Homotopical Adjoint Lifting Theorem

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.Comment: This is the final, journal versio

Smith Ideals of Operadic Algebras in Monoidal Model Categories

Building upon Hovey's work on Smith ideals for monoids, we develop a homotopy theory of Smith ideals for general operads in a symmetric monoidal category. For a sufficiently nice stable monoidal model category and an operad satisfying a cofibrancy condition, we show that there is a Quillen equivalence between a model structure on Smith ideals and a model structure on algebra maps induced by the cokernel and the kernel. For symmetric spectra this applies to the commutative operad and all Sigma-cofibrant operads. For chain complexes over a field of characteristic zero and the stable module category, this Quillen equivalence holds for all operads.Comment: Comments welcom

Arrow Categories of Monoidal Model Categories

We prove that the arrow category of a monoidal model category, equipped with the pushout product monoidal structure and the projective model structure, is a monoidal model category. This answers a question posed by Mark Hovey, and has the important consequence that it allows for the consideration of a monoidal product in cubical homotopy theory. As illustrations we include numerous examples of non-cofibrantly generated monoidal model categories, including chain complexes, small categories, topological spaces, and pro-categories.Comment: 13 pages. Comments welcome. Version 2 adds more examples, and an application to cubical homotopy theory. Version 3 is the final, journal version, accepted to Mathematica Scandinavic

Existence and Newtonian limit of nonlinear bound states in the Einstein-Dirac system

An analysis is given of particlelike nonlinear bound states in the Newtonian limit of the coupled Einstein-Dirac system introduced by Finster, Smoller and Yau. A proof is given of existence of these bound states in the almost Newtonianian regime, and it is proved that they may be approximated by the energy minimizing solution of the Newton-Schr\"odinger system obtained by Lieb