## You are here

Homedisjoint union of categories

## Primary tabs

# disjoint union of categories

Let $\{\mathcal{C}_{i}\}$ be a collection of categories, indexed by a set $I$. The *disjoint union* $\mathcal{C}$ of these categories is defined as follows:

1. 2. the class of morphisms of $\mathcal{C}$ is the disjoint union of classes of morphisms, $\operatorname{Mor}(\mathcal{C}_{i})$, for every $i\in I$.

3. 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, $\hom(A,B):=\varnothing$.

4. 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 $\varnothing$.

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 $\mathcal{C}_{i}$.

Remark. If each $\mathcal{C}_{i}$ is small, so is their disjoint union. In fact, in Cat, the category of small categories, the disjoint union of these categories is their coproduct.

## Mathematics Subject Classification

18A05*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections