nLab
module over an algebra over an operad

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 $ℰ$ be a closed symmetric monoidal category, $P$ an operad in $ℰ$ and $A$ a $P$-algebra over an operad.

A module over $A$ consists of

• an object $N\in ℰ$;

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

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

  in $ℰ$ (the action morphims)

• such that this data satisfies

  • unitality (…)

  • associativity (…)

  • equivariance (…)

Properties

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

Examples

$A_{\mathrm{\infty }}$-modules, etc.

(…)

Related concepts

• model structure on modules over an algebra over an operad

References

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

• Clemens Berger, Ieke Moerdijk, On the derived category of an algebra over an operad (arXiv)