1,589 research outputs found
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
-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
-algebra spectra and commutative differential graded
-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
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