# categorically distinguishing product and co-product of infinite family

Let $\mathcal{M}$ be the category of left $R$-modules. Then the product (direc product) and co-product (direct sum) of any family of objects $\{A_i\}_i$ in this category exists, denote by $\bigoplus_i A_{i\in I}$ and $\prod_{i\in I} A_i$.

When set $I$ is infinite, I have seen their difference only on the level of some counting arguments, and now I wanted to know, in the category theory, how they are really different.

In other words, to show that $\bigoplus_i A_i$ is not the co-product of $\{A_i\}_i$, I have to show that it violates a universal property of the product/co-product. But, I am confused here, how to proceed for it.

Can one give some hint to prove that the co-product and product are not isomorphic in the category of modules for an infinite family?

For simplicity, let's consider the category $Ab$ of abelian groups. Now, the product of countably many copies of $\mathbb Z$ is simply their direct product: the set of all sequences of integers with point-wise operation. The coproduct though is the set of all eventually $0$ such sequences with point-wise operation. To verify these claims, simply establish the universal property. For the product it is completely straightforward, while for the coprouct there is a tiny little but to pay attention to.
• Thanks for the suggestions; but still I have not solved. What I tried is to prove that $(\oplus_i A_i, \{\varphi_j\}_j)$ is not the product of $\{A_i\}$, where $\varphi_j\colon \oplus_i A_i\rightarrow A_j$ are morphisms, for this I assumed that this is product, and for $p_j:\prod_i A_i\rightarrow A_j$, the natural projections, there exists unique $\theta:\prod_i A_i\rightarrow \oplus_i A_i$ such that $\varphi_j\circ\theta=p_j$ for all j. But, taking an element $(a_i)_i \in \prod_i A_i$ with all $a_i$'s non-zero leads to contradiction. ...What I could not prove is that $\prod_i A_i$ is not sum. – p Groups Sep 23 '16 at 9:27
• If you have a cone $\mathbb Z \to A$, how will you construct a function from the product to $A$? Where will you send the element $(1,1,1,1,\cdots)$ to? – Ittay Weiss Sep 24 '16 at 3:19