nLab
coproduct

Idea

The notion of coproduct is a generalization to arbitrary categories of the notion of disjoint union in the category Set.

Definition

For $C$ a category and $x,y\in \mathrm{Obj}(C)$ two objects, their coproduct is an object $x\coprod y$ in $C$ equipped with two morphisms

$$\begin{array}{ccccc}x& & & & y\\ & {}_{i_x}\searrow & & {\swarrow }_{i_y}\\ & & x\coprod y\end{array}$$\array{ x &&&& y \\ & {}_{\mathllap{i_x}}\searrow && \swarrow_{\mathrlap{i_y}} \\ && x \coprod y }

such that this is universal with this property, meaning such that for any other object with maps like this

$$\begin{array}{ccccc}x& & & & y\\ & {}_f\searrow & & {\swarrow }_g\\ & & Q\end{array}$$\array{ x &&&& y \\ & {}_{\mathllap{f}}\searrow && \swarrow_{\mathrlap{g}} \\ && Q }

there exists a unique morphism $(f,g):x\coprod y\to Q$ such that we have a commuting diagram

$$\begin{array}{ccccc}x& \stackrel{i_x}{\to }& x\coprod y& \stackrel{i_y}{\leftarrow }& y\\ & {}_f\searrow & {\downarrow }^{(f,g)}& {\swarrow }_g\\ & & Q\end{array}.$$\array{ x &\stackrel{i_x}{\to}& x \coprod y &\stackrel{i_y}{\leftarrow}& y \\ & {}_{\mathrlap{f}}\searrow & \downarrow^{\mathrlap{(f,g)}} & \swarrow_{\mathrlap{g}} \\ && Q } \,.

This morphism $(f,g)$ is called the copairing of $f$ and $g$.

Notation. The coproduct is also denoted $a+b$ or $a⨿b$, especially when it is disjoint (or $a\bigsqcup b$ if your fonts don't include ‘$⨿$’). The copairing is also denoted $[f,g]$ or (when possible) given vertically: $\left\{\genfrac{}{}{0ex}{}{f}{g}\right\}$.

A coproduct is thus the colimit over the diagram that consists of just two objects.

More generally, for $S$ any set and $F:S\to C$ a collection of objects in $C$ indexed by $S$, their coproduct is an object

$$\coprod _{s\in S}F(s)$$\coprod_{s \in S} F(s)

equipped with maps

$$F(s)\to \coprod _{s\in S}F(s)$$F(s) \to \coprod_{s \in S} F(s)

such that this is universal among all objects with maps from the $F(s)$.

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.

  This makes the coproduct a categorification of the operation of addition of natural numbers and more generally of cardinal numbers: for $S,T\in \mathrm{FinSet}$ two finite sets and $\mid -\mid :\mathrm{FinSet}\to \mathbb{N}$ the cardinality operation, we have

  $$\mid S\coprod T\mid =\mid S\mid +\mid T\mid .$$|S \coprod T| = |S| + |T| \,.

• 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 Ab, in Vect, the coproduct is the subobject of the product consisting of those tuples of elements for which only finitely many are not 0.

• 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.

Properties

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

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

• A coproduct indexed by the empty set is an initial object in $C$.

Related concepts

• sum type

• base change

  • dependent sum, dependent product

  • dependent sum type, dependent product type