Coproduct & Direct Sum: Understanding the Set (1,1,1...)

[Sumanta]

Hi ,

This is not a homework problem as I have long passed out of college. 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.

(1, 1, 1, ...)$\rightarrow$ Y

$\uparrow$

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

Now if I say that f$_{i}$ acting on (e$_{i}$) 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$_{i}$$\rightarrow$ Y does not give the same value as the other path. I am really not sure which is the one and why.

Thx

 

[Valkarie]

for your helpThe problem with the definition of the direct sum for the coproduct/direct sum for the set (1, 1, 1, ...) is that there is no unique map from the set (1, 1, 1, ...) to Y. The map from f_{i} to Y does not give the same value as the other path because each element in the set can be mapped to a different element in Y, which will result in a different value.