曲 operad 的概念，并在这一新的层次上发展 operad 演算。弯曲 operad 演算的代数方面为我们提供了泛构造：借助弯曲 operad 双模这一新概念，我们构造了弯曲泛包络代数。由于在弯曲语境中不存在拟同构概念，我们采用新方法发展弯曲 operad 的同伦理论。这一路径使我们引入了弯曲绝对 operad 这一新概念，它是与不必余幂零的余余单位余 operad 呈 Koszul 对偶的概念，并且我们在二者之间构造了完备的 Bar–Cobar 伴随。我们赋予弯曲绝对 operad 一个合适的模型范畴结构。我们建立了一个由对偶函子组成的对偶方块，它将这一完备 Bar–Cobar 构造与单位 operad 和余幂零弯曲余 operad 之间的 Bar–Cobar 伴随交织起来。这使我们能够计算各类弯曲绝对 operad 的最小余纤维分解。利用完备 Bar 构造，我们证明了关于弯曲代数的一个一般同伦转移定理。在此过程中，我们构造了不必余幂零的余自由余 operad。
英文摘要：
Curved algebras are a generalization of differential graded algebras which have found numerous applications recently. The goal of this foundational article is to introduce the notion of a curved operad, and to develop the operadic calculus at this new level. The algebraic side of the curved operadic calculus provides us with universal constructions: using a new notion of curved operadic bimodules, we construct curved universal enveloping algebras. Since there is no notion of quasi-isomorphism in the curved context, we develop the homotopy theory of curved operads using new methods. This approach leads us to introduce the new notion of a curved absolute operad, which is the notion Koszul dual to counital cooperads non-necessarily conilpotent, and we construct a complete Bar-Cobar adjunction between them. We endow curved absolute operads with a suitable model category structure. We establish a duality square of duality functors which intertwines this complete Bar-Cobar construction with the Bar-Cobar adjunction between unital operads and conilpotent curved cooperads. This allows us to compute minimal cofibrant resolutions for various curved absolute operads. Using the complete Bar construction, we show a general Homotopy Transfer Theorem for curved algebras. Along the way, we construct the non-necessarily conilpotent cofree cooperad.
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