nLab
2-functor

Idea

A $2$-functor is a functor between $2$-categories.

Of course, any sort of morphism between $2$-categories, if it deserves to be called anything like ‘functor’, will be a $2$-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 $2$-categories rather than an underlying $1$-category; for instance, not every $1$-functor $\mathrm{Cat}\to \mathrm{Cat}$ extends to a $2$-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 $2$-categories: strict and weak. So for the definitions, see:

• strict 2-functor
• pseudo functor

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 $2$-functors between weak $2$-categories¹, but it does make sense to speak of weak $2$-functors between strict $2$-categories. Indeed, the weak $3$-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 $3$-category Str2Cat? of strict $2$-categories, strict $2$-functors, transformations, and modifications.

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

Related concepts

• functor

• 2-functor / pseudofunctor

• n-functor

• (∞,1)-functor

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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. ↩