Showing the product and coproduct of two abelian groups is in the category of abelian groups and are isomorphic. Considering infinite (co)products.

Question

So we have the category of abelian groups $Ab$ $ Grp$. How do I show that the product and coproduct of two objects in this category, namely abelian groups, belongs to the category and that they are actually isomorphic? The definition of coproduct is so new to me and I am not the best at this study of categories in my classes right now.

Once this is shown, for clarification does it even make sense to consider the product and coproduct of infinitely many abelian groups? Would these still be isomorphic?

To show that the product and coproducts exist and are isomorphic, you just need to pick some object and show that it satisfies both the product and coproduct universal properties. — D. Brogan, Oct 05 '21 at 13:55

score 1 · Answer 1 · answered Oct 05 '21 at 17:21

Hint: Show that the inclusions $A\to A\times B,\ a\mapsto(a,0)$ and $B\to A\times B,\ b\mapsto (0,b)$ satisfy the universal property of the coproduct.

The coproduct of Abelian groups $A_i$ is also known as their direct sum $$\bigoplus_iA_i:=\{a\in\prod_iA_i\mid \{i: a_i\ne 0\}\text{ is finite}\}\,.$$ If the index set is infinite, it's a proper subgroup of the direct product $\prod_iA_i$.

Showing the product and coproduct of two abelian groups is in the category of abelian groups and are isomorphic. Considering infinite (co)products.

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