epi, mono, and solutions

The fundamental reason for the interest in epics and monics is b/c
this is related to -solutions-.

The equation $f(x)=y$
has a solution for all $y$ iff $f$ is epi;
  -existence-
a solution, if it exists, is unique iff $f$ is mono;
  -uniqueness-
A solution always exists and is unique if $f$ is both,
  
This is especially relevant in physics and PDEs:
you want to know if a particular setup -can be solved-,
and if the solution is unique (which it better be for a
physical system).

Nomenclature:
casual: onto, 1-to-1, correspondence
Latin: surjection, injection, bijection
Greek: epimorphism, monomorphism, isomorphism
formal definition: $\forall y \exists x$ etc.
picture (dots mapping to dots)

One discusses this in 3 basic settings:
- sets
  (the formal picture)

- calculus: $\bR \to \bR$ non-linear
  ``horizontal line test'' for 1-1

- linear algebra: $\bR^m \to \bR^n$ linear
  here you can just count rank and rows and columns,
    which is too easy
  ...hence you should use abstract transforms
    (EG on function spaces or polynomial spaces)
    to help people actually understand

Beyond that one should really talk -categorically-:
monic is left-cancellable;
  (monic as pushforward on Hom)
epic is right-cancellable;
  (monic as pullback on Hom)
iso is invertible
...and this really clarifies what we mean by ``losing data''
  (if not monic)
  or ``not detecting everything'' (if not epic)

    %    %    %    %    %    %    %    %

``well-defined''

One common confusion is:
``for every $x$, there is a unique $f(x)$,
  so $f$ is 1-to-1''
No. That just means that $f$ is -well-defined-,
that it's -single valued-.
...but this is related: it's the opposite/dual statement

One should presented ill-defined ``functions''
  (like $1/x$)
and multiple-valued ones (like $\sqrt$)
...this is all part of a circle of ideas.
