Product category

Short description: Product of two categories, in category theory

"Product of categories" redirects here. For the operation on objects of a category, see Product (category theory).

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.

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 $\mathsf{C}\mathsf{a}\mathsf{t}$. Thus, for example,

$\mathrm{Fct}(A,\prod _i{B}_i)\simeq \prod _i\mathrm{Fct}(A,B_i).$

Functoriality

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

$(f\times g)(x,x^{\prime })=(f(x),f(x^{\prime }))$

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

$\times :\mathsf{C}\mathsf{a}\mathsf{t}\times \mathsf{C}\mathsf{a}\mathsf{t}\to \mathsf{C}\mathsf{a}\mathsf{t}.$

It satisfies the tensor-hom adjunction in the sense

${\mathrm{Hom}}_{\mathsf{C}\mathsf{a}\mathsf{t}}(A\times B,C)\simeq {\mathrm{Hom}}_{\mathsf{C}\mathsf{a}\mathsf{t}}(A,\mathsf{F}\mathsf{c}\mathsf{t}(B,C))$

where $\mathsf{F}\mathsf{c}\mathsf{t}$ denotes a functor category.

Example: C × 2

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

$h:C\times \underset{\_}{2}\to D$

such that

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

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

$h(x,{\mathrm{id}}_0)=f(x),h(x,{\mathrm{id}}_1)=g(x),h(x,⇝)=g(x)\circ {\phi }_a={\phi }_b\circ f(x).$

Conversely, given $h:C\times \underset{\_}{2}\to D$, we get $f,g,\phi$ by $f=h(\cdot ,0),g=h(\cdot ,1)$ and ${\phi }_a=h({\mathrm{id}}_a,⇝)$.

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 —  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 $F_{b}f=F(f,{\mathrm{id}}_b)$

for $f:a\to a^{\prime }$ and similarly for $F_a$. They commute in the sense:

$F_{a^{\prime }}g\circ F_{b}f=F_{b^{\prime }}f\circ F_{a}g$.

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

$F(f,g)=F_{b^{\prime }}f\circ F_{a}g$.

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

$\mathrm{Hom}(-,b):C^{op}\to \mathsf{S}\mathsf{e}\mathsf{t}$

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

$f^{*}:\mathrm{Hom}(a^{\prime },b)\to \mathrm{Hom}(a,b)$

defined by $f^{*}g=g\circ f$. On the other hand, $\mathrm{Hom}(a,-):C\to \mathsf{S}\mathsf{e}\mathsf{t}$ 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

$\mathrm{Hom}(-,-):C^{op}\times C\to \mathsf{S}\mathsf{e}\mathsf{t},$

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 —  Let $F,G:A,B\to C$ be bifunctors and

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

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

• Definition 1.6.5 in Borceux, Francis (1994). Handbook of categorical algebra. Encyclopedia of mathematics and its applications 50-51, 53 [i.e. 52]. 1. Cambridge University Press. p. 22. ISBN 0-521-44178-1. https://archive.org/details/handbookofcatego0000borc/page/22. 
• Product category in 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.