Universal properties. Quotient.
A construction is said to satisfy a universal property or be the solution to a univeral problem when it may be viewed as a terminal object of a certain category.
The concept of a quotient can be viewed from this perspective. The idea is expressed (somewhat loosely) as follows:
Quotients, as universal properties. Proposition.
Let \( \sim \) be an equivalence relation defined on a set \( A \).
The quotient \( A / \sim \) is universal with respect to the property of [mapping something to something] in such a way that [some property holds].
There is quite a lot that is implicit in this statement. A more explicit definition is as follows:
Quotients, as initial objects. Proposition.
Let \( \sim \) be an equivalence relation defined on a set \( A \). Formulate a category \( \cat{C} \) as follows:
- An object of \( \cat{C} \) is any function [\( \phi : \; ? \to \; ?\) ] such that [for any something, something implies something].
- A morphism \( \sigma \in \cathom{C}(\phi_1, \phi_2) \), for objects \( \phi_1 : A \to Z_1 \) and \( \phi_2 : A \to Z_2 \), is a function [\( \sigma : ? \to \; ? \)] such that [\( ? = \; ? \) ].
Proposition: the function \( A \to A / \sim \) is an initial object of the category \( \cat{C} \).
The proof of this proposition is on the reverse side.
The flip side also has a diagram highlights the nature of the above category. It's a good exercise to try and guess it's form.