product

Definition

Product of two objects

Fix a category

C

.

{\displaystyle C.}

Let

X

1

{\displaystyle X_{1}}

and

X

2

{\displaystyle X_{2}}

be objects of

C

.

{\displaystyle C.}

A product of

X

1

{\displaystyle X_{1}}

and

X

2

{\displaystyle X_{2}}

is an object

X

,

{\displaystyle X,}

typically denoted

X

1

×

X

2

,

{\displaystyle X_{1}\times X_{2},}

equipped with a pair of morphisms

π

1

:

X

→

X

1

,

{\displaystyle \pi _{1}:X\to X_{1},}

π

2

:

X

→

X

2

{\displaystyle \pi _{2}:X\to X_{2}}

satisfying the following universal property:

For every object

Y

{\displaystyle Y}

and every pair of morphisms

f

1

:

Y

→

X

1

,

{\displaystyle f_{1}:Y\to X_{1},}

f

2

:

Y

→

X

2

,

{\displaystyle f_{2}:Y\to X_{2},}

there exists a unique morphism

f

:

Y

→

X

1

×

X

2

{\displaystyle f:Y\to X_{1}\times X_{2}}

such that the following diagram commutes:

Whether a product exists may depend on

C

{\displaystyle C}

or on

X

1

{\displaystyle X_{1}}

and

X

2

.

{\displaystyle X_{2}.}

If it does exist, it is unique up to canonical isomorphism, because of the universal property, so one may speak of the product. This has the following meaning: if

X

′

,

π

1

′

,

π

2

′

{\displaystyle X',\pi _{1}',\pi _{2}'}

is another product, there exists a unique isomorphism

h

:

X

′

→

X

1

×

X

2

{\displaystyle h:X'\to X_{1}\times X_{2}}

such that

π

1

′

=

π

1

∘

h

{\displaystyle \pi _{1}'=\pi _{1}\circ h}

and

π

2

′

=

π

2

∘

h

{\displaystyle \pi _{2}'=\pi _{2}\circ h}

.

The morphisms

π

1

{\displaystyle \pi _{1}}

and

π

2

{\displaystyle \pi _{2}}

are called the canonical projections or projection morphisms; the letter

π

{\displaystyle \pi }

alliterates with projection. Given

Y

{\displaystyle Y}

and

f

1

,

{\displaystyle f_{1},}

f

2

,

{\displaystyle f_{2},}

the unique morphism

f

{\displaystyle f}

is called the product of morphisms

f

1

{\displaystyle f_{1}}

and

f

2

{\displaystyle f_{2}}

and may be denoted

⟨

f

1

,

f

2

⟩

{\displaystyle \langle f_{1},f_{2}\rangle }

,

f

1

×

f

2

{\displaystyle f_{1}\times f_{2}}

, or

f

1

⊗

f

2

{\displaystyle f_{1}\otimes f_{2}}

.

Product of an arbitrary family

Instead of two objects, we can start with an arbitrary family of objects indexed by a set

I

.

{\displaystyle I.}

Given a family

(

X

i

)

i

∈

I

{\displaystyle \left(X_{i}\right)_{i\in I}}

of objects, a product of the family is an object

X

{\displaystyle X}

equipped with morphisms

π

i

:

X

→

X

i

,

{\displaystyle \pi _{i}:X\to X_{i},}

satisfying the following universal property:

For every object

Y

{\displaystyle Y}

Equational definition

Alternatively, the product may be defined through equations. So, for example, for the binary product:

Existence of

f

{\displaystyle f}

is guaranteed by existence of the operation

⟨

⋅

,

⋅

⟩

.

{\displaystyle \langle \cdot ,\cdot \rangle .}

Commutativity of the diagrams above is guaranteed by the equality: for all

f

1

,

f

2

{\displaystyle f_{1},f_{2}}

and all

i

∈

{

1

,

2

}

,

{\displaystyle i\in \{1,2\},}

π

As a limit

The product is a special case of a limit. This may be seen by using a discrete category (a family of objects without any morphisms, other than their identity morphisms) as the diagram required for the definition of the limit. The discrete objects will serve as the index of the components and projections. If we regard this diagram as a functor, it is a functor from the index set

I

{\displaystyle I}

considered as a discrete category. The definition of the product then coincides with the definition of the limit,

{

f

}

i

{\displaystyle \{f\}_{i}}

being a cone and projections being the limit (limiting cone).

Universal property

Just as the limit is a special case of the universal construction, so is the product. Starting with the definition given for the universal property of limits, take

J

{\displaystyle \mathbf {J} }

as the discrete category with two objects, so that

C

J

{\displaystyle \mathbf {C} ^{\mathbf {J} }}

is simply the product category

C

×

C

.

{\displaystyle \mathbf {C} \times \mathbf {C} .}

The diagonal functor

Δ

:

C

→

C

×

C

{\displaystyle \Delta :\mathbf {C} \to \mathbf {C} \times \mathbf {C} }

assigns to each object

X

{\displaystyle X}

the ordered pair

(

X

Examples

In the category of sets, the product (in the category theoretic sense) is the Cartesian product. Given a family of sets

X

i

{\displaystyle X_{i}}

the product is defined as

∏

i

∈

I

X

i

:=

{

(

x

i

)

i

∈

I

:

x

i

∈

X

i

for all

i

∈

I

}

{\displaystyle \prod _{i\in I}X_{i}:=\left\{\left(x_{i}\right)_{i\in I}:x_{i}\in X_{i}{\text{ for all }}i\in I\right\}}

with the canonical projections

Discussion

An example in which the product does not exist: In the category of fields, the product

Q

×

F

p

{\displaystyle \mathbb {Q} \times F_{p}}

does not exist, since there is no field with homomorphisms to both

Q

{\displaystyle \mathbb {Q} }

and

F

p

.

{\displaystyle F_{p}.}

Another example: An empty product (that is,

I

{\displaystyle I}

is the empty set) is the same as a terminal object, and some categories, such as the category of infinite groups, do not have a terminal object: given any infinite group

G

{\displaystyle G}

there are infinitely many morphisms

Z

→

G

,

{\displaystyle \mathbb {Z} \to G,}

Distributivity

For any objects

X

,

Y

,

and

Z

{\displaystyle X,Y,{\text{ and }}Z}

of a category with finite products and coproducts, there is a canonical morphism

X

×

Y

+

X

×

Z

→

X

×

(

Y

+

Z

)

,

{\displaystyle X\times Y+X\times Z\to X\times (Y+Z),}

where the plus sign here denotes the coproduct. To see this, note that the universal property of the coproduct

X

×

Y

+

X