# nLab module over an algebra over an operad

### Context

#### Higher algebra

higher algebra

universal algebra

# Contents

## Idea

The notion of module over an associative algebra has a generalization to a notion of modules over an algebra that is an algebra over an operad.

Note that sometimes an algebra over an operad is called a module over the operad, so here we have a module over a module. (Whether algebras/modules over operads are more like algebras or more like modules depends on your point of view, so both terms are used.)

## Definition

Let $\mathcal{E}$ be a closed symmetric monoidal category, $P$ an operad in $\mathcal{E}$ and $A$ a $P$-algebra over an operad.

A module over $A$ consists of

• an object $N \in \mathcal{E}$;

• for all $1 \leq k \leq n \in \mathbb{N}$ a morphism

$\mu_{n,k} : P(n) \otimes A^{\otimes^{k-1}} \otimes N \otimes A^{\otimes^{n-k}} \to N$

in $\mathcal{E}$ (the action morphims)

• such that this data satisfies

## Properties

Under suitable conditions there is a model structure on modules over an algebra over an operad.

## Examples

$A_\infty$-modules, etc.

(…)

## References

A review of modules over algebras over operads is at the beginning of

Revised on September 1, 2012 19:50:17 by Urs Schreiber (89.204.130.105)