disjoint union of categoriesDisjointUnionOfCategories2008-10-01 16:38:292008-12-24 21:36:53Examplecategorydisjoint union\usepackage{amssymb,amscd}
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\newcommand{\mpdiff}[3]{\frac{\partial^#1 #2}{\partial #3^#1}}Let $\lbrace \mathcal{C}_i\rbrace$ be a collection of categories, indexed by a set $I$. The \PMlinkescapetext{\emph{disjoint union}} $\mathcal{C}$ of these categories is defined as follows:
\begin{enumerate}
\item the class of objects of $\mathcal{C}$ is the disjoint union of classes of objects, $\operatorname{Ob}(\mathcal{C}_i)$, for every $i\in I$,
\item the class of morphisms of $\mathcal{C}$ is the disjoint union of classes of morphisms, $\operatorname{Mor}(\mathcal{C}_i)$, for every $i\in I$.
\item for objects $A,B$ in $\mathcal{C}$, if they are objects of $\mathcal{C}_i$, then $\hom(A,B)$ is the set of morphisms from $A$ to $B$ in $\mathcal{C}_i$, otherwise, $\hom(A,B):=\varnothing$.
\item given $\hom(A,B)$ and $\hom(B,C)$, the composition of morphisms is defined so that, if $A,B,C$ are all objects of some $\mathcal{C}_i$, the composition is the same as the composition of morphisms defined in $\mathcal{C}_i$. Otherwise, it is defined as $\varnothing$.
\end{enumerate}
With the above conditions, one immediately sees that $\mathcal{C}$ is a category, as each $\hom(A,B)$ is a set, associativity of morphism composition and identity morphisms all inherit from the individual categories $\mathcal{C}_i$.
\textbf{Remark}. If each $\mathcal{C}_i$ is small, so is their disjoint union. In fact, in \textbf{Cat}, the category of small categories, the disjoint union of these categories is their coproduct.