# nLab coproduct

category theory

## Applications

#### Limits and colimits

limits and colimits

# Coproducts

## 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 $x,y\in \mathrm{Obj}\left(C\right)$ two objects, their coproduct is an object $x\coprod y$ in $C$ equipped with two morphisms

$\begin{array}{ccccc}x& & & & y\\ & {}_{{i}_{x}}↘& & {↙}_{{i}_{y}}\\ & & x\coprod y\end{array}$\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

$\begin{array}{ccccc}x& & & & y\\ & {}_{f}↘& & {↙}_{g}\\ & & Q\end{array}$\array{ x &&&& y \\ & {}_{\mathllap{f}}\searrow && \swarrow_{\mathrlap{g}} \\ && Q }

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

$\begin{array}{ccccc}x& \stackrel{{i}_{x}}{\to }& x\coprod y& \stackrel{{i}_{y}}{←}& y\\ & {}_{f}↘& {↓}^{\left(f,g\right)}& {↙}_{g}\\ & & Q\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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 $\left(f,g\right)$ 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 $\left[f,g\right]$ or (when possible) given vertically: $\left\{\genfrac{}{}{0}{}{f}{g}\right\}$.

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

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

equipped with maps

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

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

## Examples

• In Set, the coproduct of a family of sets $\left({C}_{i}{\right)}_{i\in I}$ 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 $\mid -\mid :\mathrm{FinSet}\to ℕ$ the cardinality operation, we have

$\mid S\coprod T\mid =\mid S\mid +\mid T\mid \phantom{\rule{thinmathspace}{0ex}}.$|S \coprod T| = |S| + |T| \,.
• In Top, the coproduct of a family of spaces $\left({C}_{i}{\right)}_{i\in I}$ 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 $\left({C}_{i}{\right)}_{i\in I}$ is the category with

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

and

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

Revised on February 14, 2013 23:47:41 by Anonymous Coward (173.64.113.16)