$ and
$\$ and two order-preserving functions $P --f,g-> Q$.
One classically defines $f\le g$, if this ist pointwise the case, i.e., if
$xf\sqq xg$ for every $x\in P$. Natural transformations extend
this notion of pointwise comparison to general functors. Notice that the
commutativity condition is automatically satisfied in the
pre-ordered case.
- Exa (naturality, allegedly the starting point of category theory).
Consider the category $Vec(F)$ of vectorspaces over a field
$F$. From linear algebra we know that any finite-dimensional
vectorspace $V$ is isomorphic to F^n for suitable $n in\IN$. In
particular, this applies to the dual space $V^*$ of linear functions
$V --l-> F$ (not the free monoids!), and consequently to the double
dual space $V^{**}$.
Now the vector spaces $V$ and $V^{**}$ turn out to be ``naturally
isomorphic'', while this is not the case for $V$ and $V^*$.
A first indication for this is that any isomorphism between
between $V$ and $V^*$ needs to be defined using bases of these
space, while there is a canonical isomorphism between $V$ and
$V^{**}$ given by evaluation: map a vector $v\in V$ to the
function $V^* --> F$, that evaluates a function $V --l->F$ at $v$.
Now consider, how other linear functions $V --f-> W$ interact with
these isomorphisms. Taking dual spaces leads to a reversal of the
linear function, i.e., we have $W^* --f^*-> W^*$ and then $V^{**}
--f^{**}-> V^{**}$. And in the latter case we obtain a nice
commuting diagram involving $f$, $f^{**}$ and the two evaluation
isomorphisms, whereas in the former case we cannot even form such
a diagram.
- Many important categorical concepts rely on this 2-dimensional
structure, most notably adjoints and monads. We consider this for
the specific case of the word-functor $(-)^*$ and the (finite)
power set functor $\mathbb P$ resp. $\mathbb F$.