 Math and science::Algebra::Aluffi

# 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.