Definitions
Transfors between 2-categories
Morphisms in 2-categories
Structures in 2-categories
Limits in 2-categories
Structures on 2-categories
The notion of a 2-category generalizes that of category: a 2-category is a higher category, where on top of the objects and morphisms, there are also 2-morphisms. In old texts, strict 2-categories are occasionally called hypercategories.
A 2-category consists of
1-morphisms between objects;
2-morphisms between morphisms.
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 are also a horizontal categorification of monoidal categories: they are like monoidal categories with many objects.
2-categories provide the context for discussing
monads.
The concept of 2-category generalizes further in higher category theory to n-categories, which have k-morphisms for all $k\le n$.
2-categories form a 3-category, 2Cat.
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 $hom(x,y)$. 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 $\infty$-category, however weak or strict we wish, then we can define a $2$-category to be an $\infty$-category such that every 3-morphism is an equivalence, and all parallel pairs of $j$-morphisms are equivalent for $j \geq 3$. It follows that, up to equivalence, there is no point in mentioning anything beyond $2$-morphisms, except whether two given parallel $2$-morphisms are equivalent. In some models of $\infty$-categories, it is possible to make this precise by demanding that all parallel pairs of $j$-morphisms are actually equal for $j\geq 3$, 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 $2$-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;
horizontal composition of 2-morphisms is the Godement product.
This happens to be a strict 2-category.
More generally, for $V$ any enriching category (such as a Benabou cosmos), there is a 2-category $V Cat$ whose
On the other hand, for any such $V$ we also have a bicategory $V$-Prof whose
If $C$ is a category with pullbacks, then there is a bicategory Span$(C)$ whose
Every monoidal category $C$ may be thought of as a bicategory $\mathbf{B}C$ (its delooping). This has
a single object $\bullet$;
morphisms are the objects of $C$: $(\mathbf{B}C)_1 = C_0$;
2-morphisms are the morphisms of $C$ : $(\mathbf{B}C)_2 = C_1$;
horizontal composition in $\mathbf{B}C$ is the tensor product in $C$ and vertical composition in $\mathbf{B}C$ is composition in $C$.
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 $A$ any abelian group, the double delooping $\mathbf{B}^2 A$ is the strict 2-category with a single object, a single 1-morphisms, set of 2-moprhisms being $A$ and both horizontal composition as well as vertical composition being the product in $A$.
for $G$ any 2-group, its single delooping is a 2-groupoid with a single object.
Every topological space has a path 2-groupoid.
Every (∞,2)-category has a homotopy 2-category, obtained by dividing out all 3-morphisms and higher.
An ordinary category has a nerve which is a simplicial set. For 2-categories one may consider their double nerve which is a bisimplicial set.
There is also a 2-nerve. (LackPaoli)
(…)
There is a model category structure on 2-categories – sometimes known as the folk model structure – that models the (2,1)-category underlying 2Cat (Lack).
For strict 2-categories this is the restriction of the corresponding folk model structure on strict omega-categories.
The weak equivalences are the 2-functors that are equivalences of 2-categories.
The acyclic fibrations are the k-surjective functors for all $k$.
Theorem A strict 2-category $C$ is cofibrant precisely if the underlying 1-category $C_1$ is a free category.
This is theorem 4.8 in (LackStrict). This is a special case of the more general statement that free strict $\omega$-categories are given by computads.
Example (free resolution of a 1-category). Let $C$ be an ordinary category (a 1-category) regarded as a strict 2-category. Then the cofibrant resolution $\hat C \stackrel{\simeq}{\to} C$ is the strict 2-category given as follows:
the objects of $\hat C$ are those of $C$;
the morphisms of $\hat C$ are finite sequences of composable morphisms of $C$, and composition is concatenation of such sequences
(hence $(\hat C)_1$ is the free category on the quiver underlying $C$);
the 2-morphisms of $\hat C$ are generated from 2-morphisms $c_{f,g}$ of the form
and their formal inverses
for all composable $f,g \in Mor(C)$ with composite (in $C$!) $g \circ_C f$;
subject to the relation that for all composable triples $f,g,h \in Mor(C)$ the following equation of 2-morphisms holds
Observation Let $D$ be any strict 2-catgeory. Then a pseudofunctor $C \to D$ is the same as a strict 2-functor $\hat C \to D$.
constructions
extra properties
types of morphisms
specific versions
limit notions
model structures
2-category
Early texts
The terminology hypercategory was introduced in
Exposition and review:
Max Kelly and Ross Street, Review of the elements of 2-categories, Sydney Category Seminar 1972/1973, in G. M. Kelly (ed.) Lecture Notes in Mathematics 420, Springer 1974 (doi:10.1007/BFb0063101)
Ross Street, Encyclopedia article on 2-categories and bicategories (pdf)
Tom Leinster, Basic bicategories (arXiv:9810017)
Steve Lack, A 2-categories companion, In: Baez J., May J. (eds.) Towards Higher Categories. The IMA Volumes in Mathematics and its Applications, vol 152. Springer 2010 (arXiv:math.CT/0702535, doi:10.1007/978-1-4419-1524-5_4)
John Power, 2-Categories, BRICS Notes Series 1998 (pdf)
Comprehensive textbook accounts:
Ofer Gabber, Lorenzo Ramero, Chapter 2 of: Foundations for almost ring theory (arXiv:math/0409584)
Niles Johnson, Donald Yau, 2-Dimensional Categories, Oxford University Press 2021 (arXiv:2002.06055, doi:10.1093/oso/9780198871378.001.0001)
Relation between bicategories and Tamsamani weak 2-categories:
Steve Lack, Simona Paoli, 2-nerves for bicategories (arXiv)
Simona Paoli, From Tamsamani weak 2-categories to bicategories (arXiv)
There is a model category structure on 2-categories – the canonical model structure – that models the (2,1)-category underlying 2Cat:
Steve Lack, A Quillen Model Structure for 2-Categories, K-Theory 26: 171–205, 2002. (website)
Steve Lack, A Quillen Model Structure for Biategories, K-Theory 33: 185-197, 2004. (website)
Discussion of weak 2-categories in the style of A-infinity categories is (using dendroidal sets to model the higher operads) in
Andor Lucacs, Dendroidal weak 2-categories (arXiv:1304.4278)
Jonathan Chiche, La théorie de l’homotopie des 2-catégories, thesis, arXiv.
Last revised on March 5, 2023 at 13:59:32. See the history of this page for a list of all contributions to it.