module over an algebra over an operad



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.)


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

A module over AA consists of

  • an object NN \in \mathcal{E};

  • for all 1kn1 \leq k \leq n \in \mathbb{N} a morphism

    μ n,k:P(n)A k1NA nkN \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


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


A A_\infty-modules, etc.



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

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