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So I just started in on Category theory (reading the quintessential text, "Categories for the Working Mathematician"), and I am trying to get my head around the difference between Products and Coproducts. Can someone explain or discuss the conceptual and practical difference between them to me? They are seeming remarkably similar.

For example, I recall that in the category of abelian groups, the product and coproduct of a finite number of objects are the exact same, and they are remarkably similar for infinite groups, just that coproducts have a limit on their terms, so should I be lead to think they are similar in they way? It's similar in the non-abelian case (though free products look quite a bit more messy), the coproduct is kinda like a limited version of the product, right?

Does anyone have an explanation? I would really appreciate anything, thanks!

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    $\begingroup$ Can you describe the product and coproduct of the sets $\{1,2,3\}$ and $\{10,11\}$ in the category of sets? I don't believe it is possible to explain to anyone anything before they have tried to compute explicitely simple examples. $\endgroup$ – Mariano Suárez-Álvarez Jul 19 '13 at 19:20
  • $\begingroup$ What do you mean by «a limited version of the product»? $\endgroup$ – Mariano Suárez-Álvarez Jul 19 '13 at 19:20
  • $\begingroup$ I know I'm being extremely vague, and the product of the sets is the cartesisan product, the coproduct is the disjoint union. So first has 6, the latter 5, elements. I guess statemtent doesn't make sense in this context. $\endgroup$ – Pax Kivimae Jul 19 '13 at 19:34
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    $\begingroup$ @MarianoSuárez-Álvarez The product is just the Cartesian product : $\{(1, 10), (1, 11), (2, 10), (2, 11), (3, 10), (3, 11)\} $; and the coproduct is the disjoint union and since the two sets are already disjoint, we can forget about the disjoint union tag and just write $\{1, 2,3, 10, 11\}$. Is this correct ? $\endgroup$ – Al Jebr Aug 8 '18 at 3:24
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Products and coproducts are dual concepts. If you look at their defining universal properties, they are identical, except that all arrows are turned around.

For abelian groups, finite products and coproducts happen to be "the same." This is a general phenomenon in additive categories.

For general groups, one has the cartesian versus the free product. I would argue that these are very different.

It is also illustrative to work out what products and coproducts in the category of sets are.

(For those who know what a groupoid is, it is interesting to observe that coproducts of groupoids are much simpler than coproducts of groups.)

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