A 2-category consists of
The morphisms can be composed along the objects, while the 2-morphisms can be composed in two different directions: along objects – called horizontal composition – and along morphisms – called vertical composition. The composition of morphisms is allowed to be associative only up to coherent associator 2-morphisms.
2-Categories provide the context for discussing
The easiest definition of 2-category is that it is a category enriched over the cartesian monoidal category Cat. Thus it has a collection of objects, and for each pair of objects a category . The objects of these hom-categories are the morphisms, and the morphisms of these hom-categories are the 2-morphisms. This produces the classical notion of strict 2-category.
For some purposes, strict 2-categories are too strict: one would like to allow composition of morphisms to be associative and unital only up to coherent invertible 2-morphisms. A direct generalization of the above “enriched” definition produces the classical notion of bicategory.
One can also obtain notions of 2-category by specialization from the case of higher categories. Specifically, if we fix a meaning of -category, however weak or strict we wish, then we can define a -category to be an -category such that every 3-morphism is an equivalence, and all parallel pairs of -morphisms are equivalent for . It follows that, up to equivalence, there is no point in mentioning anything beyond -morphisms, except whether two given parallel -morphisms are equivalent. In some models of -categories, it is possible to make this precise by demanding that all parallel pairs of -morphisms are actually equal for , producing a simpler notion of 2-category in which we can speak about equality of 2-morphisms instead of equivalence. (This is the case for both strict -categories and bicategories.)
All of the above definitions produce “equivalent” theories of 2-category, although in some cases (such as the fact that every bicategory is equivalent to a strict 2-category) this requires some work to prove. On the nLab, we often use the word “2-category” in the general sense of referring to whatever model one may prefer, but usually one in which composition is weak; a bicategory is an adequate definition. One should beware, however, that in the literature it is common for “2-category” to refer only to strict 2-categories.
A 2-category in which all 1-morphisms and 2-morphisms are invertible is a 2-groupoid.
The archetypical 2-category is Cat, the 2-category whose
objects are categories;
morphisms are functors;
2-morphisms are natural transformation;
vertical composition of 2-morphisms is the Godement product.
This happens to be a strict 2-category.
More generally, for any enriching category (such as a Benabou cosmos), there is a 2-category whose
On the other hand, for any such we also have a bicategory -Prof whose
If is a category with pullbacks, then there is a bicategory whose
a single object ;
morphisms are the objects of : ;
2-morphisms are the morphisms of : ;
Conversely, every 2-category with a single object comes from a monoidal category this way, so the concepts are effectively equivalent. (Precisely: the 2-category of pointed 2-categories with a single object is equivalent to that of monoidal categories). For more on this relation see delooping hypothesis, k-tuply monoidal n-category, and periodic table.
Every 2-groupoid is a 2-category. For instance
for any abelian group, the double delooping is the strict 2-category with a single object, a single 1-morphisms, set of 2-moprhisms being and both horizontal composition as well as vertical composition being the product in .
Every (∞,2)-category has a homotopy 2-category, obtained by dividing out all 3-morphisms and higher.
There is also a 2-nerve. (LackPaoli)
The weak equivalences are the 2-functors that are equivalences of 2-categories.
The acyclic fibrations are the k-surjective functors for all .
Theorem A strict 2-category is cofibrant precisely if the underlying 1-category is a free category.
Example (free resolution of a 1-category). Let be an ordinary category (a 1-category) regarded as a strict 2-category. Then the cofibrant resolution is the strict 2-category given as follows:
the objects of are those of ;
the morphisms of are finite sequences of composable morphisms of , and composition is concatenation of such sequences
the 2-morphisms of are generated from 2-morphisms of the form
and their formal inverses
for all composable with composite (in !) ;
subject to the relation that for all composable triples the following equation of 2-morphisms holds
Observation Let be any strict 2-catgeory. Then a pseudofunctor is the same as a strict 2-functor .
types of morphisms
A brief account of the definition is in
A more detailed account of the definition, including a discussion of its coherence theorem, is in
Some 2-category theory, including 2-limits/2-colimit is discussed in
An older reference which introduces some of the basic notions is
A relation between bicategories and Tamsamani weak 2-categories is established in
The reverse construction is in
Andor Lucacs, Dendroidal weak 2-categories (arXiv:1304.4278)