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
The quotient
There is quite a lot that is implicit in this statement. A more explicit definition is as follows:
Quotients, as initial objects. Proposition.
Let
- An object of
is any function to any set such that for any . - A morphism
, for objects and , is a function such that .
Proposition: the function
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.
Category diagram
![](coslice_example_p32.png)
Are there any final objects?
The category formulated above does have final objects, but they are not interesting: a function to a singleton set is a final object of the category.
TODO: proof
Essence
Any object in the category can be placed into a one-to-one correspondence with the quotient object and a morphism from the quotient. In other words, in the original set, any function