Is There a Limit to the Number of Elements in a Coproduct/Direct Sum?

In summary, the conversation discusses the problem of defining the direct sum for an infinite set (1, 1, 1, ...) and the potential mistakes that can arise in this process. The conversation also mentions the concept of a coproduct and the need to define a homomorphism g to make it a coproduct, with the limitation of only being able to use finite sums and products. The speaker asks for clarification on their understanding of the problem and the potential gaps in their reasoning.
  • #1
Sumanta
26
0
Hi ,

I was trying to understand why or where would the problem arise in the definition of the direct sum for the coproduct/direct sum for the set (1, 1, 1, ...) infinite number of times. I was trying to reason out as follows. I posted this on the set theory but I am not sure how to tag it to algebra hence I decided to copy paste the same.

(1, 1, 1, ...)[itex]\rightarrow[/itex] Y

[itex]\uparrow[/itex]

f[itex]_{i}[/itex][itex]\rightarrow[/itex] YPls note that the Y is the same as I cannot write the angular arrow.

Now if I say that f[itex]_{i}[/itex] acting on (e[itex]_{i}[/itex]) maps it to ( 0, 0, ...1 at the ith coordinate , 0, ...) then what is the place where I am making a mistake. The problem as I see is that either the map from the set (1, 1, 1, ...) to Y is either not unique or map from f[itex]_{i}[/itex][itex]\rightarrow[/itex] Y does not give the same value as the other path. I am really not sure which is the one and why.

Now Y can be any kind of general space. So I first thought if Y was just the {0} or the whole space R[itex]^{\infty}[/itex]. But I see very clearly that {0} choice does not give a counter example. Not sure how to handle the second case. Even thought with Y= N ( countable ) but still could not get to any kind of solution.
Thx
 
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  • #2
Hi,

I thought about the problem and I think this could be thought of as a way of thinking.

So what we need is the following.

Given data:

f[itex]_{i}[/itex]: X[itex]_{i}[/itex][itex]\stackrel{injective}{\rightarrow}[/itex] P
g[itex]_{i}[/itex]: X[itex]_{i}[/itex] [itex]\stackrel{homomorphism}{\rightarrow}[/itex] X

Now define in someway

g: P [itex]\stackrel{homomorphism}{\rightarrow}[/itex] X



P is the coproduct we are trying to define. Pls note that we need to define g given g[itex]_{i}[/itex]. g is actually to be defined so that it completes the commutative diagram and then only P is the coproduct.

The most obvious way of defining g on each of the basis elements is g := [itex]\sum g_{i}[/itex] where i ranges over the index set. Now all our group, ring homomorphism operations are defined over finite sums or finite products. So there is no way we could make sense of an infinite sum in the definition of g. Thus we have to restrict it to finite sum. Hence if we have to restrict it to finite sum then only a finite number of elements could be non zero and the rest are all 0s.

This is my understanding and if somebody could point out that there is some gaps it would be very helpful.

Thx
 

1. What is a coproduct?

A coproduct, also known as a disjoint union or sum, is a mathematical operation that combines two objects to create a new object that contains all the elements of the original objects.

2. How is a coproduct different from a direct sum?

A coproduct and a direct sum are both mathematical operations that combine objects, but they differ in how they handle repeated elements. In a coproduct, repeated elements are considered distinct, while in a direct sum they are merged together.

3. Can a coproduct be applied to any mathematical objects?

Yes, a coproduct is a general operation that can be applied to various mathematical objects such as sets, groups, vector spaces, and more.

4. How is a coproduct represented in notation?

In mathematical notation, a coproduct is typically denoted by the symbol ⊔ or ⊎, with the objects being combined written on either side of the symbol.

5. What is the purpose of using coproducts?

Coproducts are useful in mathematics as they allow us to combine objects in a way that preserves their individual properties, while also creating a new object that contains all the elements of the original objects. This can be helpful in understanding the structure and relationships between different mathematical objects.

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