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 $x, y \in Obj(C)$ two objects, their coproduct is an object $x \coprod y$ in $C$ equipped with two morphisms

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

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

This morphism $(f,g)$ is called the copairing of $f$ and $g$. The morphisms $x\to x\coprod y$ and $y\to x\coprod y$ are called coprojections or sometimes “injections” or “inclusions”, although in general they may not be monomorphisms.

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

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

equipped with maps

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 $S,T \in FinSet$ two finite sets and $|-| : FinSet \to \mathbb{N}$ the cardinality operation, we have

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

• 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 of groups”, 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

  $$Obj(\coprod_{i\in I} C_i) = \coprod_{i\in I} Obj(C_i)$$

  and

  $$Hom_{\coprod_{i\in I} C_i}(x,y) = \left\{ \begin{aligned} Hom_{C_i}(x,y) & if x,y \in C_i \\ \emptyset & otherwise \end{aligned} \right.$$

• 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 $C^{op}$.

• 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

• quasi-coproduct

• sum type

• free coproduct completion

• base change

  • dependent sum, dependent product

  • dependent sum type, dependent product type

References

Textbook account:

• Francis Borceux, Section 2.2 in Vol. 1: Basic Category Theory of: Handbook of Categorical Algebra, Encyclopedia of Mathematics and its Applications 50 Cambridge University Press (1994) (doi:10.1017/CBO9780511525858)