Math and science::Algebra::Aluffi
Canonical decomposition
Any function can be decomposed into a [type of function], followed
by an [type of function], followed by an [type of function].
The surjection will be a projection to a partition of induced by . The isomorphism will be from this set to [what set?]. The
injection will be [what function?] back to .
Equivalence relation induced by a function
Let be a function. induces an equivalence
relation on as follows: For all ,
Theorem. Canonical decomposition
Let be any function, and define as above.
Then decomposes as follows:
First we have the canonical projection . The last
function is the inclusion . The bijection
in the middle is defined as:
[]
This theorem states that the above diagram for 's decomposition computes and that is a valid function and is a bijection.
Well-defined
There is ambiguity as to which elements of are chosen to represent
the equivalence classes in . 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.