# 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.