Strict 2-category

For the notion of a weak 2-category, see bicategory.

In category theory, a strict 2-category is a category with "morphisms between morphisms", that is, where each hom-set itself carries the structure of a category. It can be formally defined as a category enriched over Cat (the category of categories and functors, with the monoidal structure given by product of categories).

The concept of 2-category was first introduced by Charles Ehresmann in his work on enriched categories in 1965. ^[1] 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. ^[2]

Definition

A 2-category C consists of:

• A class of 0-cells (or objects ) A, B, ....
• For all objects A and B, a category ${\displaystyle \mathbf {C} (A,B)}$. The objects ${\displaystyle f,g:A\to B}$ of this category are called 1-cells and its morphisms ${\displaystyle \alpha :f\Rightarrow g}$ are called 2-cells; the composition in this category is usually written ${\displaystyle \circ }$ or ${\displaystyle \circ _{1}}$ and called vertical composition or composition along a 1-cell.
• For any object A there is a functor from the terminal category (with one object and one arrow) to ${\displaystyle \mathbf {C} (A,A)}$ that picks out the identity 1-cell id_A on A and its identity 2-cell id_idA. In practice these two are often denoted simply by A.
• For all objects A, B and C, there is a functor ${\displaystyle \circ _{0}\colon \mathbf {C} (B,C)\times \mathbf {C} (A,B)\to \mathbf {C} (A,C)}$, called horizontal composition or composition along a 0-cell, which is associative and admits^{[ clarification needed ]} the identity 1 and 2-cells of id_A as identities. Here, associativity for ${\displaystyle \circ _{0}}$ means that horizontally composing ${\displaystyle \mathbf {C} (C,D)\times \mathbf {C} (B,C)\times \mathbf {C} (A,B)}$ twice to ${\displaystyle \mathbf {C} (A,D)}$ is independent of which of the two ${\displaystyle \mathbf {C} (C,D)\times \mathbf {C} (B,C)}$ and ${\displaystyle \mathbf {C} (B,C)\times \mathbf {C} (A,B)}$ are composed first. The composition symbol ${\displaystyle \circ _{0}}$ is often omitted, the horizontal composite of 2-cells ${\displaystyle \alpha \colon f\Rightarrow g\colon A\to B}$ and ${\displaystyle \beta \colon f'\Rightarrow g'\colon B\to C}$ being written simply as ${\displaystyle \beta \alpha \colon f'f\Rightarrow g'g\colon A\to C}$.

The 0-cells, 1-cells, and 2-cells terminology is replaced by 0-morphisms, 1-morphisms, and 2-morphisms in some sources ^[3] (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 id_f.
• Horizontal composition is also (strictly) associative and unital, the units being the identity 2-cells id_idA on the identity 1-cells id_A.
• The interchange law holds; i.e. it is true that for composable 2-cells ${\displaystyle \alpha ,\beta ,\gamma ,\delta }$

${\displaystyle (\alpha \circ _{0}\beta )\circ _{1}(\gamma \circ _{0}\delta )=(\alpha \circ _{1}\gamma )\circ _{0}(\beta \circ _{1}\delta )}$

The interchange law follows from the fact that ${\displaystyle \circ _{0}}$ is a functor between hom categories. It can be drawn as a pasting diagram as follows:

	=		=		${\displaystyle \circ _{0}}$
${\displaystyle \circ _{1}}$

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 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 ${\displaystyle \alpha }$ above) for this reason.

The objects (0-cells) are all small categories, and for all objects A and B the category ${\displaystyle \mathbf {C} (A,B)}$ is a functor category. In this context, vertical composition is ^[4] the composition of natural transformations.

Duskin nerve

The Duskin nerve${\displaystyle N^{hc}(C)}$ of a 2-category C is a simplicial set where each n-simplex is determined by the following data: n objects ${\displaystyle x_{1},\dots ,x_{n}}$, morphisms ${\displaystyle f_{ij}:x_{i}\to x_{j},\,i<j}$ and 2-morphisms ${\displaystyle \mu _{ijk}:f_{jk}\circ f_{ij}\rightarrow f_{ik},\,i<j<k}$ that are subject to the (obvious) compatibility conditions. ^[5] Then the following are equivalent: ^[6]

• ${\displaystyle C}$ is a (2, 1)-category; i.e., each 2-morphism is invertible.
• ${\displaystyle N^{hc}(C)}$ is a weak Kan complex.

The Duskin nerve is an instance of the homotopy coherent nerve.

See also

• n-category
• 2-category at the nLab
• Doctrine (mathematics)

References

1. Charles Ehresmann, Catégories et structures, Dunod, Paris 1965.
2. Jean Bénabou, Introduction to bicategories, in Reports of the Midwest Category Seminar, Springer, Berlin, 1967, pp. 1--77.
3. "2-category in nLab". ncatlab.org. Retrieved 2023-02-20.
4. "vertical composition in nLab". ncatlab.org. Retrieved 2023-02-20.
5. Khan 2023 , Construction 1.5.1. harvnb error: no target: CITEREFKhan2023 (help)
6. Khan 2023 , Theorem 1.5.2. harvnb error: no target: CITEREFKhan2023 (help)

• Adeel A. Khan, A modern introduction to algebraic stacks, https://www.preschema.com/lecture-notes/2022-stacks/
• Generalised algebraic models, by Claudia Centazzo.
• Garth Warner: Fibrations and Sheaves, EPrint Collection, University of Washington (2012)

External links

• Media related to Strict 2-category at Wikimedia Commons