Does the notion of $\omega$-monoid exist, analogous to the notions of $\omega$-groupoid and $\omega$-category? If so, some references would be appreciated.


This is an attempted rephrasing of question: [Chain/Hierarchy of Monoids](https://mathoverflow.net/questions/24723/chain-hierarchy-of-monoids). My application domain is reasoning about modifiers of modifiers in software product line engineering, thus lacking established mathematical background. I find it easier to adapt existing results, even if the application domain is significantly different, so any help would be appreciated. 



$\omega$-monoids
Sure. A monoid is the same as a (pointed) category with a single object.


So an $n$-monoid is the same as a pointed $n$-category with a single object. 


These usually go by names like $A\_\infty$-algebras (mostly if they are linear) or similar. 


If you want *strict* $\infty$-monoids, then the notion of a strict $\omega$-category with a single object will be all you want. For more general notions, see the links at


<http://ncatlab.org/nlab/show/algebra+in+an+(infinity,1)-category>


