counterexamples for products and coproduct
1 Direct sum is not always a coproduct
For groups the notion of a direct sum (http://planetmath.org/DirectProductAndRestrictedDirectProductOfGroups) 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 difference.
Let denote the disjoint union of sets.
The direct sum is a subgroup of consisting of all with the added property that
is a finite set, that is, has finite support. 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
The direct sum and direct product are equal whenever is a finite set. That is, for any family with finite, then
Certainly is a subset of and so is finite. ∎
Claim: The direct sum is not a coproduct in the category of all groups.
Example. Let and . We observe that and is a group of order (http://planetmath.org/OrderGroup) 12. Now suppose that is a coproduct for the category of groups. The canonical inclussion maps are
If indeed is a coproduct in the category of groups then their exists a unique homomorphism such that , . This means that
Notice then that the image of in is all of since . But this is impossible since and . Hence there cannot exist such a homomorphism and so is not a categorical coproduct.
2 Infinite products and coproducts
“ In an abelian category the product and coproduct are equivalent. ”
3 Common categories without (co)products
Let FinGrp be the category of all finite groups. This category does not inherit the standard products and coproduct of the category of all groups Grp. For example,
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 groups AbGrp. However, the coproduct for groups is the free product but the coproduct for abelian groups is direct sum . These are inequivelent.
|Title||counterexamples for products and coproduct|
|Date of creation||2013-03-22 15:52:31|
|Last modified on||2013-03-22 15:52:31|
|Last modified by||Algeboy (12884)|