3-category

In mathematics, especially in category theory, a 3-category is a 2-category together with 3-morphisms. It comes in at least three flavors

a strict 3-category,

a semi-strict 3-category also called a Gray category,

a weak 3-category.

The coherence theorem of Gordon–Power–Street says a weak 3-category is equivalent (in some sense) to a Gray category.

Strict and weak 3-categories

A strict 3-category is defined as a category enriched over 2Cat, the monoidal category of (small) strict 2-categories. A weak 3-category is then defined roughly by replacing the equalities in the axioms by coherent isomorphisms.

Gray tensor product

Introduced by Gray, a Gray tensor product is a replacement of a product of 2-categories that is more convenient for higher category theory. Precisely, given a morphism

f

:

x

→

y

{\displaystyle f:x\to y}

in a strict 2-category C and

g

:

a

→

b

{\displaystyle g:a\to b}

in D, the usual product is given as

f

×

g

:

(

x

,

a

)

→

(

y

,

b

)

{\displaystyle f\times g:(x,a)\to (y,b)}

that factors both as

u

=

(

id

,

g

)

∘

(

f

,

id

)

{\displaystyle u=(\operatorname {id} ,g)\circ (f,\operatorname {id} )}

and

v

=

(

f

,

id

)

∘

(

id

,

g

)

{\displaystyle v=(f,\operatorname {id} )\circ (\operatorname {id} ,g)}

. The Gray tensor product

f

⊗

g

{\displaystyle f\otimes g}

weakens this so that we merely have a 2-morphism from

u

{\displaystyle u}

to

v

{\displaystyle v}

. Some authors require this 2-morphism to be an isomorphism, amounting to replacing lax with pseudo in the theory.

Let Gray be the monoidal category of strict 2-categories and strict 2-functors with the Gray tensor product.