What is the Definition of a Weak Direct Product in Group Theory?

[radou]

Well, I have trouble understanding the definition of the weak product of a family of groups, which states that it is the set of all $f \in \prod_{i \in I} G_{i}$ such that $f(i) = e_{i} \in G_{i}$, for all but a finite number of i from I (Gi is a family of groups indexed by the set I).

The bolded part causes confusion.

Further on, the book says that, if I is finite, the weak direct product coincides with the direct product. Even this fact didn't help me understand the definition of the weak direct product.

Thanks in advance.

 

[StatusX]

This is usually called the direct sum. For example, the direct product of countably many copies of Z consists of all infinite sequences of integers, with addition defined pointwise. The direct sum consists of all finite sequences of integers of arbitrarily length (ie, the subset of the infinite sequences which are zero after a certain point). So (1,1,1,...) is in the direct product but not the direct sum, while (1,2,3,0,0,0,...) is in both.

 

[radou]

This is usually called the direct sum. For example, the direct product of countably many copies of Z consists of all infinite sequences of integers, with addition defined pointwise. The direct sum consists of all finite sequences of integers of arbitrarily length (ie, the subset of the infinite sequences which are zero after a certain point). So (1,1,1,...) is in the direct product but not the direct sum, while (1,2,3,0,0,0,...) is in both.

Okay, I get it now, thanks.

Btw, why is an element of a Cartesian product defined as a function? I read about it and I understand it, but where's the motivation except in formality (?) ?

 

[StatusX]

I guess it's just a clean way of generalizing beyond finite or countable index sets. Besides, how else would you (rigorously) define them?

 

[matt grime]

It is the categorical paradigm. Objects should be defined by maps into, and out of, them. It also avoids any set theory (we should be talking of morphisms and not functions).

 

[radou]

Besides, how else would you (rigorously) define them?

Good point.

I have another question. I have read about the universal mapping property of a Cartesian product of sets. Further on, I have read a bit about products in the context of categories. Finally, I learned that a direct product of groups is a product, speaking in the language of categories. So, these "universal mapping properties" occur quite frequently.

For example, is Gi is a family of groups and fi : H --> Gi is a family of group homomorphisms, the there is a unique homomorphism f : H --> $\prod_{i \in I}G_{i}$ such that pi o f = fi, for all i e I (where pi is the "projection epimorphism" with image Gi), and this determines $\prod_{i \in I}G_{i}$ uniquely up to isomorphism.

Such theorems confuse me, since I can't see the direct motivation for these observations. Is it simply to prove existence of general homomorphisms, when given some "basic tools"? Are such theorems going to be extremely useful in the future (I assume they will, but still, I'm curious)?

 

[matt grime]

What theorems? That is the definition of product (if we include te universality of the projections). Actually, these things are determined up to unique isomorphism, not uniquely up to isomorphism.

None of the definitions at all justifies why any such object might exist in any category. Generically, arbitrary categories will not have products, or coproducts, or any such constructions. The concrete things you write down show that GROUP does have products, for instance.

Further, anything you prove about GROUP and products, if it only depends on the definition of product, will hold in any category that has products. That is the beauty and power of category theory. Why bother to prove something complicated in a particular situation when it holds in entire generality for completely formal reasons?

 

[radou]

None of the definitions at all justifies why any such object might exist in any category. Generically, arbitrary categories will not have products, or coproducts, or any such constructions. The concrete things you write down show that GROUP does have products, for instance.

Further, anything you prove about GROUP and products, if it only depends on the definition of product, will hold in any category that has products. That is the beauty and power of category theory. Why bother to prove something complicated in a particular situation when it holds in entire generality for completely formal reasons?

Thanks, this was enlightening.