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In the mathematical field of category theory, the product of two categories C and D, denoted C × D and called a product category, is an extension of the concept of the Cartesian product of two sets. Product categories are used to define bifunctors and multifunctors.^[1]

Definition

The product category C × D has:

as objects:
pairs of objects (A, B), where A is an object of C and B of D;
as arrows from (A₁, B₁) to (A₂, B₂):
pairs of arrows (f, g), where f : A₁ → A₂ is an arrow of C and g : B₁ → B₂ is an arrow of D;
as composition, component-wise composition from the contributing categories:
(f₂, g₂) o (f₁, g₁) = (f₂ o f₁, g₂ o g₁);
as identities, pairs of identities from the contributing categories:
1_{(A, B)} = (1_A, 1_B).

A product of a family of categories is defined exactly the same way.

Universal property

Just like for sets, a product of a family of categories is characterized by the following universal property. Given categories $C_{i}$ indexed by a set $I$ , $P=\prod C_{i},p_{j}:P\to C_{j},j\in I$ satisfy:

given a family of functors

f_{i}:D\to C_{i}

, there exists a unique functor

f:D\to P

such that

f_{j}=p_{j}\circ f

for each

j\in I

.

Put in another way, a product of a family of small categories is exactly the categorical product of them in the category of small categories ${\mathsf {Cat}}$ . Thus, for example,

\textstyle \operatorname {Fct} (A,\prod _{i}B_{i})\simeq \prod _{i}\operatorname {Fct} (A,B_{i}).

^[2]

Functoriality

Given two functors $f:C\to D,g:C'\to D'$ , the product $f\times g:C\times C'\to D\times D'$ is defined component-wise; that is,

(f\times g)(x,x')=(f(x),f(x'))

for a pair of objects $x,x'$ and a pair of morphisms $x,x'$ .^[3] (This product may also be characterized by the universal property similar to that for categories.) This way, we get the functor

\times :{\mathsf {Cat}}\times {\mathsf {Cat}}\to {\mathsf {Cat}}.

It satisfies the tensor-hom adjunction in the sense

\operatorname {Hom} _{\mathsf {Cat}}(A\times B,C)\simeq \operatorname {Hom} _{\mathsf {Cat}}(A,{\mathsf {Fct}}(B,C))

where ${\mathsf {Fct}}$ denotes a functor category.^[4]

Example: C × 2

Let $f,g:C\to D$ be functors. Suppose there is a natural transformation $\varphi :f\to g$ . Then $\varphi$ determines the functor

h:C\times {\underline {2}}\to D

such that

h(\cdot ,0)=f,\,h(\cdot ,1)=g

,

where ${\underline {2}}=\{0,1\}$ is the category with two objects and the non-identity morphism $\rightsquigarrow :0\to 1$ .^[3] Intuitively, h is a non-invertible homotopy from $f$ to $g$ . Indeed, define $h$ by, for $x:a\to b$ in $C$ ,

h(x,\operatorname {id} _{0})=f(x),\,h(x,\operatorname {id} _{1})=g(x),\,h(x,\rightsquigarrow )=g(x)\circ \varphi _{a}=\varphi _{b}\circ f(x).

Conversely, given $h:C\times {\underline {2}}\to D$ , we get $f,g,\varphi$ by $f=h(\cdot ,0),\,g=h(\cdot ,1)$ and $\varphi _{a}=h(\operatorname {id} _{a},\rightsquigarrow )$ .^[5]

Bifunctor

A functor whose domain is a product category is called a bifunctor. A bifunctor can be defined in each variable separately in the following sense:

Proposition—^[6] Each bifunctor

F:A\times B\to C

determines the families of the functors, for objects $a$ in $A$ and $b$ in $B$ ,

F_{b}:A\to C,\,F_{a}:B\to C

given by

F_{b}a=F(a,b)

and

F_{b}f=F(f,\operatorname {id} _{b})

for $f:a\to a'$ and similarly for $F_{a}$ . They commute in the sense:

F_{a'}g\circ F_{b}f=F_{b'}f\circ F_{a}g

.

Conversely, given families of functors $F_{b},F_{a}$ as above, if they commute, they define the bifunctor $F:A\times B\to C$ by

F(f,g)=F_{b'}f\circ F_{a}g

.

For example, consider $(a,b)\mapsto \operatorname {Hom} (a,b):C^{op}\times C\to {\mathsf {Set}}$ . For each fixed $b$ in $B$ , we have the functor

\operatorname {Hom} (-,b):C^{op}\to {\mathsf {Set}}

by pullback; i.e., $f:a\to a'$ goes to the function

f^{*}:\operatorname {Hom} (a',b)\to \operatorname {Hom} (a,b)

defined by $f^{*}g=g\circ f$ . On the other hand, $\operatorname {Hom} (a,-):C\to {\mathsf {Set}}$ is defined by pushforward; i.e., $f\mapsto f_{*}=f\circ -$ . Clearly, these two functors commute (the associativity of composition) and so, by the proposition, we get the functor called the Hom functor

\operatorname {Hom} (-,-):C^{op}\times C\to {\mathsf {Set}},

which is explicitly given as: $(f,g)\mapsto (h\mapsto g\circ h\circ f).$

There is a similar result for natural transformations between bifunctors:

Proposition—^[7] Let $F,G:A,B\to C$ be bifunctors and

\alpha =\{\alpha _{a,b}:F(a,b)\to G(a,b)\mid a\in \operatorname {Ob} (A),b\in \operatorname {Ob} (B)\}

a family of morphisms. Then $\alpha :F\to G$ is a natural transformation if and only if it is natural in the first variable and the second variable separately; i,e., for each object $b$ in $B$ ,

\alpha _{-,b}:F(-,b)\to G(-,b)

is a natural transformation and similarly in the second variable.

References

^ Mac Lane 1978, p. 37.
^ Mac Lane 1978, Ch. II., § 5., Exercise 2.
^ ^a ^b Mac Lane 1978, Ch. II., § 3.
^ Mac Lane 1978, Ch. II., § 5., Exercise 1.
^ Mac Lane 1978, Ch. II., § 4., Exercise 8.
^ Mac Lane 1978, Ch. II., § 3., Proposition 1.
^ Mac Lane 1978, Ch. II., § 3., Proposition 2.

Definition 1.6.5 in Borceux, Francis (1994). Handbook of categorical algebra. Encyclopedia of mathematics and its applications 50-51, 53 [i.e. 52]. Vol. 1. Cambridge University Press. p. 22. ISBN 0-521-44178-1.
Product category at the nLab
Mac Lane, Saunders (1978). Categories for the Working Mathematician (Second ed.). New York, NY: Springer New York. pp. 36–40. ISBN 1441931236. OCLC 851741862.

[rdp-we-cite_note-FOOTNOTEMac_Lane197837-1] Mac Lane 1978, p. 37.

[rdp-we-cite_note-2] Mac Lane 1978, Ch. II., § 5., Exercise 2.

[rdp-we-cite_note-products-3] Mac Lane 1978, Ch. II., § 3.

[rdp-we-cite_note-4] Mac Lane 1978, Ch. II., § 5., Exercise 1.

[rdp-we-cite_note-5] Mac Lane 1978, Ch. II., § 4., Exercise 8.

[rdp-we-cite_note-6] Mac Lane 1978, Ch. II., § 3., Proposition 1.

[rdp-we-cite_note-7] Mac Lane 1978, Ch. II., § 3., Proposition 2.

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