counterexamples for products and coproduct


direct sumMathworldPlanetmathPlanetmathPlanetmath

1 Direct sum is not always a coproduct

For groups the notion of a direct sum ( is in conflict with the categorical direct sum. For this reason a categorical direct sum is often called a coproduct instead. The following example illustrates the differencePlanetmathPlanetmath.

Let denote the disjoint unionMathworldPlanetmathPlanetmath of sets.

The direct productMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath of a family of groups {Gi:iI} is the set of all functions f:IiIGi such that f(i)Gi. We usually denote this by


This is a productMathworldPlanetmathPlanetmath in the categoryMathworldPlanetmath of all groups. This is a group under pointwise operations: (fg)(i)=f(i)g(i) and all the group properties follow.

The direct sum is a subgroupMathworldPlanetmathPlanetmath of iIGi consisting of all f:IiIGi with the added property that


is a finite setMathworldPlanetmath, that is, f has finite supportPlanetmathPlanetmath. The notation for direct sums is experiencing a shift from the historical sigma notation to the modern circled plus; thus it is common to see any of the following two notations

iIGi or iIGi.
Proposition 1.

The direct sum and direct product are equal whenever I is a finite set. That is, for any family {Gi:iI} with I finite, then

Remark 2.

The ‘=’ here means an honest set equality even more than naturally isomorphicPlanetmathPlanetmathPlanetmath.


Certainly Suppf is a subset of I and so Suppf is finite. ∎

Claim: The direct sum is not a coproduct in the category of all groups.

Example. Let G1=S3 and G2=2. We observe that S32 and is a group of order ( 12. Now suppose that is a coproduct for the category of groups. The canonical inclussion maps are




Take the homomorphismsMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath f1:S3S4 – the natural inclusion mapMathworldPlanetmath of S3=(123),(12) treated as permutations on 4 letters fixing 4 – and f2:2S4 given by 1(14).

If indeed is a coproduct in the category of groups then their exists a unique homomorphism f:S32S4 such that fi=fιi, i=1,2. This means that


Notice then that the image of f in S4 is all of S4 since (123),(14)=S4. But this is impossible since |S4|=24 and |S32|=12. Hence there cannot exist such a homomorphism f and so is not a categorical coproduct.

2 Infinite products and coproducts

In an abelian categoryMathworldPlanetmathPlanetmathPlanetmath, for example the category of abelian groups or a category of modules, the direct sum is the categorical coproduct. Thus a common misreading of PropositionPlanetmathPlanetmath 1 is to declare

“ In an abelian category the product and coproduct are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath. ”

Indeed, this is true only if the index setMathworldPlanetmathPlanetmath I of the family of objects is finite. A simple cardinality test demonstrates the flaw.

Example. Suppose that I= and Gi=2. Then the product of i2 can be equated with the set of all functions f:2 – that is, all infiniteMathworldPlanetmath sequencesPlanetmathPlanetmath of binary digits. This has cardinality 20 which is uncountable.

On the other hand, the direct sum (coproduct in this context) of this family is the set of all finite binary strings, which is countableMathworldPlanetmath. Therefore these two objects cannot be isomorphic in the category.

3 Common categories without (co)products

Let FinGrp be the category of all finite groupsMathworldPlanetmath. This category does not inherit the standard products and coproduct of the category of all groups Grp. For example,

2 and 2

are both infinite groups and so they do not lie in the category FinGrp. Indeed, this example could be done with the category of finite sets FinSet inside the category of all sets Set, and many other such categories.

However, we have not yet demonstrated that no alternate product and/or coproduct for the category FinGrp, FinSet, etc does not exist.

4 Common subcategories with different (co)products

Consider once again the category of all groups Grp. Inside this category lies the category of all abelian groupsMathworldPlanetmath AbGrp. However, the coproduct for groups is the free productMathworldPlanetmath (*) but the coproduct for abelian groups is direct sum (). These are inequivelent.

Example. * is the free groupMathworldPlanetmath on two elements – and so non-abelianMathworldPlanetmathPlanetmath – while is abelian.

Title counterexamples for products and coproduct
Canonical name CounterexamplesForProductsAndCoproduct
Date of creation 2013-03-22 15:52:31
Last modified on 2013-03-22 15:52:31
Owner Algeboy (12884)
Last modified by Algeboy (12884)
Numerical id 16
Author Algeboy (12884)
Entry type Example
Classification msc 16B50