View source for Strict 2-category

{{Short description|Generalization of category}}
{{hatnote|For the notion of a weak 2-category, see [[bicategory]].}}
In [[category theory]], a '''strict 2-category''' is a [[category (mathematics)|category]] with "[[morphism]]s between morphisms", that is, where each [[hom-set]] itself carries the structure of a category.   It can be formally defined as a category [[enriched category|enriched]] over '''Cat''' (the [[Category of small categories|category of categories and functors]], with the [[monoidal category|monoidal]] structure given by [[product category|product of categories]]).

The concept of 2-category was first introduced  by [[Charles Ehresmann]] in his work on [[enriched categories]] in 1965.<ref>[[Charles Ehresmann]], Catégories et structures, Dunod, Paris 1965.</ref>  The more general concept of [[bicategory]] (or ''weak'' 2-''category''), where composition of morphisms is [[associative]] only up to a 2-isomorphism, was introduced in 1968 by [[Jean Bénabou]].<ref>[[Jean Bénabou]], Introduction to bicategories, in Reports of the Midwest Category Seminar, Springer, Berlin, 1967, pp. 1--77.</ref>

== Definition ==
A 2-category&nbsp;'''C''' consists of:
* A [[Class (set theory)|class]] of 0-''cells'' (or ''[[Object (category theory)|objects]]'') {{mvar|A}}, {{mvar|B}}, ....
* For all objects {{mvar|A}} and {{mvar|B}}, a category <math>\mathbf{C}(A,B)</math>. The objects <math>f,g: A \to B</math> of this category are called 1-''cells'' and its morphisms <math>\alpha: f \Rightarrow g</math> are called 2-''cells''; the composition in this category is usually written <math>\circ</math> or <math>\circ_1</math> and called ''vertical composition'' or ''composition along a'' 1-''cell''.
* For any object&nbsp;{{mvar|A}} there is a [[functor]] from the [[terminal object|terminal]] [[category (mathematics)|category]] (with one object and one arrow) to <math>\mathbf{C}(A,A)</math> that picks out the [[Identity morphism|identity]] 1-cell&nbsp;{{math|id<sub>''A''</sub>}} on {{mvar|A}} and its identity 2-cell&nbsp;{{math|id<sub>id<sub>''A''</sub></sub>}}.  In practice these two are often denoted simply by {{mvar|A}}.
* For all objects {{mvar|A}}, {{mvar|B}} and {{mvar|C}}, there is a functor <math>\circ_0 \colon \mathbf{C}(B,C) \times \mathbf{C}(A,B) \to \mathbf{C}(A,C)</math>, called ''horizontal composition'' or ''composition along a 0-cell'', which is associative and admits{{clarify |date=March 2020 |reason=the meaning of "admits" is unclear in this context}} the identity 1 and 2-cells of {{math|id<sub>''A''</sub>}} as identities. Here, associativity for <math>\circ_0</math> means that horizontally composing <math>\mathbf{C}(C,D)\times\mathbf{C}(B,C)\times\mathbf{C}(A,B)</math> twice to <math>\mathbf{C}(A,D)</math> is independent of which of the two <math>\mathbf{C}(C,D)\times\mathbf{C}(B,C)</math> and <math>\mathbf{C}(B,C)\times\mathbf{C}(A,B)</math> are composed first.  The composition symbol <math>\circ_0</math> is often omitted, the horizontal composite of 2-cells <math>\alpha\colon f\Rightarrow g\colon A\to B</math> and <math>\beta\colon f'\Rightarrow g'\colon B\to C</math> being written simply as <math>\beta\alpha\colon f'f\Rightarrow g'g\colon A\to C</math>.

The ''0-cells'', ''1-cells'', and ''2-cells'' terminology is replaced by ''0-morphisms'', ''1-morphisms'', and ''2-morphisms'' in some sources<ref>{{Cite web |title=2-category in nLab |url=https://ncatlab.org/nlab/show/2-category |access-date=2023-02-20 |website=ncatlab.org}}</ref> (see also [[Higher category theory]]).

The notion of 2-category differs from the more general notion of a [[bicategory]] in that composition of 1-cells (horizontal composition) is required to be strictly associative, whereas in a bicategory it needs only be associative up to a 2-isomorphism.  The axioms of a 2-category are consequences of their definition as '''Cat'''-enriched categories:

* Vertical composition is associative and unital, the units being the identity 2-cells {{math|id<sub>''f''</sub>}}.
* Horizontal composition is also (strictly) associative and unital, the units being the identity 2-cells {{math|id<sub>id<sub>''A''</sub></sub>}} on the identity 1-cells {{math|id<sub>''A''</sub>}}.
* The [[interchange law]] holds; i.e. it is true that for composable 2-cells <math>\alpha,\beta,\gamma,\delta</math>
::<math>(\alpha\circ_0\beta)\circ_1(\gamma\circ_0\delta) = (\alpha\circ_1\gamma)\circ_0(\beta\circ_1\delta)</math>

The interchange law follows from the fact that <math>\circ_0</math> is a functor between hom categories.  It can be drawn as a [[pasting diagram]] as follows:
{| cellpadding="2" align="center"
 |- valign="center"
 | [[Image:2-category horizontal composition upper.svg|240px]]
 | rowspan="3" style="font-size:large" |&nbsp;=&nbsp;
 | rowspan="3" |[[Image:2-category double composition.svg|240px]]
 | rowspan="3" style="font-size:large" |&nbsp;=&nbsp;
 | rowspan="3" |[[Image:2-category vertical composition.svg|132px]]
 | rowspan="3" style="font-size:large" |<math>\circ_0</math>
 | rowspan="3" |[[Image:2-category vertical composition.svg|132px]]
 |-
 | style="font-size:large" align="center" |<math>\circ_1</math>
 |-
 | [[Image:2-category horizontal composition lower.svg|240px]]
 |}

Here the left-hand diagram denotes the vertical composition of horizontal composites, the right-hand diagram denotes the horizontal composition of vertical composites, and the diagram in the centre is the customary representation of both. The ''2-cell'' are drawn with double arrows ⇒, the ''1-cell'' with single arrows →, and the ''0-cell'' with points.

== Examples ==
The category [[Category of preordered sets|'''Ord''']] (of preordered sets) is a 2-category since preordered sets can easily be interpreted as categories.

=== Category of small categories ===
The archetypal 2-category is the [[category of small categories]], with natural transformations serving as 2-morphisms; typically 2-morphisms are given by Greek letters (such as <math>\alpha</math> above) for this reason.

The objects (''0-cells'') are all small categories, and for all objects {{mvar|A}} and {{mvar|B}} the category <math>\mathbf{C}(A,B)</math> is a [[functor category]]. In this context, vertical composition is<ref>{{Cite web |title=vertical composition in nLab |url=https://ncatlab.org/nlab/show/vertical+composition |access-date=2023-02-20 |website=ncatlab.org}}</ref> the composition of natural transformations.

== Doctrines ==
In mathematics, a '''doctrine''' is simply a 2-category which is heuristically regarded as a system of theories.  For example, [[algebraic theory|algebraic theories]], as invented by [[William Lawvere]], is an example of a doctrine, as are [[multi-sorted theory|multi-sorted theories]], [[operad]]s, [[category (mathematics)|categories]], and [[topos (mathematics)|toposes]].

The objects of the 2-category are called ''theories'', the 1-morphisms <math>f\colon A\rightarrow B</math> are called ''models'' of the {{mvar|A}} in {{mvar|B}}, and the 2-morphisms are called ''morphisms between models.''

The distinction between a 2-category and a doctrine is really only heuristic: one does not typically consider a 2-category to be populated by theories as objects and models as morphisms.  It is this vocabulary that makes the theory of doctrines worth while.

For example, the 2-category '''Cat''' of categories, functors, and natural transformations is a doctrine.  One sees immediately that all [[presheaf category|presheaf categories]] are categories of models.

As another example, one may take the subcategory of '''Cat''' consisting only of categories with finite products as objects and product-preserving functors as 1-morphisms.  This is the doctrine of multi-sorted algebraic theories.  If one only wanted 1-sorted algebraic theories, one would restrict the objects to only those categories that are generated under products by a single object.

Doctrines were discovered by [[Jonathan Mock Beck]].

== See also ==
* [[n-category|''n''-category]]
* {{nlab|id=2-category}}

== References ==
{{reflist}}

==Footnotes==
* ''Generalised algebraic models'', by Claudia Centazzo.

==External links==
*{{Commonscatinline}}

{{Category theory}}

[[Category:Higher category theory]]