Product category

In the mathematical field of category theory, the product of two categories C and D, denoted Template:Nowrap 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.Template:Sfn

Definition

The product category Template:Nowrap has:

• as objects:

  pairs of objects Template:Nowrap, where A is an object of C and B of D;

• as arrows from Template:Nowrap to Template:Nowrap:

  pairs of arrows Template:Nowrap, where Template:Nowrap is an arrow of C and Template:Nowrap is an arrow of D;

• as composition, component-wise composition from the contributing categories:

  Template:Nowrap;

• 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 $𝖢𝖺𝗍$. Thus, for example, $$𝖥𝖼𝗍(A,{\prod }_i{B}_i)\simeq {\prod }_i𝖥𝖼𝗍(A,B_i)$$

where $𝖥𝖼𝗍$ denotes a functor category.

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, $$(f\times g)(x,x^{\prime })=(f(x),g(x^{\prime }))$$ for a pair of objects or 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

$\times :𝖢𝖺𝗍\times 𝖢𝖺𝗍\to 𝖢𝖺𝗍.$

It satisfies the tensor-hom adjunction in the sense

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

where $𝖥𝖼𝗍$ 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:

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

$\mathrm{Hom}(-,b):C^{op}\to 𝖲𝖾𝗍$

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

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

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

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

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:

References

• Definition 1.6.5 in Template:Cite book
• Template:Nlab
• Template:Cite book