The homotopy theory of strong homotopy algebras and bialgebras

Lada introduced strong homotopy algebras to describe the structures on a deformation retract of an algebra in topological spaces. However, there is no satisfactory general definition of a morphism of strong homotopy (s.h.) algebras. Given a monad T on a simplicial category C, we instead show how s.h. T-algebras over C naturally form a Segal space. Given a distributive monad-comonad pair (T, S), the same is true for s.h. (T, S)-bialgebras over C; in particular this yields the homotopy theory of s.h. sheaves of s.h. rings. There are similar statements for quasi-monads and quasi-comonads. We also show how the structures arising are related to derived connections on bundles.Comment: 58 pages; v2 final version, to appear in HHA

Quasi-tame substitudes and the Grothendieck construction

This paper continues the study of the homotopy theory of algebras over polynomial monads initiated by the first author and Clemens Berger. We introduce the notion of a quasi-tame polynomial monad (generalizing tame ones) and produce transferred model structures (left proper in many settings) on algebras over such a monad. Our motivating application is to produce model structures on Grothendieck categories, which are used in a companion paper to give a unified approach to the study of operads, their algebras, and their modules. We prove a general result regarding when a Grothendieck construction can be realized as a category of algebras over a polynomial monad, examples illustrating that quasi-tameness is necessary as well as sufficient for admissibility, and an extension of classifier methods to a non-polynomial situation, namely the case of commutative monoids.Comment: Comments welcome. This paper has a companion paper, "Model structures on operads and algebras from a global perspective

Lifting homotopy T-algebra maps to strict maps

The settings for homotopical algebra---categories such as simplicial groups, simplicial rings, $A_\infty$ spaces, $E_\infty$ ring spectra, etc.---are often equivalent to categories of algebras over some monad or triple $T$. In such cases, $T$ is acting on a nice simplicial model category in such a way that $T$ descends to a monad on the homotopy category and defines a category of homotopy $T$-algebras. In this setting there is a forgetful functor from the homotopy category of $T$-algebras to the category of homotopy $T$-algebras. Under suitable hypotheses we provide an obstruction theory, in the form of a Bousfield-Kan spectral sequence, for lifting a homotopy $T$-algebra map to a strict map of $T$-algebras. Once we have a map of $T$-algebras to serve as a basepoint, the spectral sequence computes the homotopy groups of the space of $T$-algebra maps and the edge homomorphism on $\pi_0$ is the aforementioned forgetful functor. We discuss a variety of settings in which the required hypotheses are satisfied, including monads arising from algebraic theories and operads. We also give sufficient conditions for the $E_2$-term to be calculable in terms of Quillen cohomology groups. We provide worked examples in $G$-spaces, $G$-spectra, rational $E_\infty$ algebras, and $A_\infty$ algebras. Explicit calculations, connected to rational unstable homotopy theory, show that the forgetful functor from the homotopy category of $E_\infty$ ring spectra to the category of $H_\infty$ ring spectra is generally neither full nor faithful. We also apply a result of the second named author and Nick Kuhn to compute the homotopy type of the space $E_\infty(\Sigma^\infty_+ \mathrm{Coker}\, J, L_{K(2)} R)$.Comment: 45 pages. Substantial revision. To appear in Advances in Mathematic

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