nLab
2-category

Idea

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.

A 2-category consists of

• objects;

• 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 vertical composition – and along morphisms – called horizontal 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

• adjunctions;

• monads.

The concept of 2-category generalizes further in higher category theory to n-categories, which have k-morphisms for all $k\le n$.

Definitions

Strict 2-categories

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 $\mathrm{hom}(x,y)$. 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.

General 2-categories

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 $\mathrm{\infty }$-category, however weak or strict we wish, then we can define a $2$-category to be an $\mathrm{\infty }$-category such that every 3-morphism is an equivalence, and all parallel pairs of $j$-morphisms are equivalent for $j\ge 3$. 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 $\mathrm{\infty }$-categories, it is possible to make this precise by demanding that all parallel pairs of $j$-morphisms are actually equal for $j\ge 3$, 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.

Examples

• 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 $V$ any enriching category (such as a Benabou cosmos), there is a 2-category $V\mathrm{Cat}$ whose

  • objects are $V$-enriched categories;
  • morphisms are $V$-enriched functors; and
  • 2-morphisms are $V$-natural transformations.

• On the other hand, for any such $V$ we also have a bicategory $V$-Prof whose

  • objects are $V$-enriched categories;
  • morphisms are $V$-profunctors; and
  • 2-morphisms are natural transformations between these.

• If $C$ is a category with pullbacks, then there is a bicategory $\mathrm{Span}(C)$ whose

  • objects are the objects of $C$;
  • morphisms are spans in $C$; and
  • 2-morphisms are morphisms of spans.

• Every monoidal category $C$ may be thought of as a bicategory $BC$ (its delooping). This has

  • a single object $•$;

  • morphisms are the objects of $C$: $(BC{)}_1=C_0$;

  • 2-morphisms are the morphisms of $C$ : $(BC{)}_2=C_1$;

  horizontal composition in $BC$ is the tensor product in $C$ and vertical composition in $BC$ 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 $B^{2}A$ 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.

Properties

Double nerve

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)

(…)

Model category structure

There is a model category structure on 2-catgories – 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 $k$.

Free resolutions

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

  $$\begin{array}{ccc}& & y\\ & {}^f\nearrow & {\Downarrow }^{c_{f,g}}& {\searrow }^g\\ x& & \underset{g{\circ }_{C}f}{\to }& & z\end{array}$$\array{ && y \\ & {}^{\mathllap{f}}\nearrow &\Downarrow^{c_{f,g}}& \searrow^{\mathrlap{g}} \\ x && \underset{g \circ_C f }{\to} && z }

  and their formal inverses

  $$\begin{array}{ccc}& & y\\ & {}^f\nearrow & {\Uparrow }^{c_{f,g}^{-1}}& {\searrow }^g\\ x& & \underset{g{\circ }_{C}f}{\to }& & z\end{array}$$\array{ && y \\ & {}^{\mathllap{f}}\nearrow &\Uparrow^{c_{f,g}^{-1}}& \searrow^{\mathrlap{g}} \\ x && \underset{g \circ_C f }{\to} && z }

  for all composable $f,g\in \mathrm{Mor}(C)$ with composite (in $C$!) $g{\circ }_{C}f$;

  subject to the relation that for all composable triples $f,g,h\in \mathrm{Mor}(C)$ the following equation of 2-morphisms holds

  $$\begin{array}{ccccccccc}y& \to & & \stackrel{g}{\to }& & \to & & & z\\ \uparrow & {⇘}^{c_{f,g}}& & & & \nearrow & & & \downarrow \\ {}^f\uparrow & & & \nearrow & & & & & {\downarrow }^h\\ \uparrow & \nearrow & & & & {\Downarrow }^{c_{h,(g{\circ }_{C}f)}}& & & \downarrow \\ x& \to & & \underset{h\circ (g{\circ }_{C}f)}{\to }& & \to & & \to & w\end{array}=\begin{array}{ccccccccc}y& \to & & \stackrel{g}{\to }& & \to & & & z\\ \uparrow & \searrow & & & & & & {⇙}_{c_{g,h}}& \downarrow \\ {}^f\uparrow & & & \searrow & & & & & {\downarrow }^h\\ \uparrow & {\Downarrow }_{c_{f,(g{\circ }_{C}h)}}& & & & \searrow & & & \downarrow \\ x& \to & & \underset{(h{\circ }_{C}g)\circ f}{\to }& & \to & & \to & w\end{array}$$\array{ y &\to& &\stackrel{g}{\to}& &\to& && z \\ \uparrow &\seArrow^{c_{f,g}}& && & \nearrow & && \downarrow \\ {}^{\mathllap{f}}\uparrow && & \nearrow & &&&& \downarrow^{\mathrlap{h}} \\ \uparrow & \nearrow & && &\Downarrow^{c_{h,(g\circ_C f)}}& && \downarrow \\ x &\to& &\underset{h \circ (g \circ_C f)}{\to}& &\to& &\to& w } \;\;\; = \;\;\; \array{ y &\to& &\stackrel{g}{\to}& &\to& && z \\ \uparrow &\searrow& && & & &\swArrow_{c_{g,h}}& \downarrow \\ {}^{\mathllap{f}}\uparrow && & \searrow & &&&& \downarrow^{\mathrlap{h}} \\ \uparrow &\Downarrow_{c_{f,(g \circ_C h)}}& && &\searrow& && \downarrow \\ x &\to& &\underset{( h \circ_C g) \circ f}{\to}& &\to& &\to& w }

Observation Let $D$ be any strict 2-catgeory. Then a pseudofunctor $C\to D$ is the same as a strict 2-functor $\hat{C}\to D$.

2-categorical concepts

extra properties

• regular 2-category
• exact 2-category
• coherent 2-category
• extensive 2-category
• 2-pretopos
• 2-topos

types of morphisms

• subcategory
• faithful morphism
• fully faithful morphism
• conservative morphism
• pseudomonic morphism
• discrete morphism

specific versions

• globular strict 2-category
• bicategory

limit notions

• 2-limit
• strict 2-limit
• flexible limit
• PIE limit

model structures

• canonical model structure
• 2-trivial model structure

Related concepts

• 0-category, (0,1)-category

• category

• 2-category

• 3-category

• n-category

• (∞,0)-category

• (∞,1)-category

• (∞,2)-category

• (∞,n)-category

• (n,r)-category

References

A brief account of the definition is in

• Ross Street, Encyclopedia article on 2-categories and bicategories (pdf)

A more detailed account of the definition, including a discussion of its coherence theorem, is in

• Tom Leinster, Basic bicategories (arXiv:9810017)

Some 2-category theory, including 2-limits/2-colimit is discussed in

• Steve Lack, A 2-categories companion (arXiv:0702535)

and

• John Power, 2-Categories (pdf)

An older reference which introduces some of the basic notions is

• Max Kelly and Ross Street, Review of the elements of 2-categories, Sydney Category Seminar 1972/1973, LNM 420

A relation between bicategories and Tamsamani weak 2-categories is established in

• Steve Lack, Simona Paoli, 2-nerves for bicategories (arXiv)

The reverse construction is in

• 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)