26 research outputs found
An operadic proof of Baez-Dolan stabilization hypothesis
We prove a stabilization theorem for algebras of n-operads in a monoidal
model category. It implies a version of Baez-Dolan stabilization hypothesis for
Rezk's weak n-categories and some other stabilization results.Comment: 14 pages, the paper is now in its final form accepted for publication
in Proceedings of AM
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
Relative left properness of colored operads
The category of -colored symmetric operads admits a cofibrantly
generated model category structure. In this paper, we show that this model
structure satisfies a relative left properness condition, i.e., that the class
of weak equivalences between -cofibrant operads is closed under cobase
change along cofibrations. We also provide an example of Dwyer which shows that
the model structure on -colored symmetric operads is not left
proper.Comment: To appear in Algebraic & Geometric Topolog
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