Homotopical Adjoint Lifting Theorem

  • David WhiteEmail author
  • Donald Yau


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 \(\Sigma \)-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 \(H\mathbb {Q}\)-algebra spectra and commutative differential graded \(\mathbb {Q}\)-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.


Algebraic-topology Category-theory Model-categories Quillen-equivalences Operads Rectification 


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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Denison UniversityGranvilleUSA
  2. 2.The Ohio State University at NewarkNewarkUSA

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