PlanetMath: product of categories

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product of categories

(Definition)

There are occasions when we need to consider objects from different categories being paired up. For example, if $\mathcal{C}$ is a category, then $\hom(A,B)$ where are objects of $\mathcal{C}$ is a set, and we can think of $\hom$ as a functor from $\mathcal{C}^{\operatorname{op}}\times \mathcal{C}$ to the category of sets. But what is $\mathcal{C}^{\operatorname{op}}\times \mathcal{C}$ exactly? We will give this a formal definition presently.

Let $\mathcal{C}$ and $\mathcal{D}$ be categories. Define the Cartesian product $\mathcal{C}\times \mathcal{D}$ of $\mathcal{C}$ and $\mathcal{D}$ as the following pair , where

is the class consisting of ordered pairs , where is an object in $\mathcal{C}$ and is an object in $\mathcal{D}$
is the class consisting of ordered pairs , where is a morphism in $\mathcal{C}$ and is a morphism in $\mathcal{D}$ .

There is a category structure on $\mathcal{C}\times \mathcal{D}$ . But several things need to be defined first.

Elements of are called the objects of $\mathcal{C}\times \mathcal{D}$ and elements of are called the morphisms of $\mathcal{C}\times\mathcal{D}$ . For each morphism $(f,g)\in M$ , we define the domain and codomain operations
$\displaystyle \operatorname{dom}(f,g):=(\operatorname{dom}(f), \operatorname{dom}(g))$ and $\displaystyle \quad \operatorname{cod}(f,g):=(\operatorname{cod}(f), \operatorname{cod}(g)).$
Note that for simplicity, we have used the same symbol $\operatorname{dom}$ and $\operatorname{cod}$ for $\mathcal{C},\mathcal{D}$ , and $\mathcal{C}\times \mathcal{D}$ .
Next, for each pair of objects $A,B\in \mathcal{C}\times\mathcal{D}$ , we have a set $\hom(A,B)$ consisting of all morphisms in $\mathcal{C}\times \mathcal{D}$ whose domain is and codomain is . Note that $\hom(A,B)$ is a set because it is $\hom(X,Z)\times \hom(Y,T)$ , where and and each component in the product is assumed to be a set.
Finally, for objects in $\mathcal{C}\times\mathcal{D}$ , we have a function $\circ$ called composition:
$\displaystyle \circ:\hom(A,B)\times \hom(B,C)\to \hom(A,C).$
To define $\circ$ , write each object as ordered pairs: , , . In addition, let $\alpha=(f,g)\in \hom(A,B)$ and $\beta=(p,q)\in \hom(B,C)$ . Then
$\displaystyle \circ(\alpha,\beta):=(\circ_1(f,p),\circ_2(g,q)),$
where $\circ_1$ and $\circ_2$ are compositions defined in $\mathcal{C}$ and $\mathcal{D}$ respectively, such that
$\displaystyle \circ_1:\hom(X,Z)\times \hom(Z,U)\to \hom(X,U)$ and $\displaystyle \circ_2:\hom(Y,T)\times \hom(T,V)\to \hom(Y,V).$
As usual, we write $\beta\circ \alpha$ for $\circ(\alpha,\beta)$ .
Now, it is not hard to see that $\mathcal{C}\times \mathcal{D}$ with $\circ$ is a category. For example, let us verify that $(A,B)\ne (C,D)$ implies $\hom(A,B)\cap \hom(C,D)=\varnothing$ . Write , , and . Suppose $\alpha=(f,g)\in \hom(A,B)\cap \hom(C,D)$ . Then $f\in \hom(X,Z)\cap \hom(T,V)$ and $g\in \hom(Y,S)\cap \hom(U,W)$ . But this implies , , , and . So and .

Remarks.

The above construction can be generalized to -fold Cartesian products. If $\mathcal{C}_1,\ldots,\mathcal{C}_n$ be categories. Then $\mathcal{C}:=\mathcal{C}_1\times\cdots \mathcal{C}_n$ can be defined much the same way as in the case . $\mathcal{C}$ is a category and is sometimes written $\prod_{i=1}^n \mathcal{C}_i$ .
Associated with this product, we can form (covariant) functors called projection functors $\Pi_i:\mathcal{C}\to \mathcal{C}_i$ , given by $\Pi_i(A)=A_i$ and $\Pi_i(\alpha)=\alpha_i$ , where $A=(A_1,\ldots,A_n)$ and $\alpha=(\alpha_1,\ldots,\alpha_n)$ .
The product $\mathcal{C}$ of $\mathcal{C}_i$ also enjoys the universal property that for every category $\mathcal{D}$ and functors $F_i: \mathcal{D}\to \mathcal{C}_i$ , there is a unique functor $F:\mathcal{D}\to \mathcal{C}$ such that $\Pi_i\circ F=F_i$ (in other words, factors through ).
If fact, any category that enjoys the universal property described above is naturally equivalent to the product of $\mathcal{C}_i$ . We may actually define product category this way, and then prove its existence using the construction that is given as the definition at the beginning of this article.
More generally, we can define arbitrary (direct) product of categories. The definition is completely similar to the one above. If $\lbrace \mathcal{C}_i\mid i\in I\rbrace$ is a family of categories indexed by a set , we often write $\prod_{i\in I} \mathcal{C}_i$ as the product category. Objects and morphisms are written $(A_i)_{i\in I}$ and $(\alpha_i)_{i\in I}$ respectively. When all the $\mathcal{C}_i$ are identical, say, equal to $\mathcal{C}$ , we also write the product as $\mathcal{C}^I$ , and call it the -fold direct product of $\mathcal{C}$ .
The existence of the product of categories indexed by an arbitrary set shows that the category of (small) categories Cat has products.
Let $\mathcal{C}=\mathcal{D}\times\mathcal{E}$ . Then we may identify $\mathcal{D}$ as a subcategory of $\mathcal{C}$ : for each object in $\mathcal{E}$ , define $F_E:\mathcal{D}\to\mathcal{C}$ , by and $F(\alpha)=(\alpha,1_E)$ . Then is a faithful functor. The image $\mathcal{C}_E$ of (with objects and morphisms $(\alpha,1_E)$ ) is a subcategory of $\mathcal{C}$ . It is not hard to see that $\mathcal{D}$ and $\mathcal{C}_E$ are isomorphic as categories.
The above also shows that for any objects in $\mathcal{E}$ , $\mathcal{C}_A$ and $\mathcal{C}_B$ are isomorphic.