Math and science::Algebra::Aluffi

Canonical decomposition

Any function $f : A \to B$ can be decomposed into a surjection, followed by an isomorphism, followed by an injection.

The surjection will be a projection to a partition of $A$ induced by $f$ . The isomorphism will be from this set to the image $im f$ . The injection will be the inclusion back to $B$ .

Equivalence relation induced by a function

Let $f : A \to B$ be a function. $f$ induces an equivalence relation $\sim$ on $A$ as follows: For all $a^{'}, a^{″} \in A$ ,

a^{'} \sim a^{″} ⟺ f (a^{'}) = f (a^{″}) .

Theorem. Canonical decomposition

Let $f : A \to B$ be any function, and define $\sim$ as above. Then $f$ decomposes as follows:

First we have the canonical projection $A \to A / \sim$ . The last function is the inclusion $im f \subseteq B$ . The bijection in the middle is defined as:

\tilde{f} ([a]_{\sim}) = f (a)

This theorem states that the above diagram for $f$ 's decomposition computes and that $\tilde{f}$ is a valid function and is a bijection.

Well-defined

There is ambiguity as to which elements of $A$ are chosen to represent the equivalence classes in $A \sim$ . As a consequence, this theorem should be accompanied by a proof to show that any choice available leads to the same result. Such a proof is said to be a verification of the theorem being well-defined. Aluffi does exactly this, and it is a good example of such proofs.

Source

Aluffi

p15

19.09.2020