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
Despite its being frequently attributed to Ehresmann, the notion of strict 2-categories is due to:
Jean Bénabou, Example (5) of: Catégories relatives, C. R. Acad. Sci. Paris 260 (1965) 3824-3827 [gallica]
(conceived as Cat-enriched categories and called 2-categories)
Jean-Marie Maranda, Def. 1 in: Formal categories, Canadian Journal of Mathematics 17 (1965) 758-801 [doi:10.4153/CJM-1965-076-0, pdf]
(conceived as Cat-enriched categories and called categories of the second type)
both apparently following or inspired by the earlier definition of double categories due to
An early definition also appears in the following, where it is mistakenly attributed to Charles Ehresmann:
Samuel Eilenberg, G. Max Kelly, Closed Categories, p. 425 in: S. Eilenberg, D. K. Harrison, S. MacLane, H. Röhrl (eds.): Proceedings of the Conference on Categorical Algebra - La Jolla 1965, Springer (1966) [doi:10.1007/978-3-642-99902-4]
(expressed entirely in components, under the name hypercategories)
The fundamental structure of the 2-category of categories (namely the vertical composition, horizontal composition and the whiskering of natural transformations) was first described in:
Early discussion of the general notion of bicategories:
Exposition and review:
Max Kelly, Ross Street, Review of the elements of 2-categories, Sydney Category Seminar 1972/1973, in G. Max Kelly (ed.) Lecture Notes in Mathematics 420, Springer (1974) [doi:10.1007/BFb0063101]
Ross Street, Categorical Structures, in Handbook of Algebra Vol. 1 (ed. M. Hazewinkel), Elsevier Science, Amsterdam (1996) [doi:10.1016/S1570-7954(96)80019-2, pdf, pdf, ISBN:978-0-444-82212-3]
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 December 11, 2023 at 13:43:04. See the history of this page for a list of all contributions to it.