The notion of coproduct is a generalization to arbitrary categories of the notion of disjoint union in the category Set.
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 $F(s)$.
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
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
and
In Grpd, the coproduct follows Cat rather than Grp. This is typical of oidifications: the coproduct becomes a disjoint union again.
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 $C$.
Textbook account:
Last revised on September 23, 2023 at 23:52:26. See the history of this page for a list of all contributions to it.