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

Quillen cohomology of enriched operads

A modern insight due to Quillen, which is further developed by Lurie, asserts that many cohomology theories of interest are particular cases of a single construction, which allows one to define cohomology groups in an abstract setting using only intrinsic properties of the category (or $\infty$-category) at hand. This universal cohomology theory is known as Quillen cohomology. In any setting, Quillen cohomology of a given object is classified by its cotangent complex. The main purpose of this paper is to study Quillen cohomology of enriched operads, when working in the model categorical framework. Our main result provides an explicit formula for computing Quillen cohomology of enriched operads, based on a procedure of taking certain infinitesimal models of their cotangent complexes. There is a natural construction of twisted arrow $\infty$-category of a simplicial operad, which extends the notion of twisted arrow $\infty$-category of an $\infty$-category introduced by Lurie. We assert that the cotangent complex of a simplicial operad can be represented as a spectrum valued functor on its twisted arrow $\infty$-category.Comment: 70 pages, substantial modifications, section 8 has been remove

Bousfield localisations along Quillen bifunctors and applications

We describe left and right Bousfield localisations along Quillen adjunctions of two variables. These localised model structures can be used to define Postnikov sections and homological localisations of arbitrary model categories, and to study the homotopy limit model structure on the category of sections of a left Quillen presheaf of localised model structures. We obtain explicit results in this direction in concrete examples of towers and fiber products of model categories. In particular, we prove that the category of simplicial sets is Quillen equivalent to the homotopy limit model structure of its Postnikov tower, and that the category of symmetric spectra is Quillen equivalent to the homotopy fiber product of its Bousfield arithmetic square. For spectral model categories, we show that the homotopy fiber of a stable left Bousfield localisation is a stable right Bousfield localisation

Quillen's relative Chern character is multiplicative

In the 80's, Quillen constructed a de Rham relative cohomology class associated to a smooth morphism between vector bundles, that we call the relative Quillen Chern character. In the first part of this paper we prove the multiplicativ property of the relative Quillen Chern character. Then we obtain a Riemann-Roch formula between the relative Chern character of the Bott morphism and the relative Thom form.Comment: 28 pages. This article will appear in the proceedings of the conference "Algebraic Analysis and Around" in honour of Professor Masaki Kashiwara's 60's birthda