Following the pattern of the notion of (n,r)-category, an (n,n)(n,n)-category is a higher category with non-trivial cells of at most dimension nn and none of them guaranteed to be reversible.

So this is what is usually simply called an n-category.

Note that it is possible to go on to an (n,n+1)(n,n+1)-category, or (n+1)(n+1)-poset. You can either consider than the nn-cells are ordered, or else consider that there are irreversible (n+1)(n+1)-cells which are indistinguishable. (Reversible indistinguishable (n+1)(n+1)-cells are all identities and so might as well not exist.)

Revised on October 20, 2009 01:38:52 by Toby Bartels (