This paper provides a homotopical version of the adjoint lifting theorem in category theory, allowing for Quillen equivalences to be lifted from monoidal model categories to categories of algebras over colored operads. The generality of our approach allows us to simultaneously answer questions of rectification and of changing the base model category to a Quillen equivalent one. We work in the setting of colored operads, and we do not require them to be $$\Sigma $$Σ-cofibrant. Special cases of our main theorem recover many known results regarding rectification and change of model category, as well as numerous new results. In particular, we recover a recent result of Richter–Shipley about a zig-zag of Quillen equivalences between commutative $$H\mathbb {Q}$$HQ-algebra spectra and commutative differential graded $$\mathbb {Q}$$Q-algebras, but our version involves only three Quillen equivalences instead of six. We also work out the theory of how to lift Quillen equivalences to categories of colored operad algebras after a left Bousfield localization.

中文翻译：

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同调伴随提升定理

本文提供了范畴论中伴随提升定理的同伦版本，允许将 Quillen 等价从幺半群模型范畴提升到彩色操作数上的代数范畴。我们方法的通用性使我们能够同时回答纠正问题和将基本模型类别更改为 Quillen 等效类别的问题。我们在彩色操作数的设置中工作，我们不要求它们是 $$\Sigma $$Σ-cofibrant。我们主定理的特例恢复了许多关于模型类别整改和改变的已知结果，以及许多新结果。特别地，我们恢复了 Richter-Shipley 最近关于可交换 $$H\mathbb {Q}$$HQ-代数谱和分级 $$\mathbb {Q}$$Q 之间的 Quillen 等价的之字形的结果-代数，但我们的版本只涉及三个 Quillen 等价，而不是六个。我们还研究了如何在左 Bousfield 定位后将 Quillen 等价提升到彩色操作代数类别的理论。