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For some objects there are alternate definitions, which are "morphism-oriented". To give some examples, there are two definitions of a prime number:

  • $p$ is prime if it is greater than $1$ and has no positive divisors other than $1$ and itself,
  • $p$ is prime if for any $a$ and $b$ we have $p \mid ab \implies p \mid a \lor p \mid b$.

Also, there are two definitions of the Cartesian product:

  • the product $A \times B$ is the set of ordered pairs $\langle a,b\rangle$ such that $a \in A$ and $b \in B$,
  • the product $A \times B$ is any set $P$ such that there exists functions $\pi_A : P \to A$ and $\pi_B : P \to B$ such that for any set $Q$ and functions $f_A : Q \to A$ and $f_B : Q \to B$ there exists a unique function $f : Q \to P$ such that $f_A = \pi_A \circ f$ and $f_B = \pi_B \circ f$.

The second definitions are those which I call "morphism-oriented". At first those might be harder to understand, but they capture some essential properties which aren't so apparent with the simpler definition.

The main question is: what is your experience using them (if you did)?

Some support questions might be as follows:

  • What are their advantages or disadvantages?
  • Are there any examples where such definitions shine or perform very badly?
  • Are such definitions suitable for younger students (i.e. pre-college).
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  • $\begingroup$ Be careful, the first defines "prime" as what is properly an irreducible element, the second one prime elements. In the case of the integers they (almost, up to sign) agree, but not always. $\endgroup$ – vonbrand Mar 17 '14 at 16:45
  • $\begingroup$ @vonbrand I know, but it is enough for natural numbers. This is one of the reasons I'm asking, that is, I've never seen the second definition being used in high school. $\endgroup$ – dtldarek Mar 17 '14 at 17:05
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    $\begingroup$ After having reflected on this question off and on for a while now, I am still not sure why you call your second definition of a prime number "morphism-oriented". As in vonbrand's comment, I rather see the difference between a prime element and an irreducible element. Thinking about ideals instead of elements is a step up in the set-theoretic hierarchy. This is of course an important maneuver in ring theory, but I don't really see why it is being considered categorical. Maybe if you keep going: after ideals you get to modules, and after that you get to categories of modules. Is that it? $\endgroup$ – Pete L. Clark Mar 17 '14 at 22:02
  • $\begingroup$ @PeteL.Clark To be honest, I called it that because I didn't have a better name, categorical wasn't my intention. Instead, I wanted to express a notion where the definition describes the object completely and only in terms of mappings. This is clear in the case of the product. In the case of the prime numbers, the mapping would be the divisibility relation, or, perhaps embedding as in "$p$ is prime if and only if $ \mathbb{Z}_p \hookrightarrow \mathbb{Z}_{ab} \implies \mathbb{Z}_p \hookrightarrow \mathbb{Z}_{a} \lor \mathbb{Z}_p \hookrightarrow \mathbb{Z}_{b}$". Sorry to disappoint you. $\endgroup$ – dtldarek Mar 17 '14 at 22:24
  • $\begingroup$ @dtldarek: Irreducibility is also defined in terms of the divisibility relation: $p$ is irreducible iff for all $x \in R$, $x \mid p \implies p \mid x$ or $x \mid 1$. $\endgroup$ – Pete L. Clark Mar 18 '14 at 1:18
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Your second definition for the Cartesian product is the definition of a product of objects in a category (as I am sure that you are aware). So the question basically (as I am hearing it) becomes:

Is teaching category theory useful and when can you start introducing it?

I think category theory is extremely helpful for a mathematics student who is trying to make sense of definitions across various areas of mathematics. The language of category theory is very helpful because it (IMO) provides a nice general approach to various definitions/ In category theory we have general definitions of products of objects and this in the category of sets becomes the usual Cartesian product. You can then show how your definition one is a way to concretely define a product in that category. But this also extends well to, for example, the product of topological spaces.

When can you start introducing this? Some mathematicians (my self included) wish that they had been introduced to category theory earlier than they were.

But, it is hard to get a grasp on categories without looking at examples. And so the student should probably be familiar with basic concepts like: sets, groups, rings, vector spaces, topological spaces. I think it becomes hard to provide examples without a good background in various topics.

In conclusion I would probably stick with defining the Cartesian product the first way you did it. But this, again, depends on the level of your students.

I think something similar can be said about the definition of a prime number. Here again the second definition is helpful to when trying to understand why prime numbers are so important. And it can be an eye opener when you see this when talking about integral domains.

So I don't think that these "morphism definitions" would be good in a pre-college setting simply because they require much to get at examples. They are great at getting a better understand of "what is really going on", but they are just not very accessible. But as an exercise in abstraction, I would think that you could get something good out of it.

I think another example of this is the question of whether or not you could introduce group theory in high school. I think you can, but again it isn't easy because it is hard to give good examples. But then again, I think that since high school students do have some familiarity with rational numbers, real numbers, and some even complex numbers, it might just work out.

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    $\begingroup$ +1 Completely agree. One of the main appeals of category theory is its power to give universal descriptions. When there is only a singleton set of examples, this "universal power" is a lot less impressive. On hindsight there are a lot of things I wish I grasped much earlier in my mathematical education, but in reality, for most of those things I could have in no way understood their significance that early in my mathematical education. $\endgroup$ – Willie Wong Mar 17 '14 at 10:03
  • $\begingroup$ Not group theory perhaps, but by then you have several examples of rings at hand. $\endgroup$ – vonbrand Mar 18 '14 at 9:33

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