An **arrow world**, W, is an arrow
land of which no arrow has empty Prior or Post: equivalently, (|W)=(W|). To
put this yet another way, every arrow appears in at least one homset.

Any Cartesian product of arrow worlds forms an arrow world and the dual of
any arrow world is an arrow world.

In an arrow world, W, we say of an arrow h
that it is

- initial
- iff, for each arrow f of W and each a in Prior(h),
Hom(a, f) has exactly one member (which is, then, initial);
- terminal
- iff, for each arrow f of M and each w in Post(h),
Hom(f, w) has exactly one member (which is, then, terminal);
- null
- iff it is both initial and terminal;
- self-composable
- iff (h, h) is composable; and
- auto
- (or an automorphism) iff there is a self-composable arrow in
Hom(h,h).

### Further Reading

Arrow worlds only get seriously interesting when we come to turn their
composability into an actual composition. They
do, however, suffice to describe transformations, from which functors spring.

## Examples

I'll start with one which I happen to like: maybe I'll add more later.
Also, any example of a category is an example
of an arrow world and many examples of arrow lands
will be arrow worlds (at least sometimes). For example, a partial order is an arrow world whenever
there's an identity composable after any maximal arrow and before any minimal
one: all other arrows are then `between' some others, and so appear in a homset;
the given identities are just enough to fit the extremal arrows into homsets.

[I used to define functions here: so if you're joining the discourse here in
search of the definitions of (partial) functions and the categories Partial and
Set, you've discovered a dangling link - Sorry, dash onwards !]

In the familiar world of sets and functions, we can define a composition on
arbitrary functions: this allows us to define an arrow world in which every
function is composable after any other. I call the resulting arrow world **Partial**.
It lacks, among other things, any identities: so it is not a category.

Written by Eddy.

$Id: world.html,v 1.3 1998/05/01 23:24:03 eddy Exp $