http://ncatlab.org/nlab/show/model+structure+on+operads
arXiv:1006.2316 A model structure for coloured operads in symmetric spectra from arXiv Front: math.CT by Javier J. Gutiérrez, Rainer M. Vogt We describe a model structure for coloured operads with values in the category of symmetric spectra (with the positive model structure), in which fibrations and weak equivalences are defined at the level of the underlying collections. This allows us to treat R-module spectra (where R is a cofibrant ring spectrum) as algebras over a cofibrant spectrum-valued operad with R as its first term. Using this model structure, we give suficient conditions for homotopical localizations in the category of symmetric spectra to preserve module structures.
arXiv:1002.0879 Coherence for Categorified Operadic Theories from arXiv Front: math.CT by M. R. Gould Given an algebraic theory which can be described by a (possibly symmetric) operad , we propose a definition of the \emph{weakening} (or \emph{categorification}) of the theory, in which equations that hold strictly for -algebras hold only up to coherent isomorphism. This generalizes the theories of monoidal categories and symmetric monoidal categories, and several related notions defined in the literature. Using this definition, we generalize the result that every monoidal category is monoidally equivalent to a strict monoidal category, and show that the “strictification” functor has an interesting universal property, being left adjoint to the forgetful functor from the category of strict -categories to the category of weak -categories. We further show that the categorification obtained is independent of our choice of presentation for , and extend some of our results to many-sorted theories, using multicategories.
arXiv:1101.1634 Homotopy theory of non-symmetric operads from arXiv Front: math.CT by Fernando Muro We endow categories of non-symmetric operads with natural model structures. We work with no restriction on our operads and only assume the usual hypotheses for model categories with a symmetric monoidal structure. We also study categories of algebras over these operads in enriched non-symmetric monoidal model categories.
arXiv:1104.0584 Transfer of algebras over operads along derived Quillen adjunctions from arXiv Front: math.AT by Javier J. Gutiérrez Let V be a cofibrantly generated monoidal model category and let M be a monoidal V-model category. Given a cofibrant C-coloured operad P in V, we give sufficient conditions for the fibrant replacement and cofibrant replacement functors in M^C to preserve P-algebra structures. In particular, we show how P-algebra structures can be transferred along derived Quillen adjunctions of monoidal V-model categories, and we apply this result to the Quillen adjunctions defined by enriched Bousfield localizations and colocalizations on M. As an application, we prove that in the category of symmetric spectra the n-connective cover functor preserves A_{\infty} and E_{\infty} module spectra over connective ring spectra, for every integer n.
arXiv:1109.1004 Dendroidal sets and simplicial operads from arXiv Front: math.CT by Denis-Charles Cisinski, Ieke Moerdijk We establish a Quillen equivalence relating the homotopy theory of Segal operads and the homotopy theory of simplicial operads, from which we deduce that the homotopy coherent nerve functor is a right Quillen equivalence from the model category of simplicial operads to the model category structure for infinity-operads on the category of dendroidal sets. By slicing over the monoidal unit, this also gives the Quillen equivalence between Segal categories and simplicial categories proved by J. Bergner, as well as the Quillen equivalence between quasi-categories and simplicial categories proved by A. Joyal and J. Lurie. We also explain how this theory applies to the usual notion of operad (i.e. with a single colour) in the category of spaces.
arXiv:1209.1087 The homotopy theory of simplicial props from arXiv Front: math.AT by Philip Hackney, Marcy Robertson The category of (colored) props is an enhancement of the category of colored operads, and thus of the category of small categories. In this paper, the second in a series on “higher props,” we show that the category of all small colored simplicial props admits a cofibrantly generated model category structure. With this model structure, the forgetful functor from props to operads is a right Quillen functor.
nLab page on Homotopy theory of operads