nLab
coproduct

Context

Category theory

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

x y i x i y xy \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

x y f g Q \array{ x &&&& y \\ & {}_{\mathllap{f}}\searrow && \swarrow_{\mathrlap{g}} \\ && Q }

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

x i x xy i y y f (f,g) g Q. \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)(f,g) is called the copairing of ff and gg.

Notation. The coproduct is also denoted a+ba+b or a⨿ba\amalg b, especially when it is disjoint (or aba \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:SCF : S \to C a collection of objects in CC indexed by SS, their coproduct is an object

sSF(s) \coprod_{s \in S} F(s)

equipped with maps

F(s) sSF(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) iI(C_i)_{i\in I} is the disjoint union iIC 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,TFinSetS,T \in FinSet two finite sets and :FinSet|-| : FinSet \to \mathbb{N} the cardinality operation, we have

    ST=S+T. |S \coprod T| = |S| + |T| \,.
  • In Top, the coproduct of a family of spaces (C i) iI(C_i)_{i\in I} is the space whose set of points is iIC i\coprod_{i\in I} C_i and whose open subspaces are of the form iIU 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, 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) iI(C_i)_{i\in I} is the category with

    Obj( iIC i)= iIObj(C i)Obj(\coprod_{i\in I} C_i) = \coprod_{i\in I} Obj(C_i)

    and

    Hom iIC i(x,y)={Hom C i(x,y) ifx,yC 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.

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