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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 be an equivalence relation defined on a set A.

The quotient A/ is universal with respect to the property of mapping A to a set in such a way that equivalent elements have the same image.

There is quite a lot that is implicit in this statement. A more explicit definition is as follows:

Quotients, as initial objects. Proposition.

Let be an equivalence relation defined on a set A. Formulate a category C as follows:

  • An object of C is any function ϕ:AZ to any set Z such that for any a,aA,aaϕ(a)=ϕ(a).
  • A morphism σHomC(ϕ1,ϕ2), for objects ϕ1:AZ1 and ϕ2:AZ2, is a function σ:Z1Z2 such that σϕ1=ϕ2.

Proposition: the function AA/ 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.


Category diagram

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 f satisfying the criteria (that elements of the same partition map to the same elements) can be uniquely decomposed into a composition of quotient function (function from set to partition) followed by a function g. In other words, there is a injection between all functions like f and the functions like g that compose with the quotient projection (not sure if projection is the right word here).


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