A -functor is the categorification of the notion of a functor to the setting of 2-categories. At the 2-categorical level there are several possible versions of this notion one might want depending on the given setting, some of which collapse to the standard definition of a functor between categories when considered on -categories with discrete hom-categories (viewed as -categories). The least restrictive of these is a lax functor, and the strictest is (appropriately) called a strict 2-functor.
For the various separate definitions that do collapse to standard functors, see:
There is also a notion of ‘lax functor’, however this notion does not necessarily yield a standard functor when considered on discrete hom-categories.
For the generalisation of this to higher categories, see semistrict higher category.
Here we present explicitly the definition for the middling notion of a pseudofunctor, and comment on alterations that yield the stronger and weaker notions.
Let and be strict 2-categories. A pseudofunctor consists of
A function .
For each pair of objects a functor
We will generally write the function and functors as .
whose components are -cell isomorphisms as below
where denotes the terminal category and is the identity-selecting functor at . Its component is a -cell isomorphism as below
These are subject to the following axioms:
where denotes vertical composition and denotes horizontal composition, as illustrated by the following commutative -cell diagram in :
as illustrated by the commutative -cell diagrams below
To obtain the notion of a lax functor we only require that the coherence morphisms and be -cells, not necessarily -cell isomorphisms. This prevents us from going back and forth between preimages and images of identity -cells and horizontally composed -cells/-cells. Similarly, to obtain an oplax functor we reverse the direction of these 2-cells.
To obtain the notion of a strict -functor we require that and are identity arrows, so horizontal composition and -cell identities literally factor through each functor in the same way vertical composition and -cell identities do.
There is a notion of a ‘weak 2-category’, however it usually doesn't make sense to speak of strict -functors between weak -categories1, but it does make sense to speak of lax (or ‘weak’) -functors between strict -categories. Indeed, the weak -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 -category Str2Cat? of strict -categories, strict -functors, transformations, and modifications. (For discussion of the terminological choice “-functor” and -functor in general, see higher functor.)
2-functor / pseudofunctor / (2,1)-functor
basic properties of…
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. ↩
Last revised on December 7, 2020 at 14:53:02. See the history of this page for a list of all contributions to it.