nLab
coproduct

Definition

A coproduct of two objects $a,b$ in a category $C$ is an object $a+b$ (also written $a⨿b$, especially when it is disjoint, or $a\bigsqcup b$ if your fonts don't include ‘$⨿$’) equipped with morphisms $i:a\to a+b$ and $j:b\to a+b$ such that for any object $c$ with morphisms $f:a\to c$ and $g:b\to c$, there is a unique morphism $p:a+b\to c$ such that $p\circ i=f$ and $p\circ j=g$.

This map $p$ is called the copairing of $f$ and $g$ and is often denoted $[f,g]$ or $\{f,g\}$ or (when possible) given vertically:

$$\left\{\genfrac{}{}{0ex}{}{f}{g}\right\}$$\left\{{f \atop g}\right\}

Examples

• In Set, the coproduct of a family of sets $(C_i{)}_{i\in I}$ is the disjoint union ${\coprod }_{i\in I}{C}_i$ of sets.

• In Top, the coproduct of a family of spaces $(C_i{)}_{i\in I}$ is the space whose set of points is ${\coprod }_{i\in I}{C}_i$ and whose open subspaces are of the form ${\coprod }_{i\in I}{U}_i$ (the internal disjoint union) where each $U_i$ is an open subspace? of $C_i$. This is typical of topological concrete categories.

• In Grp, the coproduct is the free product, whose underlying set is not a disjoint union. This is typical of algebraic categories.

• In Cat, the coproduct of a family of categories $(C_i{)}_{i\in I}$ is the category with

  $$\mathrm{Obj}(\coprod _{i\in I}{C}_i)=\coprod _{i\in I}\mathrm{Obj}(C_i)$$Obj(\coprod_{i\in I} C_i) = \coprod_{i\in I} Obj(C_i)

  and

  $${\mathrm{Hom}}_{\coprod _{i\in I}{C}_i}(x,y)=\{\begin{array}{rl}{\mathrm{Hom}}_{C_i}(x,y)& \mathrm{if}x,y\in C_i\\ \varnothing & \mathrm{otherwise}\end{array}$$Hom_{\coprod_{i\in I} C_i}(x,y) = \left\{ \begin{aligned} Hom_{C_i}(x,y) & if x,y \in C_i \\ \emptyset & otherwise \end{aligned} \right.

• In Grpd, the coproduct follows Cat rather than Grp. This is typical of oidifications: the coproduct becomes a disjoint union again.

Remarks

• When they exist, coproducts are unique up to unique canonical isomorphism, so we often say “the coproduct.”

• One can define in a similar way a coproduct of any family of objects. A coproduct of the empty family is an initial object.

• A coproduct is a special case of a colimit in which the diagram category is discrete.

• A coproduct in $C$ is the same as a product in the opposite category $C^{\mathrm{op}}$.