We define twisted composition products of symmetric sequences via classifying morphisms rather than twisting cochains. Our approach allows us to establish an adjunction that simultaneously generalizes a classic one for algebras and coalgebras, and the bar-cobar adjunction for quadratic operads. The comonad associated to this adjunction turns out to be, in several cases, a standard Koszul construction. The associated Kleisli categories are the “strong homotopy” morphism categories. In an appendix, we study the co-ring associated to the canonical morphism of cooperads , which is exactly the two-sided Koszul resolution of the associative operad , also known as the Alexander-Whitney co-ring.

中文翻译：

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扭曲结构和形态，直至强同构

我们通过分类形态来定义对称序列的扭曲组成产品，而不是扭曲共链。我们的方法使我们能够建立一个附加物，该附加物同时泛化一个经典的代数和余数代数，以及一个bar-cobar附加物用于二次运算。在某些情况下，与该附加组件关联的组件是标准的Koszul构造。关联的Kleisli类别是“强同态”态射类别。在附录中，我们研究与cooperads的规范形态相关的联环，这恰好是关联操作符的双面Koszul分辨率，也称为Alexander-Whitney联环。