# $\omega$-monoids

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. 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.

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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

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Urs, why pointed? – David Carchedi Jun 30 '10 at 20:40
"Pointed" is what ensures that the higher transfors are correct. An n-category with a single object can be pointed in an essentially unique way, and a functor between such n-categories is likewise a pointed functor in an essentially unique way, but the same is not true of natural transformations, modifications, etc. It's a nice excercise to write out what a pseudonatural transformation between one-object bicategories looks like, and see that "being pointed" is just what you need to reduce it to a monoidal transformation between monoidal categories. – Mike Shulman Jul 1 '10 at 2:10
An example to keep in mind is: for G a group, the automorphisms of G are not in general the same as those of BG, for whatever your favorite notion of BG is. In the context at hand, we may think of BG as being the one-object groupoid with G as its morphisms. Its automorphism group is a 1-type/groupoid-group given by the crossed module [G --> Aut(G)], where Aut(G) is the 0-type/set-group of ordinary automorphisms... – Urs Schreiber Jul 1 '10 at 7:49
...Even on connected components this does not coincide if G is not abelian. But if we regard BG as being pointed in the essentially unique way, then for the point-preserving automorphis 2-group we have Aut_*(BG) = Aut(G). – Urs Schreiber Jul 1 '10 at 7:49
Thanks Urs. I'll unravel these definitions and see what I come up with. – supercooldave Jul 2 '10 at 8:38