2-category

In category theory, a strict 2-category is a category with "morphisms between morphisms", called 2-morphisms. A basic example is the category Cat of all (small) categories, where a 2-morphism is a natural transformation.

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]

A (2, 1)-category is a 2-category where each 2-morphism is invertible.

Definition

A 2-category C consists of:

A class of 0-cells (or objects) $A$ , $B$ , ....
For all objects $A$ and $B$ , a category $\mathbf {C} (A,B)$ . The objects $f,g:A\to B$ of this category are called 1-cells and its morphisms $\alpha :f\Rightarrow g$ are called 2-cells; the composition in this category is usually written $\circ$ or $\circ _{1}$ 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 $\mathbf {C} (A,A)$ that picks out the identity 1-cell $id A$ on $A$ and its identity 2-cell $id id A$ . In practice these two are often denoted simply by $A$ .
For all objects $A$ , $B$ and $C$ , there is a functor $\circ _{0}\colon \mathbf {C} (B,C)\times \mathbf {C} (A,B)\to \mathbf {C} (A,C)$ , 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 $\circ _{0}$ means that horizontally composing $\mathbf {C} (C,D)\times \mathbf {C} (B,C)\times \mathbf {C} (A,B)$ twice to $\mathbf {C} (A,D)$ is independent of which of the two $\mathbf {C} (C,D)\times \mathbf {C} (B,C)$ and $\mathbf {C} (B,C)\times \mathbf {C} (A,B)$ are composed first. The composition symbol $\circ _{0}$ is often omitted, the horizontal composite of 2-cells $\alpha \colon f\Rightarrow g\colon A\to B$ and $\beta \colon f'\Rightarrow g'\colon B\to C$ being written simply as $\beta \alpha \colon f'f\Rightarrow g'g\colon A\to C$ .

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 $id id A$ on the identity 1-cells $id A$ .
The interchange law holds; i.e. it is true that for composable 2-cells $\alpha ,\beta ,\gamma ,\delta$

(\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 $\circ _{0}$ is a functor between hom categories. It can be drawn as a pasting diagram as follows:

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

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.

Bicategories may be considered as a weakening of the definition of 2-categories. A similar process for 3-categories leads to tricategories, and more generally to weak n-categories for n-categories.

Weak 2-category

Formally, a weak 2-category or bicategory B consists of:

objects a, b, ... called 0-cells;
morphisms f, g, ... with fixed source and target objects called 1-cells;
"morphisms between morphisms" ρ, σ, ... with fixed source and target morphisms (which should have themselves the same source and the same target), called 2-cells;

with some more structure:

given two objects a and b there is a category B(a, b) whose objects are the 1-cells and morphisms are the 2-cells. The composition in this category is called vertical composition;
given three objects a, b and c, there is a bifunctor $*:\mathbf {B} (b,c)\times \mathbf {B} (a,b)\to \mathbf {B} (a,c)$ called horizontal composition.

The horizontal composition is required to be associative up to a natural isomorphism α between morphisms $h*(g*f)$ and $(h*g)*f$ . Some more coherence axioms, similar to those needed for monoidal categories, are moreover required to hold: a monoidal category is the same as a bicategory with one 0-cell.

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 $\alpha$ above) for this reason.

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

Grpd

Like Cat, groupoids (categories where morphisms are invertible) form a 2-category, where a 2-morphism is a natural transformation. Often, one also considers Grpd where all 2-morphisms are invertible transformations. In the latter case, it is a (2, 1)-category.

Boolean monoidal category

Consider a simple monoidal category, such as the monoidal preorder Bool^[5] based on the monoid M = ({T, F}, ∧, T). As a category this is presented with two objects {T, F} and single morphism g: F → T.

We can reinterpret this monoid as a bicategory with a single object x (one 0-cell); this construction is analogous to construction of a small category from a monoid. The objects {T, F} become morphisms, and the morphism g becomes a natural transformation (forming a functor category for the single hom-category B(x, x)).

Duskin nerve

The Duskin nerve $N^{hc}(C)$ of a 2-category C is a simplicial set where each n-simplex is determined by the following data: n objects $x_{1},\dots ,x_{n}$ , morphisms $f_{ij}:x_{i}\to x_{j},\,i<j$ and 2-morphisms $\mu _{ijk}:f_{jk}\circ f_{ij}\rightarrow f_{ik},\,i<j<k$ that are subject to the (obvious) compatibility conditions.^[6] Then the following are equivalent: ^[7]

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

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

2-functor

A 2-functor is a morphism between 2-categories.^[8] They may be defined formally using enrichment by saying that a 2-category is exactly a Cat-enriched category and a 2-functor is a Cat-functor.^[9]

Explicitly, if C and D are 2-categories then a 2-functor $F\colon C\to D$ consists of

a function $F\colon {\text{Ob}}C\to {\text{Ob}}D$ , and
for each pair of objects $c,c'\in {\text{Ob}}C$ , a functor $F_{c,c'}\colon {\text{Hom}}_{C}(c,c')\to {\text{Hom}}_{D}(Fc,Fc')$

such that each $F_{c,c'}$ strictly preserves identity objects and they commute with horizontal composition in C and D.

See ^[10] for more details and for lax versions.

References

^ Charles Ehresmann, Catégories et structures, Dunod, Paris 1965.
^ Jean Bénabou, Introduction to bicategories, in Reports of the Midwest Category Seminar, Springer, Berlin, 1967, pp. 1--77.
^ "2-category in nLab". ncatlab.org. Retrieved 2023-02-20.
^ "vertical composition in nLab". ncatlab.org. Retrieved 2023-02-20.
^ Fong, Brendan; Spivak, David I. (2018-10-12). "Seven Sketches in Compositionality: An Invitation to Applied Category Theory". arXiv:1803.05316 [math.CT].
^ Khan 2023, Construction 1.5.1.
^ Khan 2023, Theorem 1.5.2.
^ Kelly, G. M.; Street, Ross (1974). "Review of the elements of 2-categories". In Kelly, Gregory M. (ed.). Category Seminar: Proceedings of the Sydney Category Theory Seminar, 1972/1973. Lecture Notes in Mathematics. Vol. 420. Springer. pp. 75–103. doi:10.1007/BFb0063101. ISBN 978-3-540-06966-9. MR 0357542.
^ G. M. Kelly. Basic concepts of enriched category theory. Reprints in Theory and Applications of Categories, (10), 2005.
^ 2-functor at the nLab

J. Bénabou. "Introduction to bicategories, part I". In Reports of the Midwest Category Seminar, Lecture Notes in Mathematics 47, pages 1–77. Springer, 1967.
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) [1]
Laumon, Gérard; Moret-Bailly, Laurent (2000). Champs algébriques. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics. Vol. 39. Berlin, New York: Springer-Verlag. doi:10.1007/978-3-540-24899-6. ISBN 978-3-540-65761-3. MR 1771927.

External links

Media related to Strict 2-category at Wikimedia Commons

[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] Fong, Brendan; Spivak, David I. (2018-10-12). "Seven Sketches in Compositionality: An Invitation to Applied Category Theory". arXiv:1803.05316 [math.CT].

[6] Khan 2023, Construction 1.5.1.

[7] Khan 2023, Theorem 1.5.2.

[8] Kelly, G. M.; Street, Ross (1974). "Review of the elements of 2-categories". In Kelly, Gregory M. (ed.). Category Seminar: Proceedings of the Sydney Category Theory Seminar, 1972/1973. Lecture Notes in Mathematics. Vol. 420. Springer. pp. 75–103. doi:10.1007/BFb0063101. ISBN 978-3-540-06966-9. MR 0357542.

[9] G. M. Kelly. Basic concepts of enriched category theory. Reprints in Theory and Applications of Categories, (10), 2005.

[10] 2-functor at the nLab

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