A 22-functor is a functor between 22-categories.

Of course, any sort of morphism between 22-categories, if it deserves to be called anything like ‘functor’, will be a 22-functor, so the prefix is not really necessary. (For discussion of this issue in general, see higher functor.) On the other hand, sometimes the more precise terminology helps to clarify that one really is thinking of 22-categories rather than an underlying 11-category; for instance, not every 11-functor CatCatCat \to Cat extends to a 22-functor.

In any case, one does need to take care to write the definition correctly. There are actually (at least) two ways to do this, corresponding to the two kinds of 22-categories: strict and weak. So for the definitions, see:

Compare lax 2-functor, which should not be called simply ‘functor’ (although ‘lax functor’ is usually fine).

It usually doesn't make sense to speak of strict 22-functors between weak 22-categories1, but it does make sense to speak of weak 22-functors between strict 22-categories. Indeed, the weak 33-category Bicat of bicategories, pseudofunctors, pseudonatural transformations, and modifications is equivalent to its full sub-3-category spanned by the strict 2-categories. However, it is not equivalent to the 33-category Str2Cat? of strict 22-categories, strict 22-functors, transformations, and modifications.

For the generalisation of this to higher categories, see semistrict higher category.

  1. Although there are certain contexts in which it does. For instance, there is a model structure on the category of bicategories and strict 2-functors between them, which models the homotopy theory of bicategories and weak 2-functors.

Revised on January 24, 2013 22:29:50 by Urs Schreiber (