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).