# Bar constructions and Quillen homology of modules over operads

@article{Harper2010BarCA, title={Bar constructions and Quillen homology of modules over operads}, author={John E. Harper}, journal={Algebraic \& Geometric Topology}, year={2010}, volume={10}, pages={87-136} }

We show that topological Quillen homology of algebras and modules over operads in symmetric spectra can be calculated by realizations of simplicial bar constructions. Working with several model category structures, we give a homotopical proof after showing that certain homotopy colimits in algebras and modules over operads can be easily understood. A key result here, which lies at the heart of this paper, is showing that the forgetful functor commutes with certain homotopy colimits. We also…

## 31 Citations

### Graph stable equivalences and operadic constructions in equivariant spectra

- Mathematics
- 2015

One of the key insights of the work of Blumberg and Hill in [2] is that, when dealing with operads on genuine G-spectra, the homotopy theory of algebras over an operad is not merely determined by the…

### On Quillen homology and a homotopy completion tower for algebras over operads

- Mathematics
- 2011

Working in the context of symmetric spectra, we describe and study a homotopy completion tower for algebras and left modules over operads in the category of modules over a commutative ring spectrum…

### Homotopy completion and topological Quillen homology of structured ring spectra

- Mathematics
- 2013

Working in the context of symmetric spectra, we describe and study a homotopy completion tower for algebras and left modules over operads in the category of modules over a commutative ring spectrum…

### Fibration theorems for TQ-completion of structured ring spectra

- Mathematics
- 2020

The aim of this short paper is to establish a spectral algebra analog of the Bousfield-Kan "fibration lemma" under appropriate conditions. We work in the context of algebraic structures that can be…

### Derived Koszul duality and TQ-homology completion of structured ring spectra

- MathematicsAdvances in Mathematics
- 2019

### On the chromatic localization of the homotopy completion tower for $\mathcal{O}$-algebras

- Mathematics
- 2020

. The completion tower of a nonunital commutative ring is a clas- sical construction in commutative algebra. In the setting of structured ring spectra as modeled by algebras over a spectral operad,…

### Operad bimodules, and composition products on Andre-Quillen filtrations of algebras

- Mathematics
- 2016

If O is a reduced operad in symmetric spectra, an O-algebra I can be viewed as analogous to the augmentation ideal of an augmented algebra. Implicit in the literature on Topological Andre-Quillen…

### On the chromatic localization of the homotopy completion tower for 𝒪 -algebras

- Mathematics
- 2022

. The completion tower of a nonunital commutative ring is a classical construction in commutative algebra. In the setting of structured ring spectra as modeled by algebras over a spectral operad, the…

### Spectral Algebra Models of Unstable $$v_n$$-Periodic Homotopy Theory

- MathematicsBousfield Classes and Ohkawa's Theorem
- 2020

We give a survey of a generalization of Quillen-Sullivan rational homotopy theory which gives spectral algebra models of unstable v_n-periodic homotopy types. In addition to describing and…

## References

SHOWING 1-10 OF 62 REFERENCES

### Operads, Algebras and Modules in General Model Categories

- Mathematics
- 2001

In this paper we develop the theory of operads, algebras and modules in cofibrantly generated symmetric monoidal model categories. We give J-semi model strucures, which are a slightly weaker version…

### Homological algebra of homotopy algebras

- Mathematics
- 1997

We define closed model category structures on different categories connected to the world of operad algebras over the category C(k) of (unbounded) complexes of k-modules: on the category of operads,…

### Homotopy theory of modules over operads in symmetric spectra

- Mathematics
- 2009

We establish model category structures on algebras and modules over operads in symmetric spectra and study when a morphism of operads induces a Quillen equivalence between corresponding categories of…

### On the André-Quillen cohomology of commutative F[2]-algebras

- Mathematics
- 1990

Quillen and Andre have rigorized and explored a notion of cohomology of commutative algebras or, more generally, simplicial commutative algebras. They were able to do a number of systematic…

### HOMOTOPY THEORY OF COALGEBRAS

- Mathematics
- 1986

This paper makes a study of operads and of coalgebras over operads. Certain operads En and E are defined, constituting the algebraic analogues of the "little n-cube" operads; it is then shown that…

### Homotopy theory of modules over operads and non-Sigma operads in monoidal model categories

- Mathematics
- 2007

### HOMOTOPY THEORY OF MODULES OVER OPERADS AND NON-Σ OPERADS IN MONOIDAL MODEL CATEGORIES

- Mathematics
- 2009

We establish model category structures on algebras and modules over operads and non-Σ operads in monoidal model categories. The results have applications in algebraic topology, stable homotopy…