coproduct (changes) in nLab

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Coproducts

Idea

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

Definition

For CC a category and x,y∈Obj(C)x, y \in Obj(C) two objects, their coproduct is an object x∐yx \coprod y in CC equipped with two morphisms

\begin{center} \begin{tikzcd} x \arrow[rd, i_y] i_x] & & y \arrow[ld, i_y] \ & x \amalg y &
\end{tikzcd} \end{center}

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

\begin{center} \begin{tikzcd} x \arrow[rd, f] & & y \arrow[ld, g] \ & Q &
\end{tikzcd} \end{center}

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

\begin{center} \begin{tikzcd} x \arrow[rd, f] \arrow[r, i_x] & x \amalg y \arrow[d, {(f, g)}] & y \arrow[ld, g] \arrow[l, i_y] \ & Q &
\end{tikzcd} \end{center}

This morphism (f,g)(f,g) is called the copairing of ff and gg. The morphisms x→x∐yx\to x\coprod y and y→x∐yy\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+ba+b or a⨿ba\amalg b, especially when it is disjoint (or a⊔ba \sqcup b if your fonts don't include ‘⨿\amalg’). The copairing is also denoted [f,g][f,g] or (when possible) given vertically: {fg}\left\{{f \atop g}\right\}.

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

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

∐ s∈SF(s) \coprod_{s \in S} F(s)

equipped with maps

F(s)→∐ s∈SF(s) F(s) \to \coprod_{s \in S} F(s)

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

Examples

• In Set, the coproduct of a family of sets (C i) i∈I(C_i)_{i\in I} is the disjoint union ∐ i∈IC i\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∈FinSetS,T \in FinSet two finite sets and |−|:FinSet→ℕ|-| : FinSet \to \mathbb{N} the cardinality operation, we have

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

• In Top, the coproduct of a family of spaces (C i) i∈I(C_i)_{i\in I} is the space whose set of points is ∐ i∈IC i\coprod_{i\in I} C_i and whose open subspaces are of the form ∐ i∈IU i\coprod_{i\in I} U_i (the internal disjoint union) where each U iU_i is an open subspace of C iC_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∈I(C_i)_{i\in I} is the category with

  Obj(∐ i∈IC i)=∐ i∈IObj(C i)Obj(\coprod_{i\in I} C_i) = \coprod_{i\in I} Obj(C_i)

  and

  Hom ∐ i∈IC i(x,y)={Hom C i(x,y) ifx,y∈C i ∅ otherwise 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 CC is the same as a product in the opposite category C opC^{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 CC.

• 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)