counterexamples for products and coproduct

counterexamples for products and coproduct

\PMlinkescapephrase

direct sum

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 product

of a family of groups

{

G

i

:

i

∈

I

}

is the set of all

functions

f

:

I

→

⊔

i

∈

I

G

i

such that

f

⁢

(

i

)

∈

G

i

. We usually denote this by

∏

i

∈

I

G

i

.

This is a

product

in the

category

of all groups. This is a group under

pointwise operations

:

(

f

⁢

g

)

⁢

(

i

)

=

f

⁢

(

i

)

⁢

g

⁢

(

i

)

and all the group properties follow.

The direct sum is a

subgroup

of

∏

i

∈

I

G

i

consisting of all

f

:

I

→

⊔

i

∈

I

G

i

with the added property that

S

⁢

u

⁢

p

⁢

p

⁢

f

=

{

i

∈

I

:

f

⁢

(

i

)

≠

1

}

is a

finite set

, that is,

f

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

∑

i

∈

I

G

i

⁢

or

⁢

⊕

i

∈

I

G

i

.

Proposition 1

.

The direct sum and direct product are equal whenever

I

is a finite set.
That is, for any family

{

G

i

:

i

∈

I

}

with

I

finite, then

∏

i

∈

I

G

i

=

⊕

i

∈

I

G

i

.

Remark 2

.

The ‘

=

’ here means an honest set

equality

even more than naturally

isomorphic

.

Proof.

Certainly

S

⁢

u

⁢

p

⁢

p

⁢

f

is a subset of

I

and so

S

⁢

u

⁢

p

⁢

p

⁢

f

is finite.
∎

Claim:

The direct sum is

not

a coproduct in the category of all groups.

Example.

Let

G

1

=

S

3

and

G

2

=

ℤ

2

. We observe that

S

3

⊕

ℤ

2

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

ι

1

:

S

3

→

S

3

⊕

ℤ

2

:

σ

↦

(

σ

,

0

)

and

ι

2

:

ℤ

2

→

S

3

×

ℤ

2

:

n

↦

(

1

,

n

)

.

Take the

homomorphisms

f

1

:

S

3

→

S

4

– the
natural

inclusion map

of

S

3

=

⟨

(

123

)

,

(

12

)

⟩

treated as

permutations

on 4 letters fixing 4 – and

f

2

:

ℤ

2

→

S

4

given by

1

↦

(

14

)

.

If indeed

⊕

is a coproduct in the category of groups then their exists
a unique homomorphism

f

:

S

3

⊕

ℤ

2

→

S

4

such that

f

i

=

f

⁢

ι

i

,

i

=

1

,

2

. This means that

(

123

)

=

f

1

⁢

(

(

123

)

)

=

f

⁢

(

ι

1

⁢

(

123

)

)

=

f

⁢

(

(

123

)

,

0

)

,

(

14

)

=

f

2

⁢

(

1

)

=

f

⁢

(

ι

2

⁢

(

1

)

)

=

f

⁢

(

1

,

1

)

.

Notice then that the image of

f

in

S

4

is all of

S

4

since

⟨

(

123

)

,

(

14

)

⟩

=

S

4

. But this is impossible since

|

S

4

|

=

24

and

|

S

3

⊕

ℤ

2

|

=

12

. Hence there cannot exist such a homomorphism

f

and so

⊕

is not a

categorical

coproduct.

□

2

Infinite products and coproducts

In an

abelian category

, for example the category of abelian groups or a category of modules, the direct sum is the categorical coproduct. Thus a common
misreading of

Proposition

1

is to declare

“ In an abelian category the product and coproduct are

equivalent

. ”

Indeed, this is true only if the

index set

I

of the family of objects is finite. A simple

cardinality

test demonstrates the flaw.

Example.

Suppose that

I

=

ℕ

and

G

i

=

ℤ

2

.
Then the product of

∏

i

∈

ℕ

ℤ

2

can be equated
with the set of all functions

f

:

ℕ

→

ℤ

2

– that is, all

infinite

sequences

of binary digits. This has cardinality

2

ℵ

0

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

countable

. Therefore these two objects cannot be isomorphic in the category.

□

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,

∏

ℕ

ℤ

2

⁢

and

⁢

∐

ℕ

ℤ

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 groups

AbGrp

. However, the coproduct for groups is the

free product

(

*

)

but the coproduct for abelian groups is direct sum

(

⊕

)

. These are inequivelent.

Example.

ℤ

*

ℤ

is the

free group

on two

elements

– and so

non-abelian

– while

ℤ

⊕

ℤ

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

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