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 $C$ as follows:

An object of $C$ is any function [ $ϕ : ? \to ?$ ] such that [for any something, something implies something].
A morphism $σ \in {Hom}_{C} (ϕ_{1}, ϕ_{2})$ , for objects $ϕ_{1} : A \to Z_{1}$ and $ϕ_{2} : A \to Z_{2}$ , is a function [ $σ : ? \to ?$ ] such that [ $? = ?$ ].

Proposition: the function $A \to A / \sim$ is an initial object of the category $C$ .

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.

07.10.2020