2-category

In category theory in mathematics, a 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 between functors.

The concept of a strict 2-category was first introduced by Charles Ehresmann in his work on enriched categories in 1965. 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 1967 by Jean Bénabou.

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

Definitions

A strict 2-category

By definition, a strict 2-category C consists of the data:

a class of 0-cells,

for each pairs of 0-cells

a

,

b

{\displaystyle a,b}

, a set

Hom

⁡

(

a

,

b

)

{\displaystyle \operatorname {Hom} (a,b)}

called the set of 1-cells from

a

{\displaystyle a}

to

b

{\displaystyle b}

,

for each pairs of 1-cells

f

,

g

{\displaystyle f,g}

in the same hom-set, a set

2Mor

⁡

(

f

,

g

)

{\displaystyle \operatorname {2Mor} (f,g)}

called the set of 2-cells from

f

{\displaystyle f}

to

g

{\displaystyle g}

,

ordinary compositions: maps

∘

:

Hom

⁡

(

b

,

c

)

×

Hom

⁡

(

a

,

b

)

→

Hom

⁡

(

a

,

c

)

{\displaystyle \circ :\operatorname {Hom} (b,c)\times \operatorname {Hom} (a,b)\to \operatorname {Hom} (a,c)}

,

vertical compositions: maps

∘

:

2Mor

⁡

(

g

,

h

)

×

2Mor

⁡

(

f

,

g

)

→

2Mor

⁡

(

f

,

h

)

{\displaystyle \circ :\operatorname {2Mor} (g,h)\times \operatorname {2Mor} (f,g)\to \operatorname {2Mor} (f,h)}

, where

f

,

g

,

h

{\displaystyle f,g,h}

are in the same hom-set,

horizontal compositions: maps

∗

:

2Mor

⁡

(

u

,

v

)

×

2Mor

⁡

(

f

,

g

)

→

2Mor

⁡

(

u

∘

f

,

v

∘

g

)

{\displaystyle *:\operatorname {2Mor} (u,v)\times \operatorname {2Mor} (f,g)\to \operatorname {2Mor} (u\circ f,v\circ g)}

for

f

,

g

:

a

→

b

{\displaystyle f,g:a\to b}

and

u

,

v

:

b

→

c

{\displaystyle u,v:b\to c}

that are subject to the following conditions

the 0-cells with 1-cells between them form a category under ordinary composition,

for each 0-cells

a

{\displaystyle a}

and

b

{\displaystyle b}

, the 1-cells from

a

{\displaystyle a}

to

b

{\displaystyle b}

with 2-cells between them form a category under vertical composition,

the 0-cells with 2-cells between 1-cells between them form a category under horizontal composition; namely, an object is a 0-cell and the hom-set from

a

{\displaystyle a}

to

b

{\displaystyle b}

is the set of all 2-cells of the form

α

:

f

⇒

g

{\displaystyle \alpha :f\Rightarrow g}

for some

f

,

g

:

a

→

b

{\displaystyle f,g:a\to b}

,

the interchange law:

(

δ

∗

β

)

∘

(

γ

∗

α

)

{\displaystyle (\delta *\beta )\circ (\gamma *\alpha )}

, when defined, is the same as

(

δ

∘

γ

)

∗

(

β

∘

α

)

{\displaystyle (\delta \circ \gamma )*(\beta \circ \alpha )}

.

The 0-cells, 1-cells, and 2-cells terminology is replaced by 0-morphisms, 1-morphisms, and 2-morphisms in some sources (see also Higher category theory). Vertical compositions and horizontal compositions are also written as

∘

1

,

∘

0

{\displaystyle \circ _{1},\circ _{0}}

.

The interchange law can be drawn as a pasting diagram as follows:

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.