Are there any universal properties of infinite (countable) direct sum in k-linear category? I'm looking for some properties which will help me to build mappings to and from the direct sum if I have mappings to or from all components of the sum. May be direct sums in k-linear categories are products or coproducts?
And I want to know more about direct sum in general, for example, I don't understand the motivation of this concept. Could you, please, recommend me some books, where I can find information about it?