Product category

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

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.

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

given a family of functors ${\displaystyle f_{i}:D\to C_{i}}$, there exists a unique functor ${\displaystyle f:D\to P}$ such that ${\displaystyle f_{j}=p_{j}\circ f}$ for each ${\displaystyle j\in I}$.

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 ${\displaystyle {\mathsf {Cat}}}$. Thus, for example, $${\displaystyle \textstyle {\mathsf {Fct}}(A,\prod _{i}B_{i})\simeq \prod _{i}{\mathsf {Fct}}(A,B_{i})}$$

where ${\displaystyle {\mathsf {Fct}}}$ denotes a functor category.²

Given two functors ${\displaystyle f:C\to D,g:C'\to D'}$, the product ${\displaystyle f\times g:C\times C'\to D\times D'}$ is defined component-wise; that is, $${\displaystyle (f\times g)(x,x')=(f(x),g(x'))}$$ for a pair of objects or morphisms ${\displaystyle x,x'}$.³ (This product may also be characterized by the universal property similar to that for categories.) This way, we get the functor

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

It satisfies the tensor-hom adjunction in the sense

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

where ${\displaystyle {\mathsf {Fct}}}$ denotes a functor category.⁴

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

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

such that

${\displaystyle h(\cdot ,0)=f,\,h(\cdot ,1)=g}$,

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

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

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

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:

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

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

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

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

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

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

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

There is a similar result for natural transformations between bifunctors: