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. In old texts, strict 2-categories are occasionally called hypercategories.

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 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

• adjunctions;

• monads.

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

2-categories form a 3-category, 2Cat.

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 $hom(x,y)$. 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 $\infty$-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.

Examples

• 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

  • 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 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 $\mathbf{B}C$ (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.

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-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 $k$.

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

  $$\array{ && y \\ & {}^{\mathllap{f}}\nearrow &\Downarrow^{c_{f,g}}& \searrow^{\mathrlap{g}} \\ x && \underset{g \circ_C f }{\to} && z }$$

  and their formal inverses

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

  for all composable $f,g \in Mor(C)$ 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

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

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

constructions

• opposite 2-category

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

  equivalence in a 2-category

  localization of a 2-category

  2-type theory, directed type theory

• 3-category

• n-category

• (∞,0)-category

• (∞,1)-category

• (∞,2)-category

• (∞,n)-category

• (n,r)-category

• double category

References

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

• Charles Ehresmann, Catégories double et catégories structurées, C.R. Acad. Paris 256 (1963) 1198-1201 [pdf, gallica]

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:

• Roger Godement, Appendix (pp. 269) of: Topologie algébrique et theorie des faisceaux, Actualités Sci. Ind. 1252, Hermann, Paris (1958) [webpage, pdf]

Early discussion of the general notion of bicategories:

• Jean Bénabou, Introduction to Bicategories, Lecture Notes in Mathematics 47 Springer (1967) 1-77 [doi:10.1007/BFb0074299]

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)

  (including discussion of (strict) 2-limits)

• 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]

On coherence theorems:

• A. John Power, A general coherence result. J. Pure Appl. Algebra 57 (1989), no. 2, 165–173. doi:10.1016/0022-4049(89)90113-8 MR0985657

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.