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This is the definition of direct product: let $\{X_i\}$, $i\in I$, be a family of objects in category $C$. A product $(X; p_i)$ is an object $X$, together with morphisms $p_i: X\rightarrow X_i$, with universal property: given any object $Y$ and morphisms $f_i:Y\rightarrow X_i$, there exists a unique morphism $f=\{ f_i\} :Y\rightarrow X$ with $p_if=f_i$.

Where does the motivation for this definition come from? How did people come up with it?

I don't see any natural way to go from Cartesian product (sets) or direct product of groups (groups) to this categorical definition.

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    $\begingroup$ The products you say below are precisely the categorical product (in their respective categories). That is precisely why this abstract definition was defined! It's a good exercise to actually verify that they satisfy this universal property. $\endgroup$ – user98602 Mar 4 '15 at 15:27
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    $\begingroup$ A bunch of morphisms can be replaced by one morphism. This is very fruitful. It is a good thing to look at Cartesians products as sets together with projections. $\endgroup$ – drhab Mar 4 '15 at 15:36
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If you look at Chs.7/8 of this (as yet very incomplete) Gentle Intro to Category Theory you'll find an extended discussion of the motivation for the categorical definition of products.

It's rather too long to repeat here, I'm sorry. But the basic approach is to note that the familiar pairing devices e.g. for constructing pairs via Kuratowski's device in set theory, or coding pairs by powers of primes in arithmetic, are obviously pretty arbitrary and there are loads of alternatives. But so what? These devices work in context. But ah, what does "it works" mean? Reflecting on that question tells us that what we need are matched coding-and-decoding functions which interact in certain ways. It's not so much what is "in" the pairs that matters as the pattern of morphisms in and out. Which is a rather categorical idea. Then we think how to implement in category theory proper. And lo and behold, we get the familiar categorical definition of products. But as I say, for the full dress version, see the Notes.

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  • $\begingroup$ Hi, which one do you mean from these? $\endgroup$ – mma Oct 6 '18 at 6:35
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    $\begingroup$ I've updated that old reference with a new link. $\endgroup$ – Peter Smith Oct 6 '18 at 8:06
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One motivation is that many of the different kinds of "products" one encounters in different areas of math e.g. Cartesian product of sets, product manifolds, products of groups etc. have some important properties in common that are independent of their concrete realizations (well, I guess this is the reason for the invention of category theory in the first place!), namely that they contains "projection maps", and that specifying a 'function' (more generally: morphism) from an object into a product is, in some sense, equivalent to defining separate maps going into each of the factors of the product.

For example, to give a map $f:\mathbb{R}\rightarrow{\mathbb{R}^2}$ is the same as giving two maps $f_1:\mathbb{R}\rightarrow{\mathbb{R}}$, $f_2:\mathbb{R}\rightarrow{\mathbb{R}}$, the components of the function $f$. Also, for each point $(x,y)$ in the plane, you can project it down to either one of the factors via e.g. $\pi_{1}:\mathbb{R}^2:\rightarrow{\mathbb{R}}$, $\pi_{1}(x,y)=x$.

It is an instructive exercise to show that the above mentioned examples do, in fact, satisfy the universal property you mentioned.

Of course, all of the examples I listed have the same underlying set structure (namely direct product of sets), and so it might also prove insightful to look at a more "exotic" example.

For instance, if you denote by $\mathbb{N}$ the category (show this!) where the objects are the natural numbers $\{{1,2,.\ldots\}}$, and where we have exactly one morphism from $m\rightarrow{n}$ iff $m$ divides $n$, what is the categorial product in this case? Hint: It should be a familiar notion from elementary number theory (some would say kindergarten mathematics ;) )

In any case, the usefulness of this "language" is that it singles out some very important properties that many (possibly very diverse) mathematical objects have in common, and one can then prove certain facts about all of them at once, instead of essentially giving the same proof separately in each case (one for sets, one for groups, one for manifolds etc.). In some of the more advanced areas of maths (algebraic geometry especially comes to mind), it can also greatly simplify notation, conciseness of definitions etc..

However, I'm sure someone else with more knowledge on these issues can expand on my answer!

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Other users have already given good answers, so I'll give an example showing that the categorical products and sums give the right notion of multiplication and addition of natural numbers. Natural numbers can of course be considered as the cardinalities of finite sets. Consider the category of finite sets, with functions as the arrows. Let $|X|$ denote the cardinality of a set $X$, which is of course a cardinal number. Then $|X| \times |Y|= |X \times Y|$ and $|X| + |Y| = |X + Y|$.

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